Mikael De

L A Salle

Emilie Dupre

Richard Ferrere

Odile Henry

Romain Pace

Catherine Pagani

Catherine Vuillemenot

Keywords: Von Neumann algebras, noncommutative L p -spaces, noncommutative martingales, John-Nirenberg inequality, atomic decomposition, matrix-valued Hardy spaces and BMO spaces, wavelets, Calderón-Zygmund operators, matrix-valued kernels, Haar shifts, martingales transforms, paraproducts

Je voudrais adresser mes remerciements les plus vifs à Quanhua XU pour avoir assuré la direction de ma thèse. Je le remercie pour le temps qu'il m'a consacré, la multitude des sujets qu'il a proposés ainsi que son immense savoir dont j'ai pu profiter au cours de ces trois dernières années. Je le remercie pour la manière avisée avec laquelle il m'a initié au monde des mathématiques en me faisant participer aux conférences et écoles afin d'enrichir ma culture mathématique et y faire connaître mes résultats, en m'envoyant travailler auprès de spécialistes pour élargir ma connaissance et mes expériences. Je le remercie enfin pour m'avoir donné l'opportunité d'apprendre le français et pour ses conseils sur ma vie quotidienne, qui sont indispensables pour un étudiant étranger comme moi.

C'est un honneur que Marius JUNGE et Stefanie PETERMICHL aient accepté de rapporter ce travail. Je les remercie pour leur lecture attentive du manuscrit et l'intérêt qu'ils portent à mes recherches.

Merci à Javier PARCET, qui a accepté de se déplacer depuis l'Espagne pour participer à mon jury de thèse. Je le remercie pour son chaleureux accueil lors de mes deux séjours à Madrid, et pour l'opportunité qu'il m'a donnée de collaborer avec lui sur divers sujets d'analyse harmonique non commutative. J'ai eu la chance de pouvoir profiter de ses nombreuses connaissances et de ses précieux encouragements. Mes remerciements se dirigent également vers Yong DING, qui était mon directeur en master et qui m'a permis de décourir le monde de l'analyse harmonique commutative. C'est lui qui m'a proposé de faire une thèse sous la direction de Quanhua XU. Il a encore accepté de se déplacer depuis la Chine pour participer à mon jury de thèse. Je le remercie vivement pour tout ce qu'il a fait pour moi. Je tiens aussi à remercier vivement Stéphane ATTAL, Christian LE MERDY et Stefan NEUWIRTH qui ont accepté de faire partie de mon jury. Merci tout particulièrement à Christian LE MERDY et Stefan NEUWIRTH pour leurs excellents exposés au groupe de travail qui m'ont beaucoup profité.

Je tiens à exprimer ma reconnaissance à Tao MEI, qui est un co-auteur d'une partie de ma thèse. J'ai beaucoup profité des discussions stimulantes que nous avons eues. Je tiens aussi à exprimer ma gratitude à Eric RICARD et Caleb ECKHARDT pour leur disponibilité, pour leur patience, pour avoir pris le temps de répondre à mes questions ainsi que pour m'avoir soutenu et encouragé.

Cette thèse a été effectuée au sein de l'équipe d'analyse fonctionnelle de Besançon, dont la dynamique et la bonne ambiance ont contribué au bon déroulement de ces trois dernières années. Je remercie en particulier mes

Il est temps de remercier la joyeuse bande des doctorants, sans qui la vie au labo n'aurait pas la même saveur. Commençons par Mathilde avec qui j'ai partagé séminaires, groupes de travail, conférences et qui a été là pour répondre à mes questions sur le math et la vie de thésard. Je la remercie vivement pour avoir pris le temps de corriger attentivement le français dans ma thèse. Je suis aussi reconnaissant envers Yanqi pour les nombreuses discussions mathématiquement profitables que nous avons eues. Merci aux membres de mon bureau, du meilleur bureau (le 401) et ses affiliés, Alexis, Anthony, Céline, Clément, Émilie, François, Guillaume, Hicham, Karine, Laurent, Olivier et Stéphane pour la bonne ambiance. Ils sont "mes professeurs de français". J'exprime en particulier ma gratitude à Alexis, Émilie, François, Karine, Laurent pour m'avoir aidé à corriger le français dans ma thèse, et surtout ma profonde reconnaissance à Alexis qui est "mon technicien dédié de L A T E X". Je voudrais remercie Tingjian, Xiao, Zhi et les autres étudiants chinois pour leur aide dans la vie quotidienne.

Je tiens enfin à remercier spécialement ma famille pour son soutien sans faille tout au long de ces trois dernières années. Mes derniers remerciements vont bien sûr à Xinlu, qui m'a apporté le bonheur pendant la préparation de cette thèse.

Résumé Résumé

Cette thèse présente quelques résultats de la théorie des probabilités quantiques et de l'analyse harmonique non commutative. Elle est constituée de trois parties. La première partie démontre l'analogue non commutatif de l'inégalité de John-Nirenberg et la décomposition atomique pour les martingales non commutatives. Ces résultats étendent et améliorent ceux qui existent déjà, et correspondent exactement à ceux que l'on connaît dans le cas classique. La deuxième partie est consacrée à l'étude des espaces de Hardy à valeurs opérateurs via la méthode d'ondelettes. Il est montré que les espaces de Hardy définis par ondelettes coïncident avec ceux définis par les fonctions carrées de Littlewood-Paley et Lusin. Cette approche est similaire à celle du cas des martingales non commutatives, mais l'utilisation des outils de martingales en analyse harmonique permet une démonstration plus rapide. Dans la troisième partie, nous nous tournons vers des applications de la théorie bien établie des espaces de Hardy, c'est-à-dire des opérateurs de Calderón-Zygmund (OCZ pour abréviation) associés à des noyaux à valeurs matricielles. On obtient des estimations de type faible (1, 1) pour des OCZ dyadiques parfaites et des shifts de Haar annulateurs associés à des noyaux non commutatifs, ainsi que des estimations de type H 1 → L 1 pour des OCZ arbitaires d'après une décomposition d'une fonction en ligne/colonne. En conjonction avec L ∞ → BM O, nous établissons certaines estimations de type L p . Cette approche s'applique aussi à des paraproduits et des transformées de martingales avec des symboles et coefficients non commutatifs respectivement.

Mots-clefs

Algèbres de von Neumann, espaces L p non commutatifs, martingales non commutatives, inégalité de John-Nirenberg, décomposition atomique, espaces de Hardy et BMO à valeurs matricielles, ondelettes, opérateurs de Calderón-Zygmund, noyaux à valeurs matricielles, shifts de Haar, transformées de martingale, paraproduits.

Some problems on noncommutative harmonique analysis

Abstract

Table des matières Introduction

La théorie des probabilités quantiques et l'analyse harmonique non commutative se posent dans le cadre des algèbres de von Neumann. L'algèbre de von Neumann est le cadre naturel pour la théorie d'intégration non commutative, où les fonctions de la théorie de l'intégration classique sont remplacées par des opérateurs sur un espace de Hilbert, et les mesure par des traces. Historiquement, c'est dans le but d'étudier la mécanique quantique que von Neumann et ses collaborateurs ont posé les bases de la théorie de l'intégration non commutative. Pour cette raison, l'étude des thèmes dans l'analyse non commutative était sans surprise partiellement motivée par la mécanique quantique. Néanmoins, la théorie des probabilités quantiques et l'analyse harmonique non commutative sont devenues les domaines de recherche indépendants en mathématique.

La théorie des inégalités de martingales non commutatives est importante en probabilités quantiques. Le développement moderne des inégalités de martingales non commutatives a commencé avec le papier fondateur de Pisier et Xu [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] dans lequel les inégalités de Burkholder-Gundy et le théorème de dualité de Fefferman ont été étendus au cas non commutatif. Depuis, de nombreux résultats classiques ont été tranférés avec succès dans le monde non commutatif. Il doit être souligné qu'étendre les résultats classiques au cadre non commutatif exige souvent d'attaquer le problème sous un autre angle. Par exemple, l'argument de temps d'arrêt et la fonction maximale ponctuelle, qui sont souvent utilisés dans les preuves classiques, n'existent pas dans ce cadre. Par conséquent, des techniques ou théories supplémentaires, comme par exemple la théorie des espaces d'opérateurs, sont exploitées afin de traiter les martingales non commutatives. De plus, ces techniques peuvent donner de nouveaux résultats même dans la théorie classique, comme illustré dans [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF].

Il est bien connu qu'il existe de nombreuses interactions entre la théorie des probabilités classiques et l'analyse harmonique. Ces interactions sont encore fructueuses et jouent un rôle important dans le cadre non commutatif. Ainsi, le deuxième thème de ce travail concerne l'application de la théorie des martingales non commutatives à l'analyse harmonique non commutative. Motivé par les inégalités de martingales non commutatives, Mei [START_REF] Mei | Operator valued Hardy spaces[END_REF] a effectué une étude systématique des espaces de Hardy de fonctions à valeurs dans des espaces L p non commutatifs. Ses espaces de Hardy non commutatifs sont définis par la fonction intégrale de Lusin non commutative, et sont étroitement liés à ceux définis via le semigroupe de Poisson. Ensuite, en combinant ce lien avec la propriété de dilatation Markovienne des semigroupes d'opérateurs et les propriétés des martingales non commutatives relativement à une filtration continue, obtenus récemment, Junge et Mei [START_REF] Junge | BMO spaces assocated with semigroups of operators[END_REF] et [START_REF] Mei | Tent spaces associated with semigroups of operators[END_REF] ont établi la théorie des espaces de Hardy associés aux semigroupes d'opérateurs, et ont trouvé quelques applications importantes. Certaines de ces applications sont nouvelles même dans le cadre classique.

Dans la théorie des martingales non commutatives, il existe une technique assez importante : la construction de Cuculescu, qui constitue l'analogue de l'argument de temps d'arrêt, mais sous une forme plus faible que dans le cas classique. Randrianantoanina Introduction dans [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] [START_REF] Randrianantoanina | A weak type inequality for non-commutative martingales and applications[END_REF] et [START_REF] Randrianantoanina | Conditioned square functions for noncommutative martingales[END_REF] a utilisé cette construction pour démontrer des estimations de type faible (1, 1) des transformées de martingales et des fonctions carrées (conditionnelles). La construction de Cuculescu est aussi un outil clef dans la décomposition de Gundy non commutative démontrée par Parcet et Randrianantoanina [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF]. Il est bien connu que la transformée de martingales et la fonction carré correspondent en analyse harmonique à la transformée de Hilbert et la fonction de Littlewood-Paley respectivement. Parcet [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF] a établi la décomposition de Calderón-Zygmund non commutative basée sur la construction de Cuculescu, et obtenu les estimations de type faible (1, 1) pour les opérateurs de Calderón-Zygmund. Par la suite, les estimations de type faible des fonctions carrées à valeurs opérateurs ont également été obtenues par Parcet et Mei [START_REF] Mei | Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities[END_REF]. Une nouvelle propriété découverte par Parcet est un principe de pseudo-localisation, qui est approfondi par Hytönen [START_REF] Hytönen | Pseudo-localisation of singular integrals in L p[END_REF] dans le cas classique.

Cette thèse est constituée de trois parties. Le premier chapitre présente un travail en collaboration avec Mei intitulé "John-Nirenberg inequality and atomic decomposition for noncommutative martingales", qui s'inscrit dans la théorie des martingales non commutatives. Ce travail a été accepté par J. Funct. Anal. Le contenu du deuxième chapitre concerne la théorie des espaces de Hardy mentionné dans le troisième paragraphe précédent. Ce chapitre est un travail en collaboration avec Yin intitulé "Wavelet approach to operator-valued Hardy spaces", qui a été accepté par Revista Mat. Iberoa. Le dernier chapitre est centré sur les opérateurs de Calderón-Zygmund associés à des noyaux à valeurs matricielles et la transformée de martingales. Il s'agit d'un travail effectué en collaboration avec López, Martell et Parcet intitulé "Calderón-Zygmund operators associated to matrix-valued kernels".

Avant que je détaille chaque chapitre dans la suite de cette introduction, nous rappelons l'objet principal des trois chapitres, c'est-à-dire des espaces L p non commutatifs. Soit M une algèbre de von Neumann semifinie munie d'une trace normale et fidèle τ et S + M l'ensemble des éléments positifs x de M tels que τ (s(x)) < ∞, où s(x) est la plus petite projection vérifiant exe = x. Soit S M l'espace vectoriel engendré par S + M . Alors tout élément x ∈ S M a une trace finie, et S M est une *-sous-algèbre w * -dense de M. Soit maintenant 0 < p < ∞. Pour tout x ∈ S M , l'opérateur |x| p appartient à S + M (où |x| = (x * x) 1/2 désigne le module de x). Nous définissons alors

x p = (τ (|x| p )) 1 p , ∀x ∈ S M .
On peut vérifier que • p est bien définie et est une (quasi)norme sur S M . Le complété de (S M , • p ) est noté L p (M) : c'est l'espace usuel L p non commutatif associé à (M, τ ).

Pour simplifier les notations, nous écrirons M à la place de L ∞ (M) munie de la norme d'opérateur • M . Les éléments de L p (M) peuvent être décrits comme des opérateurs fermés densément définis sur H, H étant l'espace de Hilbert sur lequel M agit.

Chapitre 1

La théorie des inégalités de martingales non commutatives a été développée au cours des dernières années. Nous renvoyons le lecteur, par exemple, à [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF], [START_REF] Junge | Doob's Inequality for Non-commutative Martingales[END_REF], [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], [START_REF] Junge | Noncommutative maximal ergodic theorems[END_REF] et [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] pour les inégalités de martingales non commutatives, à [START_REF] Musat | Interpolation Between Non-commutative BMO and Non-commutative Lpspaces[END_REF], [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] pour les interpolations des espaces de Hardy et à [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF], [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF] pour les décompositions de Gundy et Davis non commutatives. Le chapitre 1 est également motivé par les deux travaux suivants. Le premier concerne le théorème de John-Nirenberg non commutatif démontré par Junge et Musat, et le deuxième la décomposition 2-atomique des espaces de Hardy établie par Bekjan, Chen, Perrin et Yin [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. 0.1. Chapitre 1 D'abord, rappelons quelques notions élémentaires des martingales non commutatives. Dans ce chapitre M est une algèbre de von Neumann finie munie d'une trace normalisée. Soit (M n ) n≥1 une filtration croissante de sous-algèbres de von Neumann de M dont l'union est w * -dense dans M. Soit E n l'espérance conditionnelle de M relativement à M n . Une suite x = (x n ) n≥1 de L 1 (M) est une martingale non commutative relativement à (M n ) n≥1 si E n (x n+1 ) = x n pour tout n ≥ 1. Si de plus tous les x n sont dans L p (M) pour un certain 1 ≤ p ≤ ∞, alors on dit que x est une martingale L p . Dans ce cas, on considère

x p = sup n≥1 x n p .
Si x p < ∞, on dit que x est une martingale bornée dans L p . Soit x = (x n ) n≥1 une martingale non commutative relativement à (M n ) n≥1 . On définit dx n = x nx n-1 pour n ≥ 1 avec la convention x 0 = 0. La suite dx = (dx n ) n≥1 est appelée la suite des différences de la martingale x. Dans la suite, pour tout x ∈ L 1 (M) on note x n = E n (x) pour n ≥ 1. Soit 1 ≤ p < ∞. Définissons H c p (resp. H r p ) comme le complété de l'ensemble des martingales L p finies pour la norme x H c p = S c (x) p (resp. x H r p = S r (x) p ), où S c (x) et S r (x) sont définis par

S c (x) = k≥1 |dx k | 2 1/2 , S r (x) = S c (x * ).
Les espaces de Hardy non commutatifs H p (M) sont définis comme suit : si

BMO c (M) = {x ∈ L 1 (M) : x BMO c < ∞}, où x BMO c = sup n≥1 E n |x -x n-1 | 2 1/2 ∞ , et BMO r (M) = {x : x * ∈ BMO c (M)}.
On considère l'espace

BMO(M) = BMO c (M) ∩ BMO r (M)
muni de la norme x BMO = max{ x BMO c , x BMO r }.

Introduction

où par convention E 0 = E 1 . Soit alors 0 < p < ∞. On définit l'espace h c p (M) (resp. h r p (M)) comme le complété de l'ensemble des martingales L ∞ finies pour la (quasi)norme x h c p = s c (x) p (resp. x h r p = s r (x) p ). Soit h d p (M) l'espace des martingales dont la suite des differences de martingale est dans ℓ p (L p (M)), où ℓ p (L p (M)) est l'espace des suites a = (a n ) n≥1 dans L p (M) telles que

a ℓp(Lp(M)) = n≥1 a n p p 1/p < ∞.
La version conditionnelle des espaces de Hardy de martingales non commutatives est définie comme suit : si 0 < p < 2,

h p (M) = h c p (M) + h r p (M) + h d p (M)
muni de la (quasi)norme

x hp = inf x=y+z+w { y h c p + z h r p + w h d p }. Si 2 ≤ p < ∞, h p (M) = h c p (M) ∩ h r p (M) ∩ h d p (M)
muni de la norme

x hp = max{ x h c p , x h r p , x h d p }.

L'espace bmo c est défini par bmo c (M) = {x ∈ L 1 (M) :

x bmo c < ∞} où x bmo c = max E 1 (x) ∞ , sup n≥1 E n |x -x n | 2 1/2 ∞ .
On considère bmo r (M) = {x : x * ∈ bmo c (M)}.

Soit bmo d (M) l'espace des martingales dont la suite des differences de martingale est dans ℓ ∞ (L ∞ (M)), où ℓ ∞ (L ∞ (M)) est l'espace des suites a = (a n ) n≥1 dans L ∞ (M) telles que

a ℓ∞(L∞(M)) = sup n a n ∞ < ∞.

L'inégalité de John-Nirenberg

On commence par rappeler l'inégalité de John-Nirenberg de la théorie classique des martingales. Soit (Ω, F, P) un espace probabilisé et (F n ) n≥1 une suite croissante de sousσ-algèbres de F. On notera E n les espérances conditionnelles associées. L'espace BM O(Ω) est défini comme l'ensemble des fonctions x ∈ L 1 (Ω) telles que

x BM O = sup n E n |x -x n-1 | ∞ < ∞.
Le théorème de John-Nirenberg classique dit qu'il existe deux constantes universelles c 1 , c 2 > 0 telles que si x BM O < c 2 , alors

sup n E n (e c 1 |x-x n-1 | ) ∞ < 1.
(0.1.1)

Ce résultat est équivalent à la propriété suivante : pour tout n ≥ 1, E ∈ F n et λ > 0, il existe une constante universelle c > 0 telle que

1 P(E) P ω ∈ E : |x(ω) -x n-1 (ω)| > λ ≤ c 2 exp(-cλ/ x BM O ). (0.1.2)
Il y a encore une caractérisation équivalente : il existe une constante universelle c > 0 telle que pour tout 1 ≤ p < ∞,

x BM O ≤ sup n sup E∈Fn 1 P(E) 1/p (x -x n-1 )1 E p ≤ cp x BM O . (0.1.3)
Junge et Musat dans [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] ont démontré une version non commutative du théorème de John-Nirenberg similaire à (0.1.3), en prouvant qu'il existe une constante universelle c > 0 telle que pour tout 2 ≤ p < ∞, Cependant, ce théorème n'est plus vrai (voir Remark 2.14 pour un contre-example) si on considère BMO c (M) et BMO r (M) séparément. D'autre part, il ne correspond pas à la forme utilisée couramment de l'inégalité de John-Nirenberg classique. Le premier but de ce chapitre est de remédier à ces aspects du théorème de Junge and Musat. Le théorème suivant est un de nos principaux résultats. Dans ce chapitre, P(M) dénote l'ensemble des projections de M. Les deux constantes α p and β p ont les propriétés suivantes

Introduction (i) α p = 1 pour 2 ≤ p < ∞ ; (ii) α p ≤ C 1/p-1/2 pour 0 < p < 2 ; (iii) β p ≤ cp pour 2 ≤ p < ∞ ; (iv) β p = 1 pour 0 < p < 2.
Un résultat similaire est encore vrai pour BMO c (M), mais uniquement pour 2 ≤ p < ∞ (voir Remark 2.9). D'un autre côté, notre preuve du Théorème 0.1.1 peut être modifiée facilement afin d'étendre le résultat de Junge/Musat à tout 0 < p < ∞ sous la forme (0.1.3) (voir Corollary 2.19). De plus, la constante optimale cp obtenue en utilisant le résultat de Randrianantoanina [START_REF] Randrianantoanina | A weak type inequality for non-commutative martingales and applications[END_REF] nous permet de formuler l'inégalité sous la forme (0.1.1) et (0.1.2).

Dans la dernière section de ce chapitre, on donne une réponse négative à une question posée dans [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] (page 136) : étant donné 2 < p < ∞, existe-t-il une constante c p telle que

sup k E k |x -E k-1 x| p 1 p ∞ ≤ c p x BMO ? Théorème 0.1.2. Supposons que sup k E k |f -E k-1 f | p 1/p ∞ ≤ c p (n) f BMO pour un cer- tain p ≥ 3. Alors c p (n) ≥ c(log(n + 1)) 2 p .

Décomposition atomique

Nous nous tournons maintenant vers le deuxième objectif de ce chapitre : la décomposition atomique des espaces de Hardy non commutatifs. Nous rappelons que la décomposition 2-atomique a déjà été obtenue dans [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. Un élément a ∈ L 1 (M) est un (1, 2) c -atome relativement à (M n ) n≥1 s'il existe n ≥ 1 et e ∈ P(M n ) tels que (i) E n (a) = 0 ; (ii)ae = a ; (iii) a 2 ≤ (τ (e)) -1/2 . L'espace de Hardy atomique h c 1,at (M) est défini comme l'espace de tous les opérateurs x ∈ L 1 (M) tels que la norme • h c 1,at est finie, où

x h c 1,at = E 1 x 1 + inf j |λ j |.
Ici l'infimum est pris sur toutes les décompositions possibles x -E 1 x = j λ j a j telles que

λ j ∈ C, a j est un (1, 2) c -atome. Il est démontré dans [2] que x ∈ h c 1 (M) si et seulement si x ∈ h c 1,at (M), avec x h c 1 ≃ x h c 1,at . Combiné à l'equivalence H c 1 (M) = h c 1 (M) + h d 1 (M)
, les auteurs de [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] ont aussi obtenu une decomposition 2-atomique pour H c 1 (M). Nous rappelons brièvement l'argument utilisé dans [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. L'espace dual de h c 1,at (M) peut être décrit comme

Λ c (M) = {x ∈ L 2 (M) : x Λ c < ∞} avec x Λ c = max{ E 1 x ∞ , sup n≥1 sup e∈Pn ( 1 τ (e) τ (e|x -x n | 2 )) 1 2 }. 0.2. Chapitre 2
En fait, le supremum dans la définition précédente peut être pris sur tous les b ∈ L 1 (M n ), puisque les points extrémaux de la boule unité de L 1 (M n ) sont des multiples de projections. Par conséquent,

x Λ c = max{ E 1 x ∞ , sup n≥1 sup b∈Mn ( 1 b 1 τ (b|x -x n | 2 )) 1 2 } (0.1.4) = max{ E 1 x ∞ , sup n≥1 E n |x -x n | 2 1 2 ∞ } = x bmo c . Ainsi la dualité (h c 1 (M)) * = bmo c (M) implique h c 1,at (M) = h c 1 (M).
Il est bien connu dans le cas classique que les 2-atomes dans la décomposition précédente peuvent être remplacés par les q-atomes pour tout 1 < q ≤ ∞. Nous rappelons la définition de ces atomes dans le cadre classique. On dit qu'une fonction a ∈ L 1 (Ω) est un q-atome s'il existe

n ≥ 1 et E ∈ F n tels que (i) E n a = 0 ; (ii) {a = 0} ⊂ E ; (iii) a q ≤ P(E) -1+ 1
q . Nous nous référons à [START_REF] Weisz | Atomic Hardy spaces[END_REF] pour plus d'information.

La principale difficulté pour obtenir la décomposition q-atomique dans le cadre non commutatif est que l'équivalence clef (0.1.4) n'est plus vraie si on remplace dans (iii) l'indice de puissance 2 par q ′ = 2, 1 ≤ q ′ < ∞ où q' désigne l'indice conjugué de q. On surmonte cet obstacle par le Théorème 0.1.1, et on obtient le théorème suivant.

Théorème 0.1.3. Pour tout 1 < q ≤ ∞, h c 1 (M) = h c 1,atq,pr (M)
et ces normes sont équivalentes. Ici, h c 1,atq,pr (M) est l'espaces de Hardy q-atomique avec ses atomes définis ainsi : a est un q-atome s'il existe n ≥ 1 et une projection e ∈ P(M n ) tels que

(i) E n (a) = 0 ; (ii) ae = a ; (iii) a h c q ≤ (τ (e)) -1 q ′ .
Notons que h c 1,at 2,pr (M) = h c 1,at (M), et la décomposition 2-atomique est ainsi retrouvée dans [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. Par ailleurs, en appliquant la version conditionnelle du théorème de Junge et Musat de la forme (0.1.3), on obtient une décomposition q-atomique pour h 1 (M) dans laquelle les atomes sont définis d'une façon similaire, où la norme • h c q dans (iii) ci-dessus est remplacée par q et la condition de support (ii) est affaiblie à r(a) ≤ e ou l(a) ≤ e (voir Theorem 3.19). C'est exactement l'analogue non commutatif de la décomposition atomique classique.

Les inégalités de John-Nirenberg et la décomposition établie ici seront apppliquées pour démontrer les estimations de type H 1 → L 1 dans le chapitre 3.

Introduction

f à valeurs dans S M tels que S c (f ) p est finie, où S c (f ) est l'analogue non commutatif de l'intégrale de Lusin classique définie par

S c (f )(x) = Γ ∂f * ∂t ∂f ∂t + ∂f * ∂y ∂f ∂y (x + y, t) dydt 1 2 avec Γ = {(y, t) ∈ R 2 + | |y| < t} et f (y, t) = P t f (y)
∈ L ∞ (M; L c 2 (R, dt/1 + t 2 )) tels que ϕ BM O c = sup I⊂R 1 |I| I |ϕ -ϕ I | 2 1 2 M < ∞.
S c (f )(x) = I∈D | f, w I | 2 |I| 1 I (x) 1 2 , (0.2.1) et S r (f ) = S c (f * ). Les normes sont données par f H c p = S c (f ) Lp(N ) , et f H r p = S r (f ) Lp(N ) . Puis l'espace H c p (R, M) (resp. H r p (R, M)) est défini comme l'espace complété de (S N , • H c p (R,M) ) (resp. (S N , • H c p (R,M) )
). On définit alors les espaces de Hardy à valeurs opérateurs comme suit : pour 1 ≤ p < 2,

H p (R, M) = H c p (R, M) + H r p (R, M) (0.2.2)
munis de la norme

f Hp = inf{ g H c p + h H r p : f = g + h, g ∈ H c p , h ∈ H r p }, et pour 2 ≤ p < ∞, H p (R, M) = H c p (R, M) ∩ H r p (R, M) (0.2.3) munis de la norme f Hp = max{ f H c p , f H r p }. Pour ϕ ∈ L ∞ (M; L c 2 (R, dx 1+x 2 )), on pose ϕ BMO c = sup J∈D 1 |J| I⊂J | ϕ, w I | 2 1 2 M (0.2.4) et ϕ BMO r = ϕ * BMO c (R,M)
. Ce sont des normes modulo les fonctions constantes. Définissons

BMO c (R, M) = {ϕ ∈ L ∞ (M; L c 2 (R, dx 1 + x 2 )) : ϕ BMO c < ∞}, BMO r (R, M) = {ϕ : ϕ * ∈ BMO c (R, M)}, et BMO(R, M) = BMO c (R, M) ∩ BMO r (R, M).
Ensuite, on obtient la dualité de Fefferman désirée, et les résultats d'interpolation.

Theorem 0.2.1. On a

(H c 1 (R, M)) * = BMO c (R, M) (0.2.5) avec normes équivalentes. Theorem 0.2.2. Soit 1 < p < ∞, on a [BMO c (R, M), H c 1 (R, M)] 1 p = H c p (R, M) (0.2.6)
avec normes équivalentes.

Finalement, on prouve directement que notre espace BMO c (R, M) est le même que dans le cadre de l'analyse harmonique à valeurs matricielles, et que nos espaces H c p (R, M) sont les mêmes que ceux introduits par Mei par dualité et interpolation.

Theorem 0.2.3. On a BMO c (R, M) = BM O c (R, M) avec normes équivalentes. De même, H c p (R, M) = H c p (R, M
) avec normes équivalentes. En d'autres termes, on donne une autre approche pour traiter les espaces de Hardy à valeurs opérateurs. Il doit être souligné que notre méthode est très similaire à celle utilisée dans le cas des martingales non commutatives, c'est donc beaucoup plus simple que la méthode de Mei. C'est aussi la première tentative de transférer des résultats en probabilités quantiques à l'analyse harmonique à valeurs opérateurs en utilisant l'ondelette.

Chapitre 3

Les transformées de martingales non commutatives et les opérateurs de Calderón-Zygmund rencontrés précedemment peuvent être considérés comme des opérateurs d'intégrales avec des noyaux à valeurs scalaires, de même les fonctions carrées non commutatives et les fonctions carrées à valeurs opérateurs peuvent être considérées comme des noyaux à valeurs dans un espace de Hilbert. L'objectif principal de ce chapitre est d'obtenir les estimations d'endpoint pour les OCZ ayant les noyaux qui ne commutent pas avec des fonctions, motivé par une estimation récente dans [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] pour les OCZ semi-commutatifs. Dans ce chapitre, on note A = L ∞ (R n ) ⊗B(ℓ 2 ). Si k(x, y) agit linéairement sur B(ℓ 2 ) et satisfait la condition de finesse de Hörmander pour la norme des applications linéaires bornées sur B(ℓ 2 ), le contenu de [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF]Lemma 1.3] peut être résumé comme suit

Introduction • Si T est borné dans L ∞ (B(ℓ 2 ); L r 2 (R n )), alors T : L ∞ (A) → BMO r (A), • Si T est borné dans L ∞ (B(ℓ 2 ); L c 2 (R n )), alors T : L ∞ (A) → BMO c (A). Ici, on dit que T est borné dans L ∞ (B(ℓ 2 ); L c 2 (R n )) si R n T f (x) * T f (x) dx 1 2 B(ℓ 2 ) R n f (x) * f (x) dx 1 2 B(ℓ 2 )
, et BMO c (A) désigne la version dyadique de BM O c (R n , B(ℓ 2 )) définie dans le chapitre 2 avec la norme donnée par

sup Q cube dyadique - Q g(x) -g Q * g(x) -g Q dx 1 2 B(ℓ 2 )
.

En prenant les ajoints-afin que le * passe partout de gauche à droite-on obtient la définition de T borné dans L ∞ (L r 2 ) et de la norme de BMO ligne. Ainsi, les arguments standard d'interpolation et de dualité démontrent que T est borné dans L p (A) pour 1 < p < ∞ à condition que le noyau soit assez lisse par rapport aux deux variables et que T soit une application normale auto-adjointe bornée dans

L ∞ (L r 2 ) et L ∞ (L c 2 )
. En d'autre termes, les conditions de bornitude ligne/colonne jouent essentiellement le rôle de l'hypothèse de bornitude dans L 2 dans la théorie classique de Calderón-Zygmund.

Bien que cela fontionne certainement pour des noyaux non-scalaires -les actions de produit de Schur ont été utilisées dans [20, Theorem B] par exemple-les hypothèses de bornitude imposent les conditions que les noyaux commutent presque avec les fonctions, qui sont trop fortes pour les OCZ associés aux noyaux qui ne commutent pas avec les fonctions. C'est-à-dire, étant donnés k : R 2n \ ∆ → B(ℓ 2 ) lisse et x / ∈ supp R n f , on définit formellement les OCZ ligne/colonne ainsi :

T c f (x) = R n k(x, y)f (y) dy et T r f (x) = R n f (y)k(x, y) dy.
Ce n'est pas difficile de constuire des noyaux qui ne commutent pas avec les fonctions mais vérifiant i) T r et T c sont bornés dans L 2 (A), ii) T r et T c ne sont pas bornés dans L p (A) pour 1 < p = 2 < ∞, voir par exemple [43, Section 6.1] pour des exemples spécifiques. Par conséquent, les hypothèses de bornitude dans

L ∞ (L r 2 ) et L ∞ (L c
2 ) sont en général trop restrictives quand le noyau et la fonction ne commutent pas. On suppose pour ce qui suit que T r et T c sont bornés dans L 2 (A). On s'intéresse aux formes affaiblies de bornitude dans L p et aux estimations d'endpoint pour ces OCZ. Un OCZ dyadique non commutant sera une paire (T r , T c ) qui est bornée dans L 2 (A) associée à un noyau ne commutant pas avec la fonction mais vérifiant une des conditions suivantes :

a) Noyaux dyadiques Parfaits k(x, y) -k(z, y) B(ℓ 2 ) + k(y, x) -k(y, z) B(ℓ 2 ) = 0 pour tous x, z ∈ Q, y ∈ R et Q, R des cubes dyadiques disjoints.

b) Opérateurs de shifts de Haar annulateurs

k(x, y) = Q dyadic R,S dyadic ⊂ Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(S) α Q RS h R (x)h S (y), 0.3. Chapitre 3 avec r, s ∈ Z + fixés, où α Q RS ∈ B(ℓ 2 ) avec α Q RS B(ℓ 2 ) ≤ √ |R||S| |Q| . Ici les h Q sont les 2 n -
i) OCZ Dyadique non commutant. Si f ∈ L 1 (A) inf f =fr+fc T r f r 1,∞ + T c f c 1,∞ f 1 . ii) OCZ Générique non commutant. Si f ∈ H 1 (A) inf f =fr+fc T r f r 1 + T c f c 1 f H 1 (A) . H 1 (A) est la version dyadique de H 1 (R n , B(ℓ 2 )) défini dans chapitre 2.
Pour la preuve de i), on utilise la décomposition de Calderón-Zygmund non commutative et la troncature triangulaire. Pour les estimations de type ii), on se sert de la décomposition atomique et l'inégalité de John-Nirenberg dans le section 1.1. En combinant avec L ∞ → BMO, on obtient certaines estimations ligne/colonne L p .

Theorem 0.3.2. Les inégalités suivantes sont vraies pour les OCZ générique non commutant

: i) Si 1 < p < 2 et f ∈ L p (A) inf f =fr+fc T r f r p + T c f c p f p . ii) Si 2 < p < ∞ et f ∈ L p (A) T r f H r p (A) + T c f H c p (A) f p . iii) Si f ∈ L ∞ (A), on a aussi T r f BMOr(A) + T c f BMOc(A) f ∞ .
Notre approche s'applique aussi aux paraproduits et aux tranformées de martingales dont les symboles et coefficients qui ne commutent pas avec les fonctions. a) Transformées de martingales

M r ξ f = k≥1 ∆ k (f )ξ k-1 et M c ξ f = k≥1 ξ k-1 ∆ k (f ). b) Paraproduits de martingales Π r ρ (f ) = k≥1 E k-1 (f )∆ k (ρ) et Π c ρ (f ) = k≥1 ∆ k (ρ)E k-1 (f ).
Ici ∆ k dénote l'opérateur de la différence de martingale

E k -E k-1 et ξ k ∈ A k est

Introduction

The theory of quantum probability and noncommutative harmonic analysis arise from the setting of von Neumann algebras. The theory of von Neumann algebras is the natural framework for non commutative integration theory, where functions in the classical integration theory are replaced by operators on a Hilbert space, measures by traces. Historically, it is in order to study quantum mechanic that von Neumann and his collaborators laid the foundation of noncommutative integration theory. Therefore, without surprise, the study of these topics in noncommutative analysis has been partly motivated by quantum mechanics. However, the theory of quantum probability and noncommutative harmonic analysis have become independent fields of mathematical research.

The theory of noncommutative martingale inequalities is an important direction of quantum probability. The modern period of development of noncommutative martingale inequalities began with Pisier and Xu's seminal paper [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] in which the noncommutative Burkholder-Gundy inequalities and Fefferman duality theorem were established. Since then, many classical results have been successfully transferred to the noncommutative world. It should be pointed out that extending classical results to the noncommutative setting often requires additional insights. For instance, stopping time arguments and pointwise maximal function, which are often used in the classical proofs, appear unavailable in this setting. Therefore, extra techniques or theories, e.g. operator space theory, are exploited to deal with noncommutative martingales. Moreover, these techniques may yield new results even in the classical theory, as illustrated in [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF].

It is well known that there are numerous interactions between classical probability theory and harmonic analysis. This interplay continues to be fruitful and play an important role in the noncommutative setting. So the second subject of this thesis deals with applications of noncommutative martingale theory to noncommutative harmonic analysis. Motivated by noncommutative martingale inequalities, Mei [START_REF] Mei | Operator valued Hardy spaces[END_REF] gave a systematic study of Hardy spaces of functions with values in noncommutative L p -spaces. His noncommutative Hardy spaces are defined by the noncommutative Lusin integral function, which are closely related to the ones defined through Poisson semigroup. Then by this connection, combined with the recently established Markov dilation property of semigroups of operators and noncommutative continuous times martingale, Junge and Mei in [START_REF] Junge | BMO spaces assocated with semigroups of operators[END_REF] and [START_REF] Mei | Tent spaces associated with semigroups of operators[END_REF] built the theory of Hardy spaces associated with semigroups of operators, and find some important applications. Some of the applications are new even in the classical setting.

In the theory of noncommutative martingales, there is an important technique: Cuculescu's construction, which is the analogue of stopping time argument, but not strong enough as that in the classical case. However, Randrianantoanina in [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] [START_REF] Randrianantoanina | A weak type inequality for non-commutative martingales and applications[END_REF] and [START_REF] Randrianantoanina | Conditioned square functions for noncommutative martingales[END_REF] made use of this construction to prove weak (1, 1) type estimates of martingale transforms, (conditional) square functions. As applications, he obtained the optimal order of the best constants in several noncommutative martingale inequalities. Cuculescu's construction is also a key tool in Parcet-Randrianantoanina's noncommutative Gundy decomposition [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF].

Introduction

It is well known that the martingale transform and square function correspond to Hilbert transform and Littlewood-Paley function respectively in harmonic analysis. Hence Parcet [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF] built the noncommutative Calderón-Zygmund decomposition based on Cuculescu's construction and obtained the weak (1, 1) type estimate for Calderón-Zygmund operators. Later the weak type estimates for operator-valued square functions was also obtained by Parcet and Mei [START_REF] Mei | Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities[END_REF]. A new phenomenon found by Parcet is a pseudo-localization principle which is pursed further by Hytönen [START_REF] Hytönen | Pseudo-localisation of singular integrals in L p[END_REF] in the classical case.

This thesis consists of three chapters. The first chapter presents a joint work with Mei entitled "John-Nirenberg inequality and atomic decomposition for noncommutative martingales", which can be viewed as a part of noncommutative martingale theory. This work has been accepted by J. Funct. Anal. The content of the second chapter is on theory of Hardy spaces relating to the subject mentioned in the previous second paragraph. This chapter is a joint work with Yin entitled "Wavelet approach to operator-valued Hardy spaces", which has been accepted by Revista Mat. Iberoa. The last chapter is centered on Calderón-Zygmund operator with matrix-valued kernels or noncommuting martingale transforms, which is a joint work with López, Martell and Parcet entitled "Calderón-Zygmund operators associated to matrix-valued kernels".

Before I describe each chapter at length in the rest of this introduction, let us recall the main object of the three chapters, that is the noncommutative L p spaces. Let M be a semifinite von Neumann algebra equipped with a normal and faithful trace τ and S + M be the set of all positive element x in M with τ (s(x)) < ∞, where s(x) is the smallest projection e such that exe = x. Let S M be the linear span of S + M . Then any x ∈ S M has finite trace, and S M is a w * -dense * -subalgebra of M. Let 1 ≤ p < ∞. For any x ∈ S M , the operator |x| p belongs to

S + M (|x| = (x * x) 1 2
). We define

x p = τ (|x| p ) 1 p , ∀x ∈ S M .
One can check that • p is a norm on S M . The completion of (S M , • p ) is denoted by L p (M) which is the usual noncommutative L p -space associated with (M, τ ). For convenience, we usually set L ∞ (M) = M equipped with the operator norm • M . The elements of L p (M) can be described as closed densely defined operators on H (H being the Hilbert space on which M acts).

Chapter 1

The theory of noncommutative martingales inequalities has been well developed in recent years. We refer, for instance, to [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF], [START_REF] Junge | Doob's Inequality for Non-commutative Martingales[END_REF], [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], [START_REF] Junge | Noncommutative maximal ergodic theorems[END_REF], [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] for noncommutative martingales inequalities, to [START_REF] Musat | Interpolation Between Non-commutative BMO and Non-commutative Lpspaces[END_REF], [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] for interpolation of noncommutative Hardy spaces and to [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF], [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF] for the noncommutative Gundy and Davis decompositions. There are two other works that motivate the content of chapter 1. The first one is Junge and Musat's noncommutative John-Nirenberg theorem [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] and the second the 2-atomic decomposition of the Hardy spaces H 1 by Bekjan, Chen, Perrin and Yin [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. Let us recall some basic notions on noncommutative martingales. In this chapter, M is a finite von Neumann algebra equipped with a normalized trace. Let (M n ) n≥1 be an increasing sequence of von Neumann subalgebras of M such that the union of the M n 's is w * -dense in M. Let E n be the conditional expectation of M with respect to M n . A sequence x = (x n ) n≥1 in L 1 (M) is called a noncommutative martingale with respect to (M n ) n≥1 if E n (x n+1 ) = x n for every n ≥ 1. If in addition, all the x n 's are in L p (M) for 0.1. Chapter 1 some 1 ≤ p ≤ ∞, x is called an L p -martingale. In this case we set

x p = sup n≥1 x n p . If x p < ∞,
x is called a bounded L p -martingale. Let x = (x n ) n≥1 be a noncommutative martingale with respect to (M n ) n≥1 . Define dx n = x n -x n-1 for n ≥ 1 with the convention that x 0 = 0 and E 0 = E 1 . The sequence dx = (dx n ) n is called the martingale difference sequence of x. In the sequel, for any operator x ∈ L 1 (M) we denote x n = E n (x) for n ≥ 1. For 1 ≤ p < ∞. Define H c p (resp. H r p ) as the completion of all finite L p -martingales under the norm x H c p = S c (x) p (resp. x H r p = S r (x) p ), where S c (x) and S r (x) are defined as

S c (x) = k≥1 |dx k | 2 1/2 , S r (x) = S c (x * ).
The noncommutative martingale Hardy spaces H p (M) are defined as follows: if 1 ≤ p < 2,

H p (M) = H c p (M) + H r p (M)
equipped with the norm

x Hp = inf x=y+z { y H c p + z H r p }. When 2 ≤ p < ∞, H p (M) = H c p (M) ∩ H r p (M) equipped with the norm
x Hp = max{ x H c p , x H r p }. The space BMO c is defined as

BMO c (M) = {x ∈ L 1 (M) : x BMO c < ∞} where x BMO c = sup n≥1 E n |x -x n-1 | 2 1/2 ∞ , and BMO r (M) = {x : x * ∈ BMO c (M)}. Define BMO(M) = BMO c (M) ∩ BMO r (M)
equipped with the norm

x BMO = max{ x BMO c , x BMO r }.
We will also work on the conditional version of Hardy and BMO spaces developed in [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF]. Let x = (x n ) n≥1 be a finite martingale in L 2 (M). We set

s c (x) = k≥1 Introduction subspace of ℓ p (L p (M)
) consisting of all martingale difference sequences, where ℓ p (L p (M)) is the space of all sequences a = (a n ) n≥1 in L p (M) such that

a ℓp(Lp(M)) = n≥1 a n p p 1/p < ∞
with the usual modification for p = ∞. The noncommutative conditional martingale Hardy spaces are defined as follows: if 0 < p < 2,

h p (M) = h c p (M) + h r p (M) + h d p (M)
equipped with the (quasi-)norm

x hp = inf x=y+z+w { y h c p + z h r p + w h d p }. When 2 ≤ p < ∞, h p (M) = h c p (M) ∩ h r p (M) ∩ h d p (M) equipped with the norm x hp = max{ x h c p , x h r p , x h d p }.
The space bmo c is defined as

bmo c (M) = {x ∈ L 1 (M) : x bmo c < ∞} where x bmo c = max E 1 (x) ∞ , sup n≥1 E n |x -x n | 2 1/2 ∞ . Let bmo r (M) = {x : x * ∈ bmo c (M)}.
Let bmo d (M) be the subspace of ℓ ∞ (L ∞ (M)) consisting of all martingale difference sequences. Note that bmo

d (M) = h d ∞ (M). Define bmo(M) = bmo c (M) ∩ bmo r (M) ∩ bmo d (M)
equipped with the norm

x bmo = max{ x bmo c , x bmo r , x bmo d }.

John-Nirenberg inequality

We begin with recalling the classical John-Nirenberg inequalities in the martingale theory. Let (Ω, F, P) be a probability space and (F n ) n≥0 an increasing sequence of sub-σ-algebras of F with the associated conditional expectations (E n ) n≥0 . The BM O(Ω) space is defined as the set of all x ∈ L 1 (Ω) with the norm

x BM O = sup n E n |x -x n-1 | ∞ < ∞.
The classical John-Nirenberg theorem says that there exist two universal constants c 1 ,

c 2 > 0 such that if x BM O < c 2 , then sup n E n (e c 1 |x-x n-1 | ) ∞ < 1, (0.1.1)
which is equivalent to the statement: For any n ≥ 1, E ∈ F n and λ > 0, there exists a universal constant c > 0 such that

1 P(E) P ω ∈ E : |x(ω) -x n-1 (ω)| > λ ≤ c 2 exp(-cλ/ x BM O ). (0.1.2)
There remains a equivalent characterization: There exists an absolute constant c such that for all 1 ≤ p < ∞,

x BM O ≤ sup n sup E∈Fn 1 P(E) 1/p (x -x n-1 )1 E p ≤ cp x BM O . (0.1.3)
Junge and Musat [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] proved a noncommutative version of John-Nirenberg theorem resembling (0.1.3): There exists an absolute constant c such that for all 2 ≤ p < ∞,

x BMO ≤ B p (x) ≤ cp x BMO , where B p (x) = max{ sup n sup b∈Mn, b p≤1 (x -x n-1 )b p , sup n sup b∈Mn, b p≤1, b(x -x n-1 ) p }.
However, this theorem does not hold (see Remark 2.14 for a counterexample) when considering BMO c (M) or BMO r (M) separately. On the other hand, it does not correspond to the commonly used form of the classical John-Nirenberg inequality. The first purpose of this chapter is to remedy these aspects of Junge and Musat's theorem. The following is one of our main results. In this chapter, P(M) denotes the set of all projections of M. Theorem 0.1.1. For all 0 < p < ∞,

α -1 p x bmo c ≤ x bmo c p,pr ≤ β p x bmo c , where x bmo c p,pr = max E 1 (x) ∞ , sup n sup e∈P(Mn) 1 (τ (e)) 1/p (x -x n )e h c p .
The two constants α p and β p have the following properties

(i) α p = 1 for 2 ≤ p < ∞; (ii) α p ≤ C 1/p-1/2 for 0 < p < 2; (iii) β p ≤ cp for 2 ≤ p < ∞; (iv) β p = 1 for 0 < p < 2.
A similar result holds for BMO c (M) but only with 2 ≤ p < ∞ (see Remark 2.9). On the other hand, our proof of Theorem D can be easily modified to extend Junge/Musat's result to all 0 < p < ∞ and in the form (0.1.3) (see Corollary 2.19). Moreover the optimal order cp obtained using very recent results of Randrianantoanina [START_REF] Randrianantoanina | A weak type inequality for non-commutative martingales and applications[END_REF] enable us to formulate the inequality in the form (0.1.1) and (0.1.2).

In the last section of chapter 1, we give an negative answer to an open question asked in [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] (on page 136) that given 2 < p < ∞, whether there exists a constant c p such that

sup k E k |x -E k-1 x| p 1 p ∞ ≤ c p x BMO ? Introduction Theorem 0.1.2. Suppose sup k E k |f -E k-1 f | p 1/p ∞ ≤ c p (n) f BMO for some p ≥ 3. Then c p (n) ≥ c(log(n + 1)) 2 p .

Atomic decomposition

We now turn to the second objective of this chapter: the atomic decomposition of different noncommutative Hardy spaces. Let us recall the 2-atomic decomposition obtained in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. An element a ∈ L 1 (M) is said to be a (1, 2) c -atom with respect to (M n ) n≥1 , if there exist n ≥ 1 and e ∈ P(M n ) such that (i) E n (a) = 0; (ii)ae = a; (iii) a 2 ≤ (τ (e)) -1/2 . The atomic Hardy space h c 1,at (M) is defined as the space of all x ∈ L 1 (M), such that the following • h c 1,at norm is finite,

x h c 1,at = E 1 x 1 + inf j |λ j |.
Here the infimum is taken for possible decompositions x -E 1 x = j λ j a j with λ j ∈ C, a j being (1, 2) c -atom. It is proved in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] that x ∈ h c 1 (M) if and only if x ∈ h c 1,at (M) and

x h c 1 ≃ x h c 1,at .
Together with the equivalence

H c 1 (M) = h c 1 (M) + h d 1 (M)
, the authors of [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] also obtained a 2-atomic decomposition for H c 1 (M). Let us briefly recall the argument used in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. The dual space of h c 1,at (M) can be described as

Λ c (M) = {x ∈ L 2 (M) : x Λ c < ∞} with x Λ c = max{ E 1 x ∞ , sup n≥1 sup e∈Pn ( 1 τ (e) τ (e|x -x n | 2 )) 1 2 }.
Actually, the supremum in the definition above can be taken for all b ∈ L 1 (M n ) since the extreme points of the unit ball of L 1 (M n ) are all multiples of projections. Therefore,

x Λ c = max{ E 1 x ∞ , sup n≥1 sup b∈Mn ( 1 b 1 τ (b|x -x n | 2 )) 1 2 } (0.1.4) = max{ E 1 x ∞ , sup n≥1 E n |x -x n | 2 1 2 ∞ } = x bmo c .
Then the duality h c 1 (M) = bmo c (M) yields h c 1,at (M) = h c 1 (M). It is well known in the classical theory that 2-atoms in the previous atomic decomposition can be replaced by q-atoms for any 1 < q ≤ ∞. Let us recall these atoms in the commutative case. A function a ∈ L 1 (Ω) is said to be a q-atom if there exist n ≥ 1 and

E ∈ F n such that (i) E n a = 0; (ii) {a = 0} ⊂ E; (iii) a q ≤ P(E) -1+ 1
q . We refer to [START_REF] Weisz | Atomic Hardy spaces[END_REF] for more information.

The main difficulty to obtain q-atomic decompositions in the noncommutative case is that the key equivalence (1.0.8) no longer holds if one replaces the power indices 2 by q ′ = 2, 1 ≤ q ′ < ∞. We overcome this obstacle by Theorem D. Theorem 0.1.3. For all 1 < q ≤ ∞,

h c 1 (M) = h c 1,atq,pr (M)
with equivalent norms. Here h c 1,atq,pr (M) is the q-atomic Hardy spaces with its atoms defined as: a is called a q-atom if there exist n ≥ 1 and a projection e ∈ P(M n ) such that

(i)) E n (a) = 0; (ii) ae = a; (iii) a h c q ≤ (τ (e)) -1 q ′ .
Note that h c 1,at 2,pr (M) = h c 1,at (M), so we recover the 2-atomic decomposition of [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. Moreover, applying the conditional version of Junge/Musat's theorem in the form (0.1.3), we get a q-atomic decomposition for h 1 (M) in which the atoms are defined in a similar way with h c q in (iii) above replaced by q and the support condition (ii) weakened to r(a) ≤ e or l(a) ≤ e (see Theorem 3.19). This is exactly the noncommutative analogue of the classical atomic decomposition.

The John-Nirenberg inequality and atomic decomposition established here will be applied to prove the H 1 → L 1 type estimates in Chapter 3.

Chapter 2

Motivated by quantum probability mentioned in chapter 1 and matrix-valued harmonic analysis, Mei in [START_REF] Mei | Operator valued Hardy spaces[END_REF] defined H p spaces for operator-valued functions by considering the operator-valued Lusin's square function. For 1 ≤ p < ∞, his H c p (R, M) is defined to be the completion of the space of all S M -valued simple functions f 's with norm S c (f ) p finite, where S c (f ) is the noncommutative analogue of the classical Lusin integral defined by

S c (f )(x) = Γ ∂f * ∂t ∂f ∂t + j ∂f * ∂y j ∂f ∂y j (x + y, t) dydt t n-1 1 2
with Γ = {(y, t) ∈ R n+1 + | |y| < t} and f (y, t) = P t f (y) for the Poisson semigroup (P t ) t≥0 . Then H c p (R, M) and H p (R, M) are defined in the same way as those in martingale case. The remarkable result proved by Mei is that H c 1 (R, M) is the predual of the BMO space appeared in matrix-valued harmonic analysis denoted by BM O c (R, M). This space is defined to be the space of all ϕ ∈ L ∞ (M; L c 2 (R, dt/1 + t 2 )) such that

ϕ BM O c = sup I⊂R 1 |I| I |ϕ -ϕ I | 2 1 2 M < ∞.
He also obtained desired interpolation results and Littewood-Paley inequality. Mei adapted the classical approach in his study and reduced many problems to the martingale case. To do so he invented a very interesting and powerful technique: the BMO space on R (both in the classical and noncommutative cases) is the intersection of two dyadic BMO spaces.

On the other hand, in classical harmonic analysis, it is well known that H 1 (R) defined by square function and H 1 (R) defined by wavelets coincide since they admit the same atomic decomposition. As explained in chapter 1, it is very difficult to obtain the atomic decomposition for noncommutative Hardy spaces by explicit construction as in the classical situation, hence we cannot compare these two spaces by the classical way. In this chapter, we directly define H p (R, M) and BMO(R, M) via wavelets. The definitions are similar to the ones in the martingale case, but associated to a fixed wavelet basis of (w I ) I∈D . In Introduction this chapter, for simplicity, we denote L ∞ (R) ⊗M by N . For f ∈ S N , the square functions are defined as

S c (f )(x) = I∈D | f, w I | 2 |I| 1 I (x) 1 2 . (0.2.1)
and S r (f ) = S c (f * ). The norms are defined as

f H c p = S c (f ) Lp(N ) , and
f H r p = S r (f ) Lp(N ) . So the spaces H c p (R, M) (resp. H r p (R, M)) are defined to the space of the completion of (S N , • H c p (R,M) ) (resp. (S N , • H c p (R,M) ).
We then define the operator-valued Hardy spaces as follows: for 1 ≤ p < 2,

H p (R, M) = H c p (R, M) + H r p (R, M) (0.2.2)
with the norm

f Hp = inf{ g H c p + h H r p : f = g + h, g ∈ H c p , h ∈ H r p } and for 2 ≤ p < ∞, H p (R, M) = H c p (R, M) ∩ H r p (R, M) (0.2.3)
with the norm defined as

f Hp = max{ f H c p , f H r p }. For ϕ ∈ L ∞ (M; L c 2 (R, dx 1+x 2 )), set ϕ BMO c = sup J∈D 1 |J| I⊂J | ϕ, w I | 2 1 2 M (0.2.4) and ϕ BMO r = ϕ * BMO c (R,M) . These are norms modulo constant functions. Define BMO c (R, M) = {ϕ ∈ L ∞ (M; L c 2 (R, dx 1 + x 2 )) : ϕ BMO c < ∞}, BMO r (R, M) = {ϕ : ϕ * ∈ BMO c (R, M)}, and BMO(R, M) = BMO c (R, M) ∩ BMO r (R, M).
Then we obtain the desired Fefferman duality theorem, and interpolation results.

Theorem 0.2.1. We have

(H c 1 (R, M)) * = BMO c (R, M) (0.2.5)
with equivalent norms.

Theorem 0.2.2. Let 1 < p < ∞, we have

[BMO c (R, M), H c 1 (R, M)] 1 p = H c p (R, M) (0.2.6)
with equivalent norms.

0.3. Chapter 3 29
At last, we directly prove our BMO c (R, M) is as same as the one in matrix-valued harmonic analysis, hence our H c p (R, M) are the same as Mei's by duality and interpolation. Theorem 0.2.3. We have

BMO c (R, M) = BM O c (R, M) with equivalent norms. Similarly, H c p (R, M) = H c p (R, M
) with equivalent norms. In other words, we give another approach to deal with operator-valued Hardy spaces. It should be pointed out that our method is very parallel to the way in noncommutative martingale case, hence it is much simpler than Mei's method. This is also the first attempt by using wavelet to transfer the results in quantum probability to operator-valued harmonic analysis.

Chapter 3

The noncommutative martingale transforms and Calderón-Zygmund operators considered before can be viewed as integral operators with scalar-valued kernels, while noncommutative square functions and operator-valued square functions Hilbert-valued kernels. The main goal of this chapter is to obtain endpoint estimates for CZO's with noncommuting kernels, motivated by a recent estimate from [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] for semicommutative CZO's. In this chapter, we denote

A = L ∞ (R n ) ⊗B(ℓ 2 ) for simplification. If k(x, y) acts linearly on B(ℓ 2 )
and satisfies the Hörmander smoothness condition in the norm of bounded linear maps on B(ℓ 2 ), the content of [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF]Lemma 1.3] can be summarized as follows

• If T is L ∞ (B(ℓ 2 ); L r 2 (R n ))-bounded, then T : L ∞ (A) → BMO r (A), • If T is L ∞ (B(ℓ 2 ); L c 2 (R n ))-bounded, then T : L ∞ (A) → BMO c (A). Here, the L ∞ (L c
2 )-boundedness assumption refers to

R n T f (x) * T f (x) dx 1 2 B(ℓ 2 ) R n f (x) * f (x) dx 1 2 B(ℓ 2 ) , while BMO c (A) is the dyadic version of BM O c (R n , B(ℓ 2 )
) defined in chapter 2 with norms given by sup

Q dyadic cube - Q g(x) -g Q * g(x) -g Q dx 1 2 B(ℓ 2 )
.

Taking adjoints -so that the * switches everywhere from left to right-we find L ∞ (L r 2 )boundedness and the row-BMO norm. Thus, standard interpolation and duality arguments show that T :

L p (A) → L p (A) for 1 < p < ∞ provided the kernel is smooth enough in both variables and T is a normal self-adjoint map satisfying the L ∞ (L r 2 ) and L ∞ (L c
2 ) boundedness assumptions. In other words, the row/column boundedness conditions essentially play the role of the L 2 -boundedness assumption in classical Calderón-Zygmund theory.

Although this certainly works for non-scalar kernels -Schur product actions were used e.g. in [20, Theorem B]-the boundedness assumptions impose nearly commuting conditions on the kernel which are too strong for CZO's associated to noncommuting kernels. Namely, given k : R 2n \ ∆ → B(ℓ 2 ) smooth and given x / ∈ supp R n f , let us set formally the row/column CZO's [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF]Section 6.1] for specific examples. Therefore, the L ∞ (L r

T c f (x) = R n k(x, y)f (y) dy and T r f (x) = R n f (y)k(x, y) dy.

It is not difficult to construct noncommuting kernels with

Introduction i) T r and T c are L 2 (A)-bounded, ii) T r and T c are not L p (A)-bounded for 1 < p = 2 < ∞, see e.g.
2 ) and L ∞ (L c 2 ) boundedness assumption is in general too restrictive when kernel and function do not commute.

Assume for what follows that T r and T c are L 2 (A)-bounded. We are interested in weakened forms of L p boundedness and endpoint estimates for these CZO's. A dyadic noncommuting CZO will be a L 2 (A)-bounded pair (T r , T c ) associated to a noncommuting kernel satisfying one of the following conditions:

a) Perfect dyadic kernels k(x, y) -k(z, y) B(ℓ 2 ) + k(y, x) -k(y, z) B(ℓ 2 ) = 0 whenever x, z ∈ Q and y ∈ R for some disjoint dyadic cubes Q, R. b) Cancellative Haar shift operators k(x, y) = Q dyadic R,S dyadic ⊂ Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(S) α Q RS h R (x)h S (y),
for some fixed r, s ∈ Z + where the

α Q RS ∈ B(ℓ 2 ) with α Q RS B(ℓ 2 ) ≤ √ |R||S| |Q| .
Here h Q refers to any of the 2 n -1 Haar functions related to the cube Q.

We write generic noncommuting CZO for L 2 (A)-bounded pairs (T r , T c ) with a noncommuting kernel satisfying the standard smoothness. Our first result is the following. Theorem 0.3.1. The following inequalities hold :

i) Dyadic noncommuting CZO's. Given f ∈ L 1 (A) inf f =fr+fc T r f r 1,∞ + T c f c 1,∞ f 1 . ii) Generic noncommuting CZO's. Given f ∈ H 1 (A) inf f =fr+fc T r f r 1 + T c f c 1 f H 1 (A) . Again, H 1 (A) is the dyadic version of H 1 (R n , B(ℓ 2 )) defined in chapter 2.
For the proof of i), we use the noncommutative Calderón-Zygmund decomposition and triangular truncation. For the type estimates of ii), we use the atomic decomposition and John-Nirenberg inequality in section 1.1. In conjunction with L ∞ → BMO, we get certain row/column L p estimates.

Theorem 0.3.2. The following inequalities hold for generic noncommuting CZO's

: i) If 1 < p < 2 and f ∈ L p (A) inf f =fr+fc T r f r p + T c f c p f p . 0.3. Chapter 3 31 ii) If 2 < p < ∞ and f ∈ L p (A) T r f H r p (A) + T c f H c p (A) f p . iii) Given f ∈ L ∞ (A), we also have T r f BMOr(A) + T c f BMOc(A) f ∞ .
Our approach also applies to noncommutative paraproducts and martingale transforms with noncommuting symbols/coefficients.

a) Noncommuting martingale transforms

M r ξ f = k≥1 ∆ k (f )ξ k-1 and M c ξ f = k≥1 ξ k-1 ∆ k (f ). b) Paraproducts with noncommuting symbol Π r ρ (f ) = k≥1 E k-1 (f )∆ k (ρ) and Π c ρ (f ) = k≥1 ∆ k (ρ)E k-1 (f ).
Here ∆ k denotes the martingale difference operator E k -E k-1 and ξ k ∈ A k is an adapted sequence. Of course, the symbols ξ and ρ do not necessarily commute with the function.

Theorem 0.3.3. Consider the pairs :

i) Martingale transforms (M r ξ , M c ξ ), with sup k ξ k M < ∞. ii) Martingale paraproducts (Π r ρ , Π c ρ ), with Π r/c ρ L 2 (A)-bounded.
If Σ A is regular, we obtain weak type (1, 1) inequalities like in Theorem 0.3.1i) for martingale transforms and paraproducts . The estimates in Theorems 0.3.1ii) and 0.3.2 also hold for both families and for arbitrary filtrations Σ A . Moreover, the martingale paraproducts Π r ρ and Π c ρ are L p -bounded for 2 < p < ∞ and L ∞ → BMO. Our results recover those in [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF][START_REF] Randrianantoanina | Conditioned square functions for noncommutative martingales[END_REF] and are in some sense sharp, providing appropriate substitutes for noncommuting coefficients. Our result for paraproducts goes beyond [START_REF] Mei | Notes on matrix valued paraproducts[END_REF]Theorem 1.2] in two aspects. First, our estimates for p > 2 hold for arbitrary martingales, not just for regular ones. Second, we give a partial answer to Mei's question in [START_REF] Mei | Notes on matrix valued paraproducts[END_REF] after the proof of Theorem 1.2 for the case p < 2 and also for the weak type (1, 1) estimates.

Introduction

This chapter deals with BMO spaces and atomic decomposition for noncommutative martingales. The modern period of development of noncommutative martingale inequalities began with Pisier and Xu's seminal paper [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] in which they established the noncommutative Burkholder-Gundy inequalities and Fefferman duality theorem between H 1 and BMO. Since then remarkable progress has been made in the field. We refer, for instance, to [START_REF] Junge | Doob's Inequality for Non-commutative Martingales[END_REF], [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], [START_REF] Junge | Noncommutative maximal ergodic theorems[END_REF], [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] for other noncommutative martingales inequalities, to [START_REF] Musat | Interpolation Between Non-commutative BMO and Non-commutative Lpspaces[END_REF], [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] for interpolation of noncommutative Hardy spaces and to [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF], [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF] for the noncommutative Gundy and Davis decompositions. Let us also mention two other works that motivate the present chapter. The first one is Junge and Musat's noncommutative John-Nirenberg theorem [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] and the second the 2-atomic decomposition of the Hardy spaces H 1 by Bekjan, Chen, Perrin and Yin [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF].

Before describing our main results, we recall the classical John-Nirenberg inequalities in the martingale theory. Let (Ω, F, P) be a probability space and (F n ) n≥0 an increasing sequence of sub-σ-algebras of F with the associated conditional expectations (E n ) n≥0 . The BM O(Ω) space is defined as the set of all x ∈ L 1 (Ω) with the norm

x BM O = sup n E n |x -x n-1 | ∞ < ∞. (1.0.1)
The classical John-Nirenberg theorem says that there exist two universal constants c 1 ,

c 2 > 0 such that if x BM O < c 2 , then sup n E n (e c 1 |x-x n-1 | ) ∞ < 1. (1.0.2)
This statement is equivalent to the following one: there exists an absolute constant c such that for all 1 ≤ p < ∞,

x BM O ≤ sup n E n |x -x n-1 | p 1 p ∞ ≤ cp x BM O . (1.0.3)
A duality argument yields

E n |x -x n-1 | p 1 p ∞ = sup b∈L∞(Fn), b 1 ≤1 |x -x n-1 | p bdP 1 p (1.0.4)
Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

= sup b∈L∞(Fn), b p≤1 (x -x n-1 )b p . (1.0.5)
Furthermore, by the extreme point property of L 1 (F n ) and (1.0.4), the John-Nirenberg theorem (1.0.3) can be rewritten as follows

x BM O ≤ sup n sup E∈Fn 1 P(E) (x -x n-1 )1 E p ≤ cp x BM O . (1.0.6)
Accordingly, (1.0.2) can be reformulated as: For any n ≥ 1, E ∈ F n and λ > 0

1 P(E) P ω ∈ E : |x(ω) -x n-1 (ω)| > λ ≤ c 2 exp(-c 1 λ/ x BM O ). (1.0.7)
Junge and Musat [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] proved a noncommutative version of John-Nirenberg theorem corresponding to (1.0.5). To state their result we need fix some notation. Let M be a finite von Neumann algebra with a normal faithful tracial state τ . Let (M n ) n≥1 be an increasing sequence of von Neumann subalgebras of M such that the union of M n 's is w * -dense in M. Let E n be the conditional expectation of M with respect to M n . Define

x BMO c = sup n≥1 E n |x -x n-1 | 2 1 2 ∞ and BMO(M) = {x ∈ L 1 (M) : x BMO < ∞} with x BMO = max{ x BMO c , x * BMO c }.
Then Junge and Musat's John-Nirenberg inequality reads as follows: there exists an absolute constant c such that for all 2 ≤ p < ∞,

x BMO ≤ B p (x) ≤ cp x BMO , where B p (x) = max{ sup n sup b∈Mn, b p≤1 (x -x n-1 )b p , sup n sup b∈Mn, b p≤1, b(x -x n-1 ) p }.
However, this theorem does not hold (see Remark 2.14 for a counterexample) when considering BMO c (M) or BMO r (M) separately. On the other hand, it does not correspond to the commonly used form of the classical John-Nirenberg inequality. The first purpose of this paper is to remedy these aspects of Junge and Musat's theorem. The following is one of our main results. We refer to the next section for all spaces and notation used below. P(M) denotes the set of all projections of M.

Theorem D. For all 0 < p < ∞, α -1 p x bmo c ≤ x bmo c p,pr ≤ β p x bmo c , where x bmo c p,pr = max E 1 (x) ∞ , sup n sup e∈P(Mn) 1 (τ (e)) 1/p (x -x n )e h c p .
The two constants α p and β p have the following properties

(i) α p = 1 for 2 ≤ p < ∞; (ii) α p ≤ C 1/p-1/2 for 0 < p < 2; (iii) β p ≤ cp for 2 ≤ p < ∞; (iv) β p = 1 for 0 < p < 2.
A similar result holds for BMO c (M) but only with 2 ≤ p < ∞ (see Remark 2.9). On the other hand, our proof of Theorem D can be easily modified to extend Junge/Musat's result to all 0 < p < ∞ and in the form (1.0.6) (see Corollary 2.19).

We now turn to the second objective of this paper: the atomic decomposition of different noncommutative Hardy spaces. Let us recall the 2-atomic decomposition obtained in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF].

An element a ∈ L 1 (M) is said to be a (1, 2) c -atom with respect to (M n ) n≥1 , if there exist n ≥ 1 and e ∈ P(M n ) such that (i) E n (a) = 0; (ii)ae = a; (iii) a 2 ≤ (τ (e)) -1/2 .

The atomic Hardy space h c

1,at (M) is defined as the space of all x ∈ L 1 (M), such that the following • h c 1,at norm is finite,

x h c 1,at = E 1 x 1 + inf j |λ j |.
Here the infimum is taken for possible decompositions

x -E 1 x = j λ j a j with λ j ∈ C, a j being (1, 2) c -atom. It is proved in [2] that x ∈ h c 1 (M) if and only if x ∈ h c 1,at (M) and x h c 1 ≃ x h c 1,at .
Together with the equivalence

H c 1 (M) = h c 1 (M) + h d 1 (M)
, the authors of [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] also obtained a 2-atomic decomposition for H c 1 (M). Let us briefly recall the argument used in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. The dual space of h c 1,at (M) can be described as

Λ c (M) = {x ∈ L 2 (M) : x Λ c < ∞} with x Λ c = max{ E 1 x ∞ , sup n≥1 sup e∈Pn ( 1 τ (e) τ (e|x -x n | 2 )) 1 2 }.
Actually, the supremum in the definition above can be taken for all b ∈ L 1 (M n ) since the extreme points of the unit ball of L 1 (M n ) are all multiples of projections. Therefore,

x Λ c = max{ E 1 x ∞ , sup n≥1 sup b∈Mn ( 1 b 1 τ (b|x -x n | 2 )) 1 2 } (1.0.8) = max{ E 1 x ∞ , sup n≥1 E n |x -x n | 2 1 2 ∞ } = x bmo c . Then the duality h c 1 (M) = bmo c (M) yields h c 1,at (M) = h c 1 (M).
It is well known in the classical theory that 2-atoms in the previous atomic decomposition can be replaced by q-atoms for any 1 < q ≤ ∞. Let us recall these atoms in the commutative case. A function a ∈ L 1 (Ω) is said to be a q-atom if there exist n ≥ 1 and

E ∈ F n such that (i) E n a = 0; (ii) {a = 0} ⊂ E; (iii) a q ≤ P(E) -1+ 1
q . We refer to [START_REF] Weisz | Atomic Hardy spaces[END_REF] for more information.

Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

The main difficulty to obtain q-atomic decompositions in the noncommutative case is that the key equivalence (1.0.8) no longer holds if one replaces the power indices 2 by q ′ = 2, 1 ≤ q ′ < ∞. We overcome this obstacle by Theorem D.

Theorem E. For all 1 < q ≤ ∞, h c 1 (M) = h c 1,atq,pr (M)
with equivalent norms. Here h c 1,atq,pr (M) is the q-atomic Hardy spaces with its atom a defined as: there exist n ≥ 1 and a projection e ∈ P(M n ) such that

(i) E n (a) = 0; (ii) ae = a; (iii) a h c q ≤ (τ (e)) -1 q ′ . Note that h c 1,at 2,pr (M) = h c 1,at (
M), so we recover the 2-atomic decomposition of [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]. Moreover, applying the conditional version of Junge/Musat's theorem in the form (1.0.6), we get a q-atomic decomposition for h 1 (M) in which the atoms are defined in a similar way with h c q in (iii) above replaced by q and the support condition (ii) weakened to ae = a or ea = a (see Theorem 3.19). This is exactly the noncommutative analogue of the classical atomic decomposition.

This chapter is organized as follows. Section 1.1 is on preliminaries and notation. All the results on John-Nirenberg inequality will be presented in section 1.2. Section 1.3 is devoted to the atomic decomposition of Hardy spaces. In section 1.4, we answer Junge/Musat's question in [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] which implies that the John-Nirenberg inequality in the classical sense does not hold any more in the noncommutative setting.

In this chapter, the letter c always denotes an absolute positive constant, while C an absolute constant bigger than 1. They may vary from lines to lines.

Preliminaries and notations

Throughout this chapter, we will work on a von Neumann algebra M with a normal faithful normalized trace τ . For all 0 < p ≤ ∞, let L p (M, τ ) or simply L p (M) be the associated noncommutative L p spaces. For x ∈ L p (M) we denote the right and left supports of x by r(x) and l(x) respectively. r(x) (resp. l(x)) is also the least projection e such that xe = x (resp. ex = x). If x is selfadjoint, r(x) = l(x), denoted by s(x). We mainly refer the reader to [START_REF] Pisier | Handbook of the Geometry of Banach Spaces[END_REF] for more information on noncommutative L p spaces.

Let us recall some basic notions on noncommutative martingales. Let (M n ) n≥1 be an increasing sequence of von Neumann subalgebras of M such that the union of the

M n 's is w * -dense in M. Let E n be the conditional expectation of M with respect to M n . A sequence x = (x n ) in L 1 (M) is called a noncommutative martingale with respect to (M n ) n≥1 if E n (x n+1 ) = x n for every n ≥ 1.
If in addition, all the x n 's are in L p (M) for some 1 ≤ p ≤ ∞, x is called an L p -martingale. In this case we set

x p = sup n≥1 x n p . If x p < ∞, x is called a bounded L p -martingale.
Let x = (x n ) be a noncommutative martingale with respect to (M n ) n≥1 . Define dx n = x nx n-1 for n ≥ 1 with the convention that x 0 = 0 and E 0 = E 1 . The sequence 1.1. Preliminaries and notations dx = (dx n ) n is called the martingale difference sequence of x. In the sequel, for any operator x ∈ L 1 (M) we denote x n = E n (x) for n ≥ 1.

The sequence (M n ) n≥1 will be fixed throughout the chapter. All martingales will be with respect to (M n ) n≥1 . Let 1 ≤ p < ∞. Define H c p (resp. H r p ) as the completion of all finite L p -martingales under the norm x H c p = S c (x) p (resp. x H r p = S r (x) p ), where S c (x) and S r (x) are defined as

S c (x) = k≥1 |dx k | 2 1/2 , S r (x) = S c (x * ).
The noncommutative martingale Hardy spaces H p (M) are defined as follows: if

1 ≤ p < 2, H p (M) = H c p (M) + H r p (M)
equipped with the norm

x Hp = inf x=y+z { y H c p + z H r p }. When 2 ≤ p < ∞, H p (M) = H c p (M) ∩ H r p (M) equipped with the norm x Hp = max{ x H c p , x H r p }. The space BMO c is defined as BMO c (M) = {x ∈ L 1 (M) : x BMO c < ∞} where x BMO c = sup n≥1 E n |x -x n-1 | 2 1/2 ∞ , and BMO r (M) = {x : x * ∈ BMO c (M)}. Define BMO(M) = BMO c (M) ∩ BMO r (M)
equipped with the norm

x BMO = max{ x BMO c , x BMO r }.
Pisier and Xu [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] proved the two fundamental results: H p (M) = L p (M) and Fefferman duality between H 1 (M) and BMO(M). Their work triggered a rapid development of the noncommutative martingale theory.

We will also work on the conditional version of Hardy and BMO spaces developed in [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF]. Let x = (x n ) n≥1 be a finite martingale in L 2 (M). We set

s c (x) = k≥1 E k-1 |dx k | 2 1/2 and s r (x) = s c (x * ). Let 0 < p < ∞. Define h c p (M) (resp. h r p (M))
as the completion of all finite L ∞ -martingales under the (quasi-)norm x h c p = s c (x) p (resp. x h r p = s r (x) p ). Define h d p (M) as the Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales subspace of ℓ p (L p (M)) consisting of all martingale difference sequences, where ℓ p (L p (M)) is the space of all sequences a = (a n ) n≥1 in L p (M) such that

a ℓp(Lp(M)) = n≥1 a n p p 1/p < ∞
with the usual modification for p = ∞. The noncommutative conditional martingale Hardy spaces are defined as follows: if 0 < p < 2,

h p (M) = h c p (M) + h r p (M) + h d p (M)
equipped with the (quasi-)norm

x hp = inf x=y+z+w { y h c p + z h r p + w h d p }. When 2 ≤ p < ∞, h p (M) = h c p (M) ∩ h r p (M) ∩ h d p (M) equipped with the norm x hp = max{ x h c p , x h r p , x h d p }.
The space bmo c is defined as

bmo c (M) = {x ∈ L 1 (M) : x bmo c < ∞} where x bmo c = max E 1 (x) ∞ , sup n≥1 E n |x -x n | 2 1/2 ∞ . Let bmo r (M) = {x : x * ∈ bmo c (M)}. Let bmo d (M) be the subspace of ℓ ∞ (L ∞ (M)) consisting of all martingale difference se- quences. Note that bmo d (M) = h d ∞ (M). Define bmo(M) = bmo c (M) ∩ bmo r (M) ∩ bmo d (M)
equipped with the norm

x bmo = max{ x bmo c , x bmo r , x bmo d }.
We refer to [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], [START_REF] Junge | Noncommutative Burkholder/Rosenthal inequalities II : applications[END_REF], [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF], [START_REF] Randrianantoanina | Conditioned square functions for noncommutative martingales[END_REF], [START_REF] Junge | Noncommutative Riesz transforms-A probabilistic approach[END_REF], [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF] for more information on these spaces.

John-Nirenberg inequality 1.2.1 A crude version

Definition 2.1. For 0 < p < ∞, we define

(i) bmo c p (M) = x ∈ L 1 (M) : x bmo c p < ∞ with x bmo c p = max E 1 (x) ∞ , sup n sup a∈Mn, a p≤1, (x -x n )a h c p ; 1.2. John-Nirenberg inequality 39 (ii) bmo r p (M) = {x : x * ∈ bmo c p (M)}; (iii) bmo p (M) = bmo c p (M) ∩ bmo r p (M) ∩ bmo d (M) equipped with the (quasi-)norm x bmop = max{ x bmo c p , x bmo r p , x bmo d }.
Remark 2.2. When p = 2, these are exactly the spaces bmo c (M), bmo r (M) and bmo(M).

Below is our first version of the column (resp. row) John-Nirenberg inequality.

Theorem 2.3. For all 0 < p < ∞, there exist two constants α p and β p such that

α -1 p x bmo c ≤ x bmo c p ≤ β p x bmo c ,
with α p and β p satisfying

(i) α p = 1 for 2 ≤ p < ∞; (ii) α p ≤ C 1/p-1/2 for 0 < p < 2; (iii) β p ≤ cp for 2 ≤ p < ∞; (iv) β p = 1 for 0 < p < 2.
The similar inequalities hold for • bmo r p and • bmo r . Proof. We only need to prove the column case, since the row case can be done by replacing x with x * . First consider the case 2 < p < ∞. We will show the following inequalities:

x bmo c 2 ≤ x bmo c p ≤ cp x bmo c 2 .
The left inequality is obtained directly by Hölder's inequality. In fact, taking a ∈ M n with a 2 ≤ 1, there exists a factorization a = a 0 a 1 such that a 0 p = a 2/p 2 ≤ 1 and

a 1 2p/(p-2) = a (p-2)/p 2 ≤ 1, so (x -x n )a 2 h c 2 = τ (a * 1 a * 0 s 2 c (x -x n )a 0 a 1 ) ≤ a * 1 2p p-2 a * 0 s 2 c (x -x n )a 0 p 2 a 1 2p p-2 ≤ (x -x n )a 0 2 h c
p . We invoke complex interpolation to prove the right inequality.

sup t∈R B(it) 2 ≤ 1, sup t∈R B(1 + it) ∞ ≤ 1. Define f (z) = (x -x n )B(z).
Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

Then on the one hand, by the definition of bmo c 2 (M), we have

f (it) h c 2 ≤ x bmo c 2 .
On the other hand, by a simple calculation, we have

f (1 + it) bmo c 2 ≤ x -x n bmo c 2 B(1 + it) ∞ ≤ x bmo c 2 . Therefore, by interpolation, f (θ) (h c 2 ,bmo c ) θ ≤ x bmo c 2 =
x bmo c . However by [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF],

(h c 2 , bmo c ) θ ⊂ h c p
with relevant constant majorized by cp. We then deduce that

f (θ) h c p ≤ cp x bmo c , (1.2.1)
hence the desired inequality holds.

For the case 0 < p < 2. We show the following inequalities:

x bmo c p ≤ x bmo c 2 ≤ C 1/p-1/2 x bmo c p .
Again, the left inequality is obtained by Hölder's inequality. It remains to prove the right one. We choose 2 < p 1 < ∞ and 0 < θ < 1 such that 1/2 = (1θ)/p + θ/p 1 . Fix n, by the definition of bmo c p (M), we can view xx n as a bounded operator from L p (M n ) to h c p (M). Then we have the following two inequalities:

x -x n Lp(Mn)→h c p ≤ x bmo c p , x -x n Lp 1 (Mn)→h c p 1 ≤ x bmo c p 1
.

Then by interpolation, we get

x -x n L 2 (Mn)→(h c p ,h c p 1 ) θ ≤ x 1-θ bmo c p x θ bmo c p 1
.

Now by the trivial contractive inclusion

(h c p , h c p 1 ) θ ⊂ h c 2 ,
and the right inequality in the case 2 < p 1 < ∞, we get

x -x n L 2 (Mn)→h c 2 ≤ cp 1 x 1-θ bmo c p x θ bmo c 2 . Therefore, x bmo c 2 ≤ (cp 1 ) θ x 1-θ bmo c p x θ bmo c 2 , hence x bmo c 2 ≤ (cp 1 ) θ 1-θ x bmo c p . Noting that θ/(1 -θ) = (1/p -1/2)/(1/2 -1/p 1 )
, we get the desired estimate by taking C = (cp 1 ) 1/(1/2-1/p 1 ) . Remark 2.4. The constant in (1.2.1) is optimal. This can be seen as follows. By Lemma

4.3 in [2], h c p ′ (M) embeds into (h c 2 (M), h c 1 (M)) θ with constant independent of p ′ . So ((h c 2 (M)) * , (h c 1 (M)) * ) θ embeds into (h c p ′ (M))
* with constant independent of p by duality. Finally, by the optimal embedding (h

c p ′ (M)) * ⊂ h c p (M) with constant cp in [25] and bmo c (M) ⊂ (h c 1 (M)) * in [45], (h c 2 (M), bmo c (M)) θ embeds into h c p (M) with optimal constant cp.
It is natural to ask whether there is a result similar to Theorem 2.3 for BMO c by replacing h c p and xx n in the definition of bmo c p by H c p and xx n-1 respectively. Using the identity

BMO c (M) ≃ bmo c (M) ∩ bmo d (M)
proved in [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF], we are reduced to deal with the diagonal space bmo d (M). Surprisingly, the result is true only for 2 ≤ p < ∞ (see Remark 2.9).

Definition 2.5. For 1 ≤ p < ∞, we define (i)

BMO c p (M) = x ∈ L 1 (M) : x BMO c p < ∞ with x BMO c p = sup n sup a∈Mn, a p≤1 (x -x n-1 )a H c p ; (ii) BMO r p (M) = {x : x * ∈ BMO c p (M)}; (iii) BMO p (M) = BMO c p (M) ∩ BMO r p (M) equipped with the norm x BMOp = max{ x BMO c p , x BMO r p }.
Remark 2.6. For p = 2, we recover the spaces BMO c (M), BMO r (M) and BMO(M).

The following lemma will alow us to handle with the diagonal space bmo d (M).

Lemma 2.7. For 2 ≤ p < ∞, we have [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF], Remark 5.4 as a reference for the constant we use here), we have

cp -1 b ∞ ≤ sup a∈M, a p≤1 ba H c p ≤ cp 1 2 b ∞ . Proof. Note that • H c p ≤ cp 1/2 • p (see
sup a∈M, a p≤1 ba H c p ≤ cp 1 2 sup a∈M, a p≤1 ba p = cp 1 2 b ∞ .
For the first inequality, without loss of generality assume b ∞ = 1. Note that for selfadjoint x ∈ M, x p ≤ cp x H c p (see [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF], Remark 5.4). Then

b * ∞ = sup y∈M, y 2p ≤1 yb * 2p = sup y∈M, y 2p ≤1 b|y| 2 b * 1 2 p ≤ cp 1 2 sup y∈M, y 2p ≤1 b|y| 2 b * 1 2 H c p ≤ cp 1 2 sup a∈M, a p≤1 ba 1 2 H c p .
And then cp -1 b ∞ ≤ sup a∈M, a p≤1 ba H c p .

Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales Theorem 2.8. For all 2 ≤ p < ∞, we have

BMO c p (M) = BMO c (M)
with equivalent norms. More precisely,

cp -1 x BMO c ≤ x BMO c p ≤ cp x BMO c .
Similarly, BMO r p (M) = BMO r (M) with equivalent norms.

Using the previous lemma and the identity BMO c (M) ≃ bmo c (M) ∩ bmo d (M), we can easily deduce Theorem 2.8 from Theorem 2.3. We will however present a direct proof. Proof. We only prove the inequalities for the column case, the row case can be dealt with similarly. By the previous lemma and Hölder's inequality, we have

E n ∞ k=n |dx k | 2 ∞ ≤ sup b∈M + n , b 1 ≤1 τ ∞ k=n+1 |dx k | 2 b + x n -x n-1 2 ∞ ≤ sup b∈M + n , b 1 ≤1 τ ∞ k=n+1 |(dx k )b 1 p | 2 b p-2 p + cp 2 sup a∈Mn, a p≤1 (x n -x n-1 )a 2 H c p ≤ sup b∈M + n , b 1 ≤1 ∞ k=n+1 |(dx k )b 1 p | 2 p 2 b p-2 p ( p 2 ) ′ + cp 2 sup a∈Mn, a p≤1 (x n -x n-1 )a 2 H c p ≤ sup b∈M + n , b 1 ≤1 (x -x n )b 1 p 2 H c p + cp 2 sup a∈Mn, a p≤1 (x n -x n-1 )a 2 H c p .
Then by

E n x H c p ≤ x H c p , x BMO c 2 ≤ cp sup a∈Mn, a p≤1 (x -x n-1 )a H c p = cp x BMO c p .
Conversely, by the previous lemma,

x BMO c p ≤ sup n sup a∈Mn, a p≤1 (x -x n )a H c p + sup n sup a∈Mn, a p≤1 (x n -x n-1 )a H c p ≤ sup n sup a∈Mn, a p≤1 (x -x n )a H c p + cp 1 2 sup n x n -x n-1 ∞ ≤ sup n sup a∈Mn, a p≤1 (dx k a) ∞ k=n+1 Lp(ℓ c 2 ) + cp 1 2 x BMO c 2 . (1.2.2)
Note that, by the Hahn-Banach theorem and the duality between H c 1 (M) and BMO c (M), there exists a sequence

(b n ) ∞ n=1 ∈ L ∞ (M; ℓ c 2 ) such that (b n ) ∞ n=1 L∞(ℓ c 2 ) = x BMO c , dx k = E k b k -E k-1 b k .
Thus by the noncommutative Stein inequality (see [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] for the constant used below) and Hölder's inequality,

sup a∈Mn, a p≤1 (dx k a) ∞ k=n+1 Lp(ℓ c 2 ) ≤ sup a∈Mn, a p≤1 (E k (b k a)) ∞ k=n+1 Lp(ℓ c 2 ) + sup a∈Mn, a p≤1 (E k b k+1 a) ∞ k=n Lp(ℓ c 2 ) ≤ cp sup a∈Mn, a p≤1 (b k a) ∞ k=n+1 Lp(ℓ c 2 ) ≤ cp ∞ k=1 |b k | 2 1 2 ∞ = cp x BMO c 2 .
Combining this with (1.2.2) we finish the proof.

Remark 2.9. It is a bit surprising that Theorem 2.8 is actually wrong for any p < 2. Indeed, choose a filtration M 1 , M 2 , M 3 ,...,M n-1 and y ∈ M n-1 such that y p = 1 and

y H c p = c n >> 1. Let M n = L ∞ (Ω, M n-1 ) with Ω = {0, 1} with µ{1} = µ{0} = 1/2. We certainly can view M k , k < n as the space of constant functions on Ω, so M k ⊂ M n . Let x = 1 on {0} and x = -1 on {1} then x n-1 = 0. Let a = y on {0} and a = -y on {1}. Then (x -x n-1 )a = y whose H c p norm equals c n and a p = 1, so x BMO c p ≥ c n . But x BMO c 2 = 1.
In the rest of this subsection, we turn to Junge/Musat's type of John-Nirenberg inequality. In [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF], Junge and Musat established the inequality for 2 < p < ∞ in the state case. Later the second author of the present paper gave a simple proof for all 1 ≤ p < ∞ in the tracial setting (see [START_REF] Mei | Notes on matrix valued paraproducts[END_REF]). The idea of the proof of Theorem 2.3 can be applied to obtain this inequality for all 0 < p < ∞ (see Corollary 2.13). We start again with bmo(M).

Theorem 2.10. For all 0 < p < ∞, we have

α -1 p x bmo ≤ b p (x) ≤ β p x bmo where b p (x) = max{ sup n (dx n ) n ∞ , sup n sup b∈Mn, b p≤1 (x -x n )b p , sup n sup b∈Mn, b p≤1 b(x -x n ) p }.
The constant α p and β p have the same orders as those in Theorem 2.3.

Proof. We first treat the case 2 ≤ p < ∞. For p = 2, it is trivial. So we can assume 2 < p < ∞. The inequality

x bmo ≤ b p (x)
follows from Hölder's inequality. We will prove the reverse inequality by interpolation. By a simple calculation, we have the following estimates

(x -x n )b bmo c ≤ x bmo c b ∞ , (x -x n )b bmo r ≤ x bmo r b ∞ ,
Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

(x -x n )b bmo d ≤ x bmo d b ∞ .
Then it follows that

(x -x n )b bmo ≤ x bmo b ∞ .
On the other hand, it is clear that

(x -x n )b 2 = (x -x n )b h c 2 ≤ x bmo b 2 .
Then by the interpolation result of [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF], we have

(x -x n )b p ≤ cp (x -x n )b (L 2 ,bmo) θ (1.2.3) ≤ cp x bmo b p .
In the same way, we obtain

b(x -x n ) p ≤ cp x bmo b p .
Thus we prove the assertion. Now we turn to the case 0 < p < 2, by Hölder's inequality, we obtain the trivial part

b p (x) ≤ b 2 (x) = x bmo .
Let us prove the inverse one, let 2 < p 1 < ∞ and θ be such that

1 2 = 1 -θ p + θ p 1 .
We view xx n and (xx n ) * as two operators. By interpolation,

(x -x n ) L 2 (Mn)→L 2 (M) ≤ (x -x n ) 1-θ Lp(Mn)→Lp(M) (x -x n ) θ Lp 1 (Mn)→Lp 1 (M)
and similarly for (xx n ) * . By the estimate for p 1 > 2, we have

b 2 (x) ≤ (cp 1 ) θ b 1-θ p (x)b θ 2 (x).
Therefore, we obtain

x bmo ≤ (cp 1 ) θ 1-θ b p (x) = C 1/p-1/2 b p (x), with C = (cp 1 ) 1/(1/2-1/p 1 ) .
Remark 2.11. The constant in (1.2.3) is optimal. This can be seen as follows. By Lemma [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF], (h 2 (M), bmo(M)) θ embeds into L p (M) with optimal constant cp. Remark 2.12. We can directly compare the norms • bmop and b p (•) directly for 1 < p < ∞ by using Theorem 2.3.

4.3 in [2], h c p ′ (M) embeds into (h c 2 (M), h c 1 (M)) θ with constant independent of p ′ . So h p ′ (M) embeds into (h 2 (M), h 1 (M)) θ with constant independent of p ′ . Now by Theorem 4.1 in [53], L p ′ (M) embeds into h p ′ (M), hence into (h 2 (M), h 1 (M)) θ with optimal constant c/(p ′ -1). Then by duality, ((h 2 (M)) * , (h 1 (M)) * ) θ embeds into (L p ′ (M)) * = L p (M) with best constant cp. At last, by bmo(M) ⊂ (h 1 (M)) * in
Let us justify this remark. We first deal with the case 2 < p < ∞. Fix n, for any b ∈ M n with b p ≤ 1, by the noncommutative Burkholder inequality [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], we have

(x -x n )b h c p ≤ cp (x -x n )b p , b(x -x n ) h r p ≤ cp b(x -x n ) p , hence (x -x n )b h c p , b(x -x n ) h r p ≤ cpb p (x)
Then by Theorem 2.3,

x bmop ≤ cpb p (x).
Another direction can be done by the way in Theorem 2.10,

b p (x) ≤ cp x bmo ≤ cp x bmop .
For the case 1 < p < 2. The trivial part b p (x) ≤ c x bmop follows from the noncommutative Burkholder inequality in [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF]. Now let us prove the inverse one. Take b ∈ M n with b 2 ≤ 1. By Hölder's inequality, we have

(x -x n )b 2 2 = τ (b 2/p ′ (x -x n ) * (x -x n )b 2/p ) ≤ b 2/p ′ (x -x n ) * p ′ (x -x n )b 2/p p and b(x -x n ) 2 2 = τ ((x -x n ) * b 2/p ′ b 2/p (x -x n )) ≤ (x -x n ) * b 2/p ′ p ′ b 2/p (x -x n ) p . So by the result in Theorem 2.3 for 2 < p ′ < ∞, we have b(x -x n ) 2 2 , (x -x n )b 2 2 ≤ max{ b 2/p ′ (x -x n ) * p ′ , (x -x n ) * b 2/p ′ p ′ } • max{ (x -x n )b 2/p p , b 2/p (x -x n ) p } ≤ c x bmo p ′ • b p (x) ≤ cp ′ x bmo 2 • b p (x)
Then by the definition of bmo 2 (M), we finish the proof by Theorem 2.3

x bmop ≤ x bmo 2 ≤ cp ′ b p (x).
The following corollary extends Junge/Musat's theorem to all 0 < p < ∞. It can be proved similarly as Theorem 2.3. However, using the identity BMO(M) ≃ bmo(M) proved in [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF], we give a simpler proof. Proof. For 2 ≤ p < ∞, it is very easy to get

B p (x) ≤ b p (x) ≤ cp x bmo ≤ cp x BMO
from the triangular inequality

(x -x n-1 )b p ≤ (x -x n )b p + (x n -x n-1 )b p ,
with b ∈ M n and b p ≤ 1. And the rest of the proof is the same to Theorem 2.10.

Remark 2.14. The following example shows that Junge/Musat's John-Nirenberg inequality does not hold for bmo c or BMO c . The example is the same as the one given in Remark 3.20 of [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF]. Let n be a positive integer and consider the von Neumann algebra

M = L ∞ (T) ⊗M n ,
where M n is the algebra of n × n matrices with normalized trace. For k ≥ 1 let F k be the σ-algebra generated by dyadic intervals in T of length 2 -k . Denote by M k the subalgebra L ∞ (T, F k ) ⊗M n of M and let E k = E k ⊗ id Mn be the conditional expectation onto M k . Let r k be the k-th Rademacher function on T and consider

x = n k=1 r k ⊗ e 1k .
Then x is a martingale relative to the filtration (M k ) k≥1 and the martingale differences are given by dx k = r k ⊗ e 1k . A simple calculation shows that sup m

xx m p = (n -1)

1 2 n -1 p , while x bmo c = sup m n k=m+1 E m |d k x| 2 1 2 ∞ = 1.
Let p > 2. Then for any c > 0, there exists n ≥ 1 such that (n -1) 1/2 n -1/p > c. Hence

sup m sup b∈Mm, b p≤1 (x -x m )b p ≥ sup m x -x m p >> x bmo c .

A fine version

Now we can formulate the fine version of the column (resp. row) John-Nirenberg inequality.

Definition 2.15. For 0 < p < ∞, we define The fine version of the column (resp. row) John-Nirenberg inequality is stated as follows.

bmo c p,pr (M) = x ∈ L 1 (M) : x bmo c p,pr < ∞ with x bmo c p,pr = max E 1 (x) ∞ , sup n sup e∈P(Mn) (x -x n ) e (τ (e))
Theorem 2.16. For all 0 < p < ∞, we have

α -1 p x bmo c ≤ x bmo c
p,pr ≤ β p x bmo c . The constants α p and β p have the same properties as those in Theorem 2.3. The same inequalities hold for • bmo r and • bmo r p,pr . Proof. We first consider the case 0 < p ≤ 1. By Theorem 2.3, the trivial part

x bmo c
p,pr ≤ x bmo c p ≤ x bmo c follows from the fact that e/(τ (e)) 1/p ∈ M n and its L p -norm equals 1. Now we turn to the proof of the inverse inequality. Since any a ∈ M n with a p ≤ 1 can be approximated by sums k λ k e k /(τ (e k )) 1/p with e k 's in M n and k |λ k | p ≤ 1. Thus we can assume that a itself is such a sum. Then

(x -x n )a p h c p = k λ k (x -x n ) e k (τ (e k )) 1/p p h c p ≤ k |λ k | p (x -x n ) e k (τ (e k )) 1/p p h c p ≤ k |λ k | p x p bmo c p,pr ≤ x p bmo c p,pr .
Therefore by Theorem 2.3,

x bmo c ≤ C 1/p-1/2 x bmo c p ≤ C 1/p-1/2
x bmo c p,pr . Now let 1 < p < ∞. Again, because of the fact that e/(τ (e)) 1/p ∈ M n and its L p -norm equals 1, by Theorem 2.3,

x bmo c p,pr ≤ x bmo c p ≤ c 1 p x bmo c . (1.2.4)
We exploit the result for p = 1 to prove the inverse inequality. By Hölder's inequality, we have

x bmo c 1,pr ≤ x bmo c p,pr . We end the proof by Theorem 2.3 and the result for p = 1,

x bmo c ≤ C 1/p-1/2 x bmo c 1 ≤ C 1/p-1/2 x bmo c 1,pr ≤ C 1/p-1/2 x bmo c p,pr .
Now we give the distributional form of the John-Nirenberg inequality for bmo c (M) and bmo r (M).

Theorem 2.17. Let x ∈ bmo c (M). Then for all natural numbers n ≥ 1, all e ∈ P(M n ) and for all λ > 0, we have

1 τ (e) τ (1 (λ,∞) (s c ((x -x n )e))) ≤ 2 exp(- cλ x bmo c
),

with c an absolute constant. Here 1 (λ,∞) (a) denotes the spectral projection of a positive operator a corresponding to the interval (λ, ∞).

Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

Proof. By homogeneity, we can assume x bmo c = 1. We first deal with the case λ ≥ 2c 1 , where c 1 is the constant in inequality (1.2.4). Let p = λ/(2c 1 ) ≥ 1, by Chebychev's inequality and Theorem 2.16,

τ (1 (λ,∞) (s c ((x -x n )e))) ≤ τ (e) (x -x n )e p h c p λ p ≤ τ (e)(c 1 pλ -1 ) p = τ (e) exp(p ln(c 1 pλ -1 )) = τ (e) exp(- ln 2 2c 1 λ). When 0 < λ < 2c 1 , 1 τ (e) τ (1 (λ,∞) (s c ((x -x n )e))) ≤ 1 < 2 exp(- ln 2 2c 1 λ).
Therefore, we obtain the desired result by letting c = ln 2/(2c 1 ).

Based on the crude version of Junge/Musat's John-Nirenberg inequality in Theorem 2.10 (resp. Corollary 2.8) for bmo(M) (resp. BMO(M)), the argument in the proof of Theorem 2.16 can be adapted to get the fine version of Junge/Musat's John-Nirenberg inequality.

Corollary 2.18. For all 0 < p < ∞, we have Again, based on Corollary 2.19, by arguments similar to the proof of Thoerem 2.17, we obtain the exponential integrability form of the John-Nirenberg inequality for BMO(M).

α -1 p x bmo ≤ Pb p (x) ≤ β p x
Theorem 2.20. Let x ∈ BMO(M). Then for all natural numbers n ≥ 1, all e ∈ P(M n ) and for all λ > 0, we have

1 τ (e) τ (1 (λ,∞) (|(x -x n-1 )e|) + 1 (λ,∞) (|e(x -x n-1 )|)) ≤ 4 exp(- cλ x BMO )
with c an absolute constant.

1.3. Atomic decomposition 49

Atomic decomposition 1.3.1 A crude version of atoms

According to the crude version of the noncommutative John-Nirenberg inequality, we introduce the following Definition 3.1. For 1 < q ≤ ∞, a ∈ L 1 (M) is said to be a (1, q, c)-atom with respect to (M n ) n≥1 , if there exist n ≥ 1 and a factorization a = yb such that

(i) E n (y) = 0; (ii) b ∈ L q ′ (M n ) and b q ′ ≤ 1; (iii) y h c q ≤ 1 for 1 < q < ∞; y bmo c ≤ 1 for q = ∞.
Similarly, we define the notion of a (1, q, r)-atom with a = yb replaced by a = by.

Lemma 3.2. Let 1 < q ≤ ∞. If a is a (1, q, c)-atom, then a h c 1 ≤ 1.
The analogous inequality holds for (1, q, r)-atoms.

Proof. We first deal with the case 1 < q < ∞. By definition, there exists an n such that the (1, q, c)-atom a admits a factorization a = yb as in Definition 3.1. Then

s 2 c (a) = b * k>n E k-1 |dy k | 2 b = b * s 2 c (y)b.
Thus by Hölder's inequality,

a h c 1 = s c (a) 1 ≤ s c (y) q b q ′ ≤ 1.
For the case q = ∞, the calculation is a bit different,

a h c 1 = b * s 2 c (y)b 1/2 1/2 = τ (E n (b * s 2 c (y)b) 1/2 ) ≤ τ ((E n (b * s c (y)b)) 1/2 ) ≤ E n (s c (y)) ∞ b 1 ≤ y bmo c b 1 ≤ 1.
We have used the trace preserving property of conditional expectations in the fourth equality and the operator Jensen inequality in the first inequality. For the second inequality, we have used the property that E n • E k-1 = E n for all k > n and Hölder's inequality. Definition 3.3. We define h c 1,atq (M) as the Banach space of all x ∈ L 1 (M) which admit a decomposition x = k λ k a k , where for each k, a k a (1, q, c)-atom or an element in the unit ball of L 1 (M 1 ), and λ k ∈ C satisfying k |λ k | < ∞. We equip this space with the norm

x h c 1,atq = inf k |λ k |,
where the infimum is taken over all decompositions of x described above. Similarly, we define h r 1,atq (M). Now, by Lemma 3.2, we have the obvious inclusion h c 1,atq (M) ⊂ h c 1 (M). In fact, the two spaces coincide thanks to the following theorem.

Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales Theorem 3.4. For all 1 < q ≤ ∞, we have

h c 1 (M) = h c 1,atq (M)
with equivalent norms. Similarly, h r 1 (M) = h r 1,atq (M) with equivalent norms.

We prove this theorem by duality. We require the following lemmas.

Lemma 3.5. (i) For all 1 < q ≤ 2, L 2 (M) densely and continuously embeds into h c 1,atq (M).

(ii) For all 2 < q ≤ ∞, L q (M) densely and continuously embeds into h c 1,atq (M).

Proof. (i). For any x ∈ L 2 (M), we decompose it as a linear combination of two atoms:

x = x -E 1 (x) 2 x -E 1 (x) x -E 1 (x) 2 + E 1 (x) 2 E 1 (x) E 1 (x) 2 .
Indeed, on the one hand,

E 1 (x)/ E 1 (x) 2 ∈ L 2 (M 1 ) ⊂ L 1 (M 1 )
and

E 1 (x) E 1 (x) 2 1 = E 1 (x) 1 E 1 (x) 2 ≤ 1.
On the other hand,

x -E 1 (x) x -E 1 (x) 2 = x -E 1 (x) x -E 1 (x) 2 • 1 . = y • b.
Clearly, E 1 (y) = 0, b q ′ ≤ 1 and

y h c q = x -E 1 (x) x -E 1 (x) 2 h c q ≤ x -E 1 (x) x -E 1 (x) 2 h c 2 ≤ 1.
Thus x is a sum of two atoms and

x h c 1,atq ≤ x -E 1 (x) 2 + E 1 (x) 2 ≤ √ 2 x 2 .
The density is trivial.

(ii). This case is similar to the previous one. We first deal with the case 2 < q < ∞. Given x ∈ L q (M), we write again:

x = c q x -E 1 (x) q x -E 1 (x) c q x -E 1 (x) q + E 1 (x) q E 1 (x) E 1 (x) q ,
where c q is fixed below. Indeed,

E 1 (x)/ E 1 (x) q ∈ L q (M 1 ) ⊂ L 1 (M 1 )
and

E 1 (x) E 1 (x) q 1 = E 1 (x) 1 E 1 (x) q ≤ 1.
On the other hand,

x -E 1 (x) c q x -E 1 (x) q = x -E 1 (x) c q x -E 1 (x) q • 1 . = y • b, E 1 ( x -E 1 (x) c q x -E 1 (x) q ) = 0, b q ′ ≤ 1 1.3. Atomic decomposition 51
and the noncommutative Burkholder inequality in [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF] yields

y h c q = x -E 1 (x) c q x -E 1 (x) q h c q ≤ c q x -E 1 (x) c q x -E 1 (x) q q ≤ 1.
Therefore, x h c 1,atq ≤ c q x -E 1 (x) q + E 1 (x) q ≤ (2c q + 1) x q . The case q = ∞ is proved in the same way just by replacing the noncommutative Burkholder inequality by the trivial fact that

• bmo c ≤ • ∞ . The density is trivial. Lemma 3.6. Let 1 < q < ∞. Then (h c 1,atq (M)) * = bmo c q ′ (M)
with equivalent norms. More precisely, (i) Every x ∈ bmo c q ′ (M) defines a bounded linear functional on h c 1,atq (M) by

ϕ x (a) = τ (x * a), ∀a ∈ (1, q, c)-atoms. (1.3.1) (ii) Conversely, each ϕ ∈ (h c 1,atq (M)) * is given as (1.3.1) by some x ∈ bmo c q ′ (M).
Similarly, (h r 1,atq (M)) * = bmo r q ′ (M) with equivalent norms.

Proof. (i) Let x ∈ bmo c q ′ , and a = yb where a is a (1, q, c)-atom as in Definition 3.1. Then

|τ (x * a)| = |τ (E n (x * y)b)| = |τ (E n ((x * -x * n )y)b)| = |τ (((x -x n )b * ) * y)|.
Thus, by the duality identity h c q (M) = (h c q ′ (M)) * (see [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF] for the relevant constants),

|τ (x * a)| ≤ (x -x n )b * h c q ′ y h c q ≤ x bmo c q ′ .
(ii). Let ϕ be any linear functional on h c 1,atq (M). When 1 < q ≤ 2, by Lemma 3.5 we can find x ∈ L 2 (M) such that

ϕ(y) = τ (x * y), ∀y ∈ L 2 (M),
and ϕ = sup

y∈L 2 , y h c 1,atq ≤1 |τ (x * y)|.
When 2 < q < ∞, by the same Lemma 3.5, we get the same representation of ϕ with an x ∈ L q ′ (M). Then fix n and take any b ∈ M n with b q ′ ≤ 1. Again, by the duality h c q (M) = (h c q ′ (M)) * , we do the following calculation:

(x -x n )b h c q ′ = sup y (h c q ′ ) * ≤1 |τ (b * (x * -x * n )y)| ≤ sup y h c q ≤cq |τ (b * (x * -x * n )y)| = sup y h c q ≤cq |τ ((x * -x * n )(y -y n )b * )|
Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales = sup

y h c q ≤cq |τ (x * ((y -y n )b * ))| ≤ cq ϕ
Here, we have used the fact that τ (x-x n ) = τ (y -y n ) = 0 in the second and third equality respectively. The second inequality is due to the fact that (yy n )b * is a (1, q, c)-atom.

Now we are at a position to prove Theorem 3.4. Proof. We consider here only the case 1 < q < ∞ and postpone the case q = ∞ to the end of the proof of Theorem 3.12 below. We only need to show the inclusion

h c 1 (M) ⊂ h c 1,atq (M).
Take x ∈ h c 1,atq (M), by Theorem 2.3 and Lemma 3.6, we can conduct the following calculation,

x h c 1,atq = sup y (h c 1,atq ) * ≤1 |τ (x * y)| ≤ sup y bmo c q ′ ≤cq |τ (x * y)| ≤ sup y bmo c ≤cq |τ (x * y)| ≤ cq x h c 1 .
Then we end the proof with the density of h c 1,atq (M) in h c 1 (M).

Definition 3.7. We define

h 1,atq (M) = h c 1,atq (M) + h r 1,atq (M) + h d 1 (M)
equipped with the sum norm

x h 1,atq = inf x=xc+xr+x d { x c h c 1,atq + x r h r 1,atq + x d h d 1 }.
Then by Theorem 3.4, we obtain the atomic decomposition of h 1 (M).

Corollary 3.8. We have

h 1 (M) = h 1,atq (M)
with equivalent norms.

Combined with Davis' decomposition presented in [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF], the above theorem yields H 1 (M) = h 1,atq (M) with equivalent norms. In other words, we obtain an atomic decomposition for H 1 (M) too.

A fine version of atoms

Definition 3.9. For 1 < q ≤ ∞, a ∈ L 1 (M) is said to be a (1, q, c) pr -atom with respect to (M n ) n≥1 , if there exist n ≥ 1 and a projection e ∈ P(M n ) such that (i) E n (a) = 0;

(ii) r(a) ≤ e;

(iii) a h c q ≤ (τ (e)) -1 q ′ for 1 < q < ∞; a bmo c ≤ (τ (e)) -1 for q = ∞.

Atomic decomposition 53

Similarly, we define (1, q, r) pr -atoms with r(a) replaced by l(a). Remark 3.10. A (1, q, c) pr -atom a is necessarily a (1, q, c)-atom. Indeed, we can factorize a as a = yb with y = a(τ (e)) 1/q ′ and b = e(τ (e)) -1/q ′ . Definition 3.11. We define h c 1,atq,pr (M) to be the Banach space of all x ∈ L 1 (M) which admit a decomposition x = k λ k a k , where for each k, a k is a (1, q, c) pr -atom or an element in the unit ball of L 1 (M 1 ), and λ k ∈ C satisfying k |λ k | < ∞. We equip this space with the norm

x h c 1,atq,pr = inf k |λ k |,
where the infimum is taken over all decompositions of x described above. Similarly, we define h r 1,atq,pr (M). Now, by Remark 3.10 and Lemma 3.4, we have the obvious inclusion h c 1,atq,pr (M) ⊂ h c 1 (M). In fact, the two spaces coincide thanks to the following theorem. Theorem 3.12. For all 1 < q ≤ ∞, we have

h c 1 (M) = h c 1,atq,pr (M)
with equivalent norms. Similarly, h r 1 (M) = h r 1,atq,pr (M) with equivalent norms. Again, we prove this theorem for 1 < q < ∞ by showing (h c 1,atq,pr (M)) * = bmo c q ′ ,pr (M). The latter duality equality is proved in the same way as Theorem 3.6. We leave the details to the reader. However by the argument in Theorem 4.6, we can not prove the theorem in the case q = ∞, due to the lack of Riesz representation. Here we provide another way to do it, which seems new, even in the commutative case.

Let P be the set of projections of M. Given e ∈ P let n e = min{k : e ∈ P(M k )}.

Note that n e = ∞ if the set on the right hand side is empty. This case is of no interest in the discussion below. For a family (g e ) e∈P ⊂ bmo c (M) define (g e ) e L P 1 (bmo c ) = e∈P τ (e) g e bmo c .

We will consider the Banach space:

L P 1 (bmo c ) = {(g e ) e
: g e e = g e , E ne g e = 0, (g e ) e L P 1 (bmo c ) < ∞}. We will also need the following space consisting of families in h c 1 (M):

L P ∞ (h c 1 ) = {(f e ) e :
f e e = f e , E ne f e = 0, (f e ) e L P ∞ (h c 1 ) < ∞}, where

(f e ) e L P ∞ (h c 1 ) = sup e∈P 1 τ (e) f e h c 1 .
For convenience, we denote L P 1 (bmo c ) by X and L P ∞ (h c 1 ) by Z. We embed bmo Thus we get (f e ) e X * ≤ √ 2 (f e ) e Z . We turn to the proof of the inverse inequality. For any (f e ) e ∈ Z, fix e 0 ∈ P, we have

1 τ (e 0 ) f e 0 h c 1 = sup g bmo c ≤1 1 τ (e 0 ) τ ((f e 0 ) * g) = sup g bmo c ≤1 1 τ (e 0 ) τ ((f e 0 ) * (g -g ne 0 )e 0 )
≤ sup (g-gn e 0 )e 0 bmo c ≤1 1 τ (e 0 ) τ ((f e 0 ) * (gg ne 0 )e 0 ) .

Then we define (g e ) e as g e = (gg ne 0 e 0 )/τ (e 0 ) if e = e 0 , otherwise g e = 0. Thus

1 τ (e 0 )
f e 0 h c 1 ≤ (f e ) e X * (g e ) e X ≤ (f e ) e X * , which implies (f e ) e Z ≤ (f e ) e X * . (ii). Since Y is a subspace of X * , by Krein and Smulian's theorem, we only need to prove that for all t > 0, Y ∩ B t (X * ) is w*-closed in X * , where B t (X * ) is the closed ball of X * centered at the origin and with radius t. Take a net (y α ) α ⊂ bmo c 1,pr (M) such that π((y α ) α ) ⊂ Y ∩ B t (X * ). Hence (y α ) α are bounded in bmo c 1,pr (M). Suppose that, π(y α ), (g e ) e → ξ, (g e ) e , ∀(g e ) e ∈ X,

for some ξ ∈ B t (X * ). We will show that ξ ∈ Y , which will complete the proof. We need two facts. The first one is that bmo c 1,pr (M) is a dual space by Theorem 2.16, so its unit ball is w*-compact. Therefore, the bounded net (y α ) α in bmo c 1,pr (M) admits a w * -cluster point y. Without loss of generality, we assume that (y α ) α converges to y in the w * -topology:

y α , x → y, x , ∀x ∈ h c 1 (M). (1.3.3)
The second fact is that for any (g e ) e ∈ X, the sum e g e is absolutely summable in h c 1 (M). Indeed, by Lemma 3.2 We can now prove Theorem 3.12 in the case of q = ∞. Proof. Let Y ⊥ be the preannihilator of Y in X * : Y ⊥ = {(g e ) e ∈ X : π(y), (g e ) e = 0, ∀y ∈ bmo c 1,pr (M)}.

Then by the bipolar theorem

Y ≃ (X/Y ⊥ ) * .
Using the second fact in the proof of the previous lemma, we get Therefore, combined with Lemma 3.2 and Remark 3.10, the density of h c 1,at∞,pr (M) in h c 1 (M) (due to Lemma 3.5) yields the desired duality identity h c 1,at∞,pr (M) = h c 1 (M). Let us return back to the unsettled case q = ∞ in the proof of Theorem 3.4. Since a fine atom is necessarily a crude atom, we get Then by Theorem 3.12 and Perrin's noncommutative Davis decomposition (see [START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF]), we get the atomic decomposition of h 1 (M) and H 1 (M).

Y ⊥ = {(g e )
h c 1 (M) ⊂ h c 1,at∞ (M), hence h c 1 (M) = h c 1,
Corollary 3.15. We have

H 1 (M) = h 1 (M) = h 1,atq,pr (M),
for any 1 < q ≤ ∞, with equivalent norms.

However, using Corollary 2.18, we can obtain another kind of atomic decomposition for h 1 (M) or H 1 (M), which is exactly the noncommutative analogue of the classical case. Definition 3.16. For 1 < q ≤ ∞, a ∈ L 1 (M) is said to be a (1, q)-atom with respect to (M n ) n≥1 , if there exist n ≥ 1 and a projection e ∈ P(M n ) such that (i) E n (a) = 0;

(ii) r(a) ≤ e or l(a) ≤ e;

(iii) a q ≤ (τ (e))

-1 q ′ for 1 < q ≤ ∞.

Definition 3.17. We define h at 1,q (M) as the Banach space of all x ∈ L 1 (M) which admit a decomposition x = y + k λ k a k , where for each k, a k is a (1, q)-atom or an element in the unit ball of L 1 (M 1 ), λ k ∈ C satisfying k |λ k | < ∞, and where the martingale differences of y satisfy j≥1 dy j 1 < ∞. We equip this space with the norm

x h at 1,q = inf j dy j 1 + k |λ k | ,
where the infimum is taken over all decompositions of x as above.

Lemma 3.18. If a is a (1, q)-atom, then

a h 1 ≤ cq q -1 .
Proof. Without loss of generality, suppose a is a (1, q)-atom with r(a) ≤ e. We apply Corollary 3.18 and the duality (h 1 (M)) * = bmo(M).

a h 1 ≤ c sup x bmo ≤1 τ (x * a) = c sup x bmo ≤1 τ ((x -x n ) * a) = c sup x bmo ≤1 τ (((x -x n )e) * a)
≤ c a q (xx n )e q ′ ≤ cq ′ .

An open question of Junge and Musat

Theorem 3.19. For all 1 < q ≤ ∞, we have

H 1 (M) = h 1 (M) = h at 1,q (M)
with equivalent norms.

By Lemma 3.18, Corollary 2.18 and using arguments similar to those in the proof of Theorem 3.4, we can prove the theorem for the case 1 < q < ∞. For the case q = ∞, we use the argument in Theorem 3.12. Instead of L P 1 (bmo c ) and L P ∞ (h c 1 ), we consider the following two spaces: Then by Lemma 3.18 and Corollary 2.18, we get the announced results. We leave the details to the reader.

L P 1 (L ∞ ) = (g e
Remark 3.20. The part of this paper on the crude versions of the John-Nirenberg inequalities and atomic decomposition can be easily extended to the type III case with minor modifications.

An open question of Junge and Musat

It is an open question asked in [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF] (on page 136) that given 2 < p < ∞, whether there exists a constant c p such that

sup k E k |x -E k-1 x| p 1 p ∞ ≤ c p x BMO ? (1.4.1)
It is easy to see that the answer is negative for matrix-valued functions with irregular filtration. In the following, we show that the answer is negative even for matrix-valued dyadic martingales. Recall that Remark 2.14 already shows that the answer is negative if one considers the column norm • BMO c alone on the right hand side.

Let M and M k be as in Remark 2.14. We consider this special case and show that the best constant c p (n) such that (1.4.1) holds is bigger than c(log(n + 1)) 1/p for all p ≥ 3. Let b be an M n -valued function on T. We need the so-called "sweep" function of b

S(b) = ∞ k=1 |db k | 2 .
Note that it is just the square of the usual square function. Matrix-valued sweep functions have been studied in [START_REF] Blasco | Embeddings between operator-valued dyadic BMO spaces[END_REF], [START_REF] Gillespie | Logarithmic growth for martingale transform[END_REF], [START_REF] Mei | Notes on matrix valued paraproducts[END_REF] etc. It is proved in [START_REF] Mei | Notes on matrix valued paraproducts[END_REF] that the best constant c n such that S(b

) BMO c ≤ c n b 2 ∞ (1.4.2)
is c(log(n + 1)) 2 . A similar result had been proved previously by Blasco and Pott (see [START_REF] Blasco | Embeddings between operator-valued dyadic BMO spaces[END_REF]) by considering b 2 BMO c on the right side of (1.4.2).

Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

Lemma 4.1. Assume f BMO c ≤ c(n) sup k E k |f -E k-1 f | ∞ for any selfadjoint f . Then c(n) ≥ c(log(n + 1)) 2 .
Proof. Under the assumption, we have

S(b) BMO c ≤ c(n) sup m E m |S(b) -E m-1 S(b)|dt ∞ = c(n) sup m E m ∞ k=1 |db k | 2 -E m-1 ∞ k=1 |db k | 2 ∞ = c(n) sup m E m ∞ k=m |db k | 2 -E m-1 ∞ k=m |db k | 2 ∞ . Let x = ∞ k=m |db k | 2 and y = E m-1 ∞ k=m |db k | 2 . By the convexity of | • | 2 , we get x -y 2 2 ≤ |x| 2 + |y| 2 2 ≤ |x| 2 + y 2 ∞ 1 2 ≤ (|x| + y ∞ 1) 2 2 .
Then by Löwner-Heinz's inequality,

x -y 2 ≤ |x| + y ∞ 1 √ 2 .
Thus by the triangle inequality, we have

S(b) BMO c ≤ 2c(n) sup m E m x + y ∞ 1 ∞ = 2c(n) sup m E m |b -E m-1 b| 2 ∞ + 2c(n) E m-1 |b -E m-1 b| 2 ∞ ≤ 2c(n) b 2 BMO c + 2c(n) E m |b -E m-1 b| 2 ∞ ≤ 4c(n) b 2 BMO c .
We then get c(n) ≥ c(log(n + 1)) 2 by (1.4.2).

Lemma 4.2. Let 0 < p < ∞ and E m be the conditional expectation from M onto M m , we have

E m |x| p+1 2 ∞ ≤ E m |x| p 1 2 ∞ E m |x | 1 2 ∞ .
Proof. By Hölder's inequality, we get

E m |x| p+1 2 ∞ = sup a L + 1 (Mm)≤1 τ (E m |x| p+1 2 a) = sup a L + 1 (Mm)≤1 τ (a 1 2 |x| p 2 |x| 1 2 a 1 2 ) ≤ sup a L + 1 (Mm)≤1 (τ (a|x| p )) 1 2 (τ (a|x|)) 1 2 = E m |x| p 1 2 ∞ E m |x | 1 2 ∞ . Theorem 4.3. Suppose sup k E k |f -E k-1 f | p 1/p ∞ ≤ c p (n) f BMO
b 2 BMO = sup m E m |b -E m-1 b| 2 ∞ ≤ sup m E m |b -E m-1 b| p+1 2 4 p+1 ∞ ≤ sup m E m |b -E m-1 b| p 2 p+1 ∞ sup m E m |b -E m-1 b | 2 p+1 ∞ ≤ (c p (n) b BMO ) 2p p+1 sup m E m |b -E m-1 b | 2 p+1 ∞ . Then b BMO ≤ (c p (n)) p sup

Introduction

In this chapter, we exploit Meyer's wavelet methods to the study of the operator-valued Hardy spaces. We are motivated by two rapidly developed fields. The first one is the theory of noncommutative martingales inequalities. This theory had been already initiated in the 1970's. Its modern period of development has begun with Pisier and Xu's seminal paper [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] in which the authors established the noncommutative Burkholder-Gundy inequalities and Fefferman duality theorem between H 1 and BM O. Since then many classical results have been successfully transferred to the noncommutative world (see [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], [START_REF] Junge | Noncommutative Burkholder/Rosenthal inequalities II : applications[END_REF], [START_REF] Mei | Operator valued Hardy spaces[END_REF], [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF]).

In particular, motivated by [START_REF] Junge | H ∞ functional calculus and square functions on noncommutative Lp-spaces[END_REF], Mei [START_REF] Mei | Operator valued Hardy spaces[END_REF] developed the theory of Hardy spaces on R n for operator-valued functions.

Our second motivation is the theory of wavelets founded by Meyer. It is nowadays well known that this theory is important for many domains, in particular in harmonic analysis. For instance, it provides powerful tools to the theory of Calderón-Zygmund singular integral operators. More recently, Meyer's wavelet methods were extended to study more sophistical subjects in harmonic analysis. For example, the authors of [START_REF] Ferguson | A characterization of product BMO by commutators[END_REF] exploited the properties of Meyer's wavelets to give a characterization of product BM O by commutators; [START_REF] Mucalu | Bi-parameter paraproducts[END_REF] deals with the estimates of bi-parameter paraproducts.

It is in this spirit that we wish to understand how useful wavelet methods are for noncommutative analysis. The most natural and possible way would be first to do this in the semi-commutative case. This is exactly the purpose of the present chapter which could be viewed as the first attempt towards the development of wavelet techniques for noncommutative analysis.

A wavelet basis of L 2 (R) is a complete orthonormal system (w I ) I∈D , where D denotes the collection of all dyadic intervals in R, w is a Schwartz function satisfying the properties needed for Meryer's construction in [START_REF] Meyer | Wavelets and Operators[END_REF], and

w I (x) . = 1 |I| 1 2 w x -c I |I| ,
where c I is the center of I. The central facts that we will need about the wavelet basis are the orthogonality between different w I 's, w L 2 (R) = 1 and the regularity of w,

max(|w(x)|, |w ′ (x)|) (1 + |x|) -m , ∀m ≥ 2.
Chapter 2. Wavelet approach to operator-valued Hardy spaces

The analogy between wavelets and dyadic martingales is well known. The key observation is the following parallelism:

|I|=2 -n+1 f, w I w I ∼ df n ,
where df n denotes the n-th dyadic martingale difference of f . As dyadic martingales are much easier to handle, this parallelism explains why wavelet approach to many problems in harmonic analysis is usually simple and efficient. On the other hand, it also indicates that martingale methods may be used to deal with wavelets. With this in mind, we develop the operator-valued Hardy spaces based on the wavelet methods in the way which is well known in the noncommutative martingales case. Then we show that our Hardy and BMO spaces coincide with Mei's. In other words, we provide another approach, which is much simpler than Mei's original one, to recover all the results of [START_REF] Mei | Operator valued Hardy spaces[END_REF].

This chapter is organized as follows. In section 2.1, we will give some preliminaries on noncommutative analysis, the definition of H p (R, M) with 1 ≤ p < ∞ and L q MO(R, M) with 2 < q ≤ ∞ in our setting. In section 2.2, we are concerned with three duality results. The most important one is the noncommutative analogue of the famous Fefferman duality theorem between H c 1 (R, M) and BMO c (R, M). The second one is the duality between H c p (R, M) and L c p ′ MO(R, M) with 1 < p < 2, where we need the noncommutative Doob's inequality, this is why we consider the case 1 < p < 2 independently. The last one is the duality between H c p (R, M) and H c p ′ (R, M) with 1 < p < ∞. As a corollary of the last two results, we identify H c q (R, M) and L c q MO(R, M) with 2 < q < ∞. Section 2.3 deals with the interpolation of our Hardy spaces. In section 2.4, we show that our Hardy spaces coincide with those of [START_REF] Mei | Operator valued Hardy spaces[END_REF]. So, we can give an explicit completely unconditional basis for the space H 1 (R), when H 1 (R) is equipped with an appropriate operator space structure.

We end this introduction by the convention that throughout the chapter the letter c will denote an absolute positive constant, which may vary from lines to lines, and c p a positive constant depending only on p.

Preliminaries

Operator-valued noncommutative L p -spaces

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ and S + M be the set of all positive element x in M with τ (s(x)) < ∞, where s(x) is the smallest projection e such that exe = x. Let S M be the linear span of S + M . Then any x ∈ S M has finite trace, and S M is a w * -dense * -subalgebra of M.

Let 1 ≤ p < ∞. For any x ∈ S M , the operator |x| p belongs to

S + M (|x| = (x * x) 1
2 ). We define

x p = τ (|x| p ) 1 p , ∀x ∈ S M .
One can check that • p is well defined and is a norm on S M . The completion of (S M , • p ) is denoted by L p (M) which is the usual noncommutative L p -space associated with (M, τ ).

For convenience, we usually set L ∞ (M) = M equipped with the operator norm • M . The elements of L p (M, τ ) can be described as closed densely defined operators on H (H being the Hilbert space on which M acts). We refer the reader to [START_REF] Pisier | Handbook of the Geometry of Banach Spaces[END_REF] for more information on noncommutative L p -spaces.

In this chapter, we are concerned with three operator-valued noncommutative L pspaces. The first one is the noncommutative space L p (M; ℓ c

2 ) (resp. L p (M; ℓ r 2 )), which is studied at length in [START_REF] Junge | H ∞ functional calculus and square functions on noncommutative Lp-spaces[END_REF]. For this space, we need the following properties. In the sequel, p ′ will always denote the conjugate index of p.

Lemma 1.1. Let 1 ≤ p < ∞. Then (L p (M; ℓ c 2 )) * = L p ′ (M; ℓ c 2 ).
(2.1.1)

Thus, for f = (f k ) k ∈ L p (M; ℓ c 2 ) and g = (g k ) k ∈ L p ′ (M; ℓ c 2 ), we have |τ ( f, g )| ≤ f Lp(M;ℓ c 2 ) g L p ′ (M;ℓ c 2 )
,

where f, g = k f k g * k . Lemma 1.2. Let 1 ≤ p 0 < p < p 1 ≤ ∞, 0 < θ < 1, 1 p = 1-θ p 0 + θ p 1 . Then [L p 0 (M; ℓ c 2 ), L p 1 (M; ℓ c 2 )] θ = L p (M; ℓ c 2 ). ( 2

.1.2)

A similar equality holds for row spaces.

The second one is the ℓ ∞ -valued noncommutative space L p (M; ℓ ∞ ), which is studied by Pisier [START_REF] Pisier | Non-commutative Vector Valued Lp-Spaces and Completely p-Summing Maps[END_REF] for an injective M and Junge [START_REF] Junge | Doob's Inequality for Non-commutative Martingales[END_REF] for a general M (see also [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF] and [START_REF] Junge | Noncommutative maximal ergodic theorems[END_REF] for more properties). About this one, we need the following property:

Lemma 1.3. Let 1 ≤ p < ∞. Then (L p (M; ℓ 1 )) * = L p ′ (M; ℓ ∞ ). Thus, for x = (x n ) n ∈ L p (M; ℓ 1 ) and y = (y n ) n ∈ L p ′ (M; ℓ ∞ ), we have n≥1 τ (x n y n ) ≤ x Lp(M;ℓ 1 ) y L p ′ (M;ℓ∞) , (2.1.3)
where L p (M; ℓ 1 ) is the space of all sequences x = (x n ) n such that

(x k ) k Lp(N;ℓ 1 ) = sup xn= k a * n,k b n,k n, k |a n,k | 2 1/2 p n, k |b n,k | 2 1/2 p
The third one is L p (M; ℓ c ∞ ) for 2 ≤ p ≤ ∞, which was introduced in [START_REF] Defant | Maximal theorems of Menchoff-Rademacher type in noncommutative L q -spaces[END_REF] and is related with the second one by

(x n ) n Lp(M;ℓ c ∞ ) = (|x n | 2 ) n L p 2 (M;ℓ∞) .
And these are normed spaces by the following characterization

(x n ) n Lp(M;ℓ c ∞ ) = inf xn=yna (y n ) ℓ∞(L∞(M)) a Lp(M) .
We need the interpolation results about these spaces (see [START_REF] Musat | Interpolation Between Non-commutative BMO and Non-commutative Lpspaces[END_REF]):

Lemma 1.4. Let 2 ≤ p 0 < p < p 1 ≤ ∞, 0 < θ < 1, 1 p = 1-θ p 0 + θ p 1 . Then [L p 0 (M; ℓ c ∞ ), L p 1 (M; ℓ c ∞ )] θ = L p (M; ℓ c ∞ ). (2.1.4)

Operator-valued Hardy spaces

In this chapter, for simplicity, we denote L ∞ (R) ⊗M by N . As indicated in the introduction, one can observe that we have the following operator-valued Calderón identity

f (x) = I∈D f, w I w I (x), (2.1.5)
which holds when f ∈ L 2 (N ). As in the classical case, for f ∈ S N , we define the two Littlewood-Paley square functions as

S c (f )(x) = I∈D | f, w I | 2 |I| 1 I (x) 1 2 . (2.1.6) S r (f )(x) = I∈D | f * , w I | 2 |I| 1 I (x) 1 2 . (2.1.7) For 1 ≤ p < ∞, define f H c p = S c (f ) Lp(N) , f H r p = S r (f ) Lp(N)
. These are norms, which can be seen easily from the space L p (N ; ℓ c 2 (D)). So we define the spaces

H c p (R, M) (resp. H r p (R, M)) as the completion of (S N , • H c p (R,M) ) (resp. (S N , • H c p (R,M)
). Now, we define the operator-valued Hardy spaces as follows: for

1 ≤ p < 2, H p (R, M) = H c p (R, M) + H r p (R, M) (2.1.8)
with the norm

f Hp = inf{ g H c p + h H r p : f = g + h, g ∈ H c p , h ∈ H r p } and for 2 ≤ p < ∞, H p (R, M) = H c p (R, M) ∩ H r p (R, M) (2.1.9)
with the norm defined as

f Hp = max{ f H c p , f H r p }. We can identify H c p (R, M) as a subspace of L p (N ; ℓ c 2 (D))
, which is related with the two maps below.

Definition 1.5. (i)

The embedding map Φ is defined from

H c p (R, M) to L p (N ; ℓ c 2 (D)) by Φ(f ) = f, w I |I| 1 2 1 I I∈D . (2.1.10) (ii) The projection map Ψ is defined from L 2 (N ; ℓ c 2 (D)) to H c 2 (R, M) by Ψ((g I )) = I∈D g I |I| 1 2
1 I dy • w I .

(2.1.11)

Operator-valued BMO spaces

For

ϕ ∈ L ∞ (M; L c 2 (R, dx 1+x 2 )), set ϕ BMO c = sup J∈D 1 |J| I⊂J | ϕ, w I | 2 1 2 M
(2.1.12) and

ϕ BMO r = ϕ * BMO c (R,M
) . These are again norms modulo constant functions. Define

BMO c (R, M) = {ϕ ∈ L ∞ (M; L c 2 (R, dx 1 + x 2 )) : ϕ BMO c < ∞} and BMO r (R, M) = {ϕ ∈ L ∞ (M; L r 2 (R, dx 1 + x 2 )) : ϕ BMO r < ∞} Now we define BMO(R, M) = BMO c (R, M) ∩ BMO r (R, M).
As in the martingale case [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], we can also define

L c p MO(R, M) for all 2 < p ≤ ∞. For ϕ ∈ L p (M; L c 2 (R, dx 1+x 2 )), set ϕ L c p MO = ( 1 |I x k | I⊂I x k | ϕ, w I | 2 ) k 1 2 L p 2 (N ;ℓ∞) (2.1.13)
and

ϕ L r p MO = ϕ * L c
p MO , where I x k denotes the unique dyadic interval with length 2 -k+1 that containing x. We will use the convention adopted in [START_REF] Junge | Noncommutative maximal ergodic theorems[END_REF] for the norm in L p 2 (N ; ℓ ∞ ). Thus

( 1 |I x k | I⊂I x k | ϕ, w I | 2 ) k 1 2 L p 2 (N ;ℓ∞) = sup k + 1 |I x k | I⊂I x k | ϕ, w I | 2 1 2 L p 2 (N )
These are norms, which can be seen from the Banach spaces L p (N ⊗B(ℓ 2 (D)); ℓ c ∞ ). Again, we can define

L c p MO(R, M) = {ϕ ∈ L p (M; L c 2 (R, dx 1 + x 2 )) : ϕ L c p MO < ∞} and L r p MO(R, M) = {ϕ ∈ L p (M; L r 2 (R, dx 1 + x 2 )) : ϕ L c r MO < ∞} Define L p MO(R, M) = L c p MO(R, M) ∩ L r p MO(R, M). Note that L c ∞ MO(R, M) = BMO c (R, M).
It is easy to check all the spaces we defined here with respect to the relevant norms are Banach spaces.

Duality

To prove the first two duality results in this section, we need the following noncommutative Doob's inequality from [START_REF] Junge | Doob's Inequality for Non-commutative Martingales[END_REF].

Let (E n ) n be the conditional expectation with respect to a filtration

(N n ) n of N . Lemma 2.1. Let 1 < p ≤ ∞ and f ∈ L p (N ). Then sup n + E n (f ) Lp(N ) ≤ c p f Lp(N ) . (2.2.1)
Theorem 2.2. We have

(H c 1 (R, M)) * = BMO c (R, M) (2.2.2) with equivalent norms. That is, every ϕ ∈ BMO c (R, M) induces a continuous linear functional l ϕ on H c 1 (R, M) by l ϕ (f ) = τ ϕ * f, ∀f ∈ S N . (2.2.3) Conversely, for every l ∈ (H c 1 (R, M)) * , there exists a ϕ ∈ BMO c (R, M) such that l = l ϕ . Moreover, c -1 ϕ BMO c ≤ l ϕ (H c 1 ) * ≤ c ϕ BMO c where c > 0 is a universal constant.
Similarly, the duality holds between H r 1 and BMO r , between H 1 and BMO with equivalent norms.

In order to adapt the arguments in the martingale case, we need to define the truncated square functions for n ∈ Z,

S c,n (f )(x) = n k=-∞ |I|=2 -k+1 | f, w I | 2 |I| 1 I (x) 1 2 .
Proof. Since S N is dense in H c 1 (R, M), by an approximation argument, we only need to prove the inequality

|l ϕ (f )| ≤ c ϕ BMO c f H c 1 for f ∈ S N .
By approximation we may assume that S c,n (f )(x) is invertible in M for all x ∈ R and n ∈ Z. Then we have

|l ϕ (f )| = |τ ϕ * f dx| = n τ |I|=2 -n+1 ϕ, w I * w I |I ′ |=2 -n+1 f, w I ′ w I ′ dx = n τ |I|=2 -n+1 ϕ, w I * |I| 1 2 1 I |I ′ |=2 -n+1 f, w I ′ |I| 1 2 1 I ′ dx ≤ n τ |I|=2 -n+1 f, w I |I| 1 2 1 I 2 S -1 c,n (f ) 1 2 • τ |I|=2 -n+1 ϕ, w I |I| 1 2 1 I 2 S c,n (f ) 1 2 2.2. Duality 67 ≤ n τ |I|=2 -n+1 | f, w I | 2 |I| 1 I S -1 c,n (f ) 1 2 • n τ |I|=2 -n+1 | ϕ, w I | 2 |I| 1 I S c,n (f ) 1 2 = A • B.
In the above estimates, the first equality has used the orthogonality of the w I 's on different levels, the second one the orthogonality of the w I 's on the same level and the disjoint of different dyadic I's on the same level; the first inequality has used the Hölder inequality in Lemma 1.1, and the second one the Cauchy-Schwarz inequality and the disjointness of different I's on the same level. Now, let us estimate A:

A 2 = n τ (S 2 c,n (f ) -S 2 c,n-1 (f ))S -1 c,n (f ) = n τ (S c,n (f ) -S c,n-1 (f ))(1 + S c,n-1 (f )S -1 c,n (f )) ≤ n τ (S c,n (f ) -S c,n-1 (f )) 1 + S c,n-1 (f )S -1 c,n (f ) ∞ ≤ 2 n τ (S c,n (f ) -S c,n-1 (f )) = 2 f H c 1 .
For the first inequality, we have used the Hölder inequality and the positivity of S c,n (f ) -S c,n-1 (f ).

The second term is estimated as follows:

B 2 = k τ (S c,k (f ) -S c,k-1 (f )) n≥k |I|=2 -n+1 | ϕ, w I | 2 |I| 1 I = k τ j (S c,k (f ) -S c,k-1 (f )) I j k n≥k |I|=2 -n+1 | ϕ, w I | 2 |I| 1 I = k τ j I j k (S c,k (f ) -S c,k-1 (f )) 1 |I j k | I⊂I j k | ϕ, w I | 2 ≤ k j τ I j k (S c,k (f ) -S c,k-1 (f )) 1 |I j k | I⊂I j k | ϕ, w I | 2 ∞ ≤ ϕ 2 BMO c k j τ I j k (S c,k (f ) -S c,k-1 (f )) = ϕ 2 BMO c f H c 1
The fist equality has used the Fubini theorem, the second one the fact that S c,k-1 (f ) and S c,k (f ) are constant on the dyadic interval I j k = [j2 -k+1 , (j + 1)2 -k+1 ); the first inequality has used the Hölder inequality and the positivity of S c,n (f ) -S c,n-1 (f ).

Now, let us begin to deal with the other direction, i.e. suppose that l is a bounded linear functional on H c 1 (R, M), we want to find an operator-valued function ϕ in BMO c (R, M), such that l = l ϕ and l ϕ (f ) = τ ϕ * f for f ∈ S N . By the embedding operator Φ in (2.1.10) and by the Hahn-Banach theorem, l extends to a bounded continuous functional on L 1 (N ; ℓ c 2 (D)) of the same norm. Then by the results in Lemma 1.1 there exists g = (g I ) I∈D such that g L∞(N ;ℓ c 2 (D)) = l , and

l(f ) = τ I∈D g * I f, w I |I| 1 2 1 I , ∀f ∈ S N .
Now let ϕ = Ψ(g), where Ψ is defined as (2.1.11). The orthogonality of the w I 's yields

I⊂J | ϕ, w I | 2 M = I⊂J | g I |I| 1 2 1 I | 2 M ≤ I⊂J J |g I | 2 M ≤ |J| I⊂J |g I | 2 L∞(N ) ≤ |J| (g I ) I L∞(N ;ℓ c 2 (D)) ,
where the first inequality has used the Kadison-Schwartz inequality. Also thanks to the orthogonality of the w I 's, we get

l(f ) = τ I∈D g * I f, w I |I| 1 2 1 I = τ ϕ * f
for all f ∈ S N . Therefore, we complete the proof about H c 1 (R, M) and BMO c (R, M). Passing to adjoint, we have the conclusion concerning H r 1 and BMO r . Finally, by the classical fact that the dual of a sum space is the intersection space, we obtain the duality between H 1 and BMO. Theorem 2.3. Let 1 < p < 2. We have

(H c p (R, M)) * = L c p ′ MO(R, M) (2.2.4) with equivalent norms. That is, every ϕ ∈ L c p ′ MO(R, M) induces a continuous linear functional l ϕ on H c p (R, M) by l ϕ (f ) = τ ϕ * f, ∀f ∈ S N . (2.2.5)
Conversely, for every l ∈ (H c p (R, M)) * , there exists an operator-valued function

ϕ ∈ L c p ′ MO(R, M) such that l = l ϕ and c -1 p ϕ L c p ′ MO ≤ l ϕ (H c p ) * ≤ √ 2 ϕ L c p ′ MO
Similarly, the duality holds between H r p and L r p ′ , between H p and L p ′ MO with equivalent norms.

We need the following lemma of [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF]. We write it down for the reader's convenience but without proof. Lemma 2.4. Let s, t be two real numbers such that s < t and 0 ≤ s ≤ 1 ≤ t ≤ 2. Let x, y be two positive operators such that x ≤ y and x t-s , y t-s ∈ L 1 (N). Then τ y -s/2 (y tx t )y -s/2 ≤ 2τ y -(s+1-t)/2 (yx)y -(s+1-t)/2 .

Duality 69

Proof. We need only to prove the first assertion on H c p . Since S N is dense in H c p (R, M), by an approximation argument, we only need to prove the inequality

|l ϕ (f )| ≤ c ϕ L c p ′ MO f H c p for f ∈ S N
. By approximation we may assume that S c,n (f )(x) is invertible in M for all x ∈ R and n ∈ Z. By the similar principle as in the noncommutative martingale case as in [START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF], we have

|l ϕ (f )| = |τ ϕ * f dx| = n τ |I|=2 -n+1 ϕ, w I * w I |I ′ |=2 -n+1 f, w I ′ w I ′ dx = n τ |I|=2 -n+1 ϕ, w I * |I| 1 2 1 I |I ′ |=2 -n+1 f, w I ′ |I| 1 2 1 I ′ dx ≤ n τ |I|=2 -n+1 f, w I |I| 1 2 1 I 2 S p-2 c,n (f ) 1 2 • τ |I|=2 -n+1 ϕ, w I |I| 1 2 1 I 2 S 2-p c,n (f ) 1 2 ≤ n τ |I|=2 -n+1 | f, w I | 2 |I| 1 I S p-2 c,n (f ) 1 2 • n τ |I|=2 -n+1 | ϕ, w I | 2 |I| 1 I S 2-p c,n (f ) 1 2 = A • B.
Now we need the above lemma to estimate the first term. Take s = 2p and t = 2, the lemma yields

A 2 = n τ (S 2 c,n (f ) -S 2 c,n-1 (f ))S p-2 c,n (f ) = n τ S -(2-p)/2 c,n (f )(S 2 c,n (f ) -S 2 c,n-1 (f ))S -(2-p)/2 c,n (f ) ≤ 2 n τ S -(1-p)/2 c,n (f )(S c,n (f ) -S c,n-1 (f ))S -(1-p)/2 c,n (f ) = 2 n τ S c,n (f ) -S c,n-1 (f )S p-1 c,n (f ) ≤ 2 n τ S p c,n (f ) -S p c,n-1 (f ) = 2 f p H c p .
The last inequality has used two elementary inequalities:

0 ≤ S c,n-1 (f ) ≤ S c,n (f ) implies S p-1 c,n-1 (f ) ≤ S p-1 c,n (f ) for 0 < p -1 < 1; and τ (S p c,n-1 (f )) ≤ τ (S 1 2 c,n-1 (f )S p-1 c,n (f )S 1 2
c,n-1 (f )). The second term can be deduced from the nontrivial duality results in Lemma 2.1.3 for 1 < p < ∞ as follows.

Chapter 2. Wavelet approach to operator-valued Hardy spaces

B 2 = k τ S 2-p c,k (f ) -S 2-p c,k-1 (f ) n≥k |I|=2 -n+1 | ϕ, w I | 2 |I| 1 I = k τ j S 2-p c,k (f ) -S 2-p c,k-1 (f ) I j k n≥k |I|=2 -n+1 | ϕ, w I | 2 |I| 1 I = k τ j 1 I j k (x)S 2-p c,k (f )(x) -S 2-p c,k-1 (f )(x) 1 |I j k | I⊂I j k | ϕ, w I | 2 dx = k τ S 2-p c,k (f )(x) -S 2-p c,k-1 (f )(x) 1 |I x k | I⊂I x k | ϕ, w I | 2 dx ≤ k S 2-p c,k (f ) -S 2-p c,k-1 (f ) L (p ′ /2) ′ sup k 1 |I x k | I⊂I x k | ϕ, w I | 2 L p ′ /2 = ϕ 2 L c p ′ MO f 2-p H c p
The fist equality has used the Fubini theorem, the second one the fact that S c,k-1 (f ) and S c,k (f ) are constant on the dyadic intervals with length 2 -k+1 .

For the other direction, we can carry out the proof as that in the case p = 1. Suppose that l is a bounded linear functional on H c p (R, M). By the embedding operator Φ and by Hahn-Banach theorem, and the results in Lemma 1.1, we can find g = (g I ) I∈D such that

g L p ′ (N ;ℓ c 2 (D)) = l and l(f ) = τ I∈D g * I f, w I |I| 1 2 1 I , ∀f ∈ S N .
Now let ϕ = Ψ(g) defined in (2.1.11), the orthogonality of the w I 's yields

sup n + 1 |I x n | I⊂I x n | ϕ, w I | 2 L p ′ /2 (N ) = sup n + 1 |I x n | I⊂I x n | g I |I| 1 2 1 I | 2 L p ′ /2 (N ) ≤ sup n + 1 |I x n | I⊂I x n I x n |g I | 2 L p ′ /2 (N ) ≤ sup n + 1 |I x n | I x n I⊂I x n |g I | 2 L p ′ /2 (N ) ≤ sup n + 1 |I x n | I x n I∈D |g I | 2 L p ′ /2 (N ) ≤ c I∈D |g I | 2 L p ′ /2 (N ) = c (g I ) I L p ′ (N ;ℓ c 2 (D))
, where for the first inequality we have used the Kadison-Schwartz inequality, and the last inequality is (2.2.1). Also due to the orthogonality of the w I 's, we get Instead of using the noncommutative Doob's inequality, we will use the following noncommutative Stein inequality from [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] to prove the duality between the spaces H c p , 1 < p < ∞.

l(f ) = τ I∈D g * I f, w I |I| 1 2 1 I = τ ϕ * f,
Let (E n ) n be the conditional expectation with respect to a filtration (N n ) n of N .

Lemma 2.5. Let 1 < p < ∞ and a = (a n ) n ∈ L p (N ; ℓ c 2 ). Then there exists a constant depending only on p such that

n |E n a n | 2 1 2 Lp(N ) ≤ c p n |a n | 2 1 2 Lp(N )
.

(2.2.6)

Theorem 2.6. For any 1 < p < ∞, we have

(H c p (R, M)) * = H c p ′ (R, M), (2.2.7)
Proof. By a similar reasoning as in the corresponding part of the proof of Theorem 2.2, we can carry out the following calculation,

|l ϕ (f )| = |τ ϕ * f dx| = n τ |I|=2 -n+1 ϕ, w I * w I |I ′ |=2 -n+1 f, w I ′ w I ′ dx = n τ |I|=2 -n+1 ϕ, w I * |I| 1 2 1 I f, w I |I| 1 2 1 I dx ≤ I∈D | f, w I | 2 |I| 1 I 1 2 Lp(R,M) • I∈D | ϕ, w I | 2 |I| 1 I 1 2
L p ′ (R,M) . Now, we turn to the proof of the inverse direction. Take a bounded linear functional l ∈ (H c p (R, M)) * , by the embedding operator Φ and the Hahn-Banach extension theorem, l extends to a bounded linear functional on L p (N ; ℓ c

2 ) with the same norm. Thus by (1.1), there exists a sequence g = (g I ) I such that

g Lq(N ;l c 2 (D)) = l and l(f ) = τ I∈D g * p f, w I |I| 1 2 1 I , ∀f ∈ S N .
Now let ϕ = Ψ(g) where Ψ is defined in (2.1.11), then applying the Stein inequality (2.5) to the conditional expectation

E I (h) = J 1 |J| J h(y)dy • 1 J ,
where J is dyadic interval with the same length as I, we get

ϕ H c p ′ (R,M) = I∈D | 1 |I| I g I dy • 1 I | 2 1 2 L p ′ (N ) ≤ I∈D |E I (g I )| 2 1 2 L p ′ (N ) ≤ c p ′ I∈D |g I | 2 1 2 L p ′ (N ) .
Chapter 2. Wavelet approach to operator-valued Hardy spaces By the orthogonality of the w I 's, we have

l(f ) = τ I∈D g * I f, w I |I| 1 2 1 I = τ ϕ * f, for all f ∈ S N .
From the proof of the second part of Theorem 2.2, Theorem 2.3 and Theorem 2.6, we state the boundedness of Ψ as a corollary.

Corollary 2.7. (i) Let 1 < p < ∞, Ψ is a projection map from L p (N ; ℓ c 2 (D)) onto H c p (R, M)
if we identify the latter as a subspace of the former.

(ii) Let 2 < p ≤ ∞, Ψ is also a bounded map from L p (N ; ℓ c 2 (D)) to L c p MO(R, M
). Theorem 2.3 and Theorem 2.6 immediately imply the following corollary:

Corollary 2.8. Let 2 < p < ∞. Then H c p (R, M) = L c p MO(R, M), ∀2 < p < ∞ with equivalent norms.
However, for the part L c p MO(R, M) ⊂ H c p (R, M), we can give another proof. The idea is essentially similar to that in [START_REF] Mei | Operator valued Hardy spaces[END_REF], the good news is that in our case, the argument seems very elegant. Now we give the detailed proof. proof. Our tent space is defined as

T c p = f = {f I } I ∈ L p (M; ℓ c 2 (D)) : τ I∈D f 2 I |I| 1 I p 2 < ∞
We claim that every ϕ ∈ L c p MO(R, M) induces a bounded linear functional on T c p ′ ,

l ϕ (f ) = τ I∈D ϕ, w I * |I| 1 2 1 I f I |I| 1 2
1 I dx and l ϕ ≤ ϕ L c p MO(R,M) . The proof is just the copy of the proof of the first part in the

last theorem. Now T c p ′ is naturally embedded into L p ′ (N ; ℓ c 2 (D)) by (f I ) I → ( f I |I| 1 2
1 I ) I . So by the Hahn-Banach extension theorem, l ϕ extends to an bounded linear functional on L p ′ (N ; ℓ c 2 (D)) with the same norm. Then by the duality between

(L p ′ (N ; ℓ c 2 (D))) * = L p (N ; ℓ c 2 (D)). there exists a unique h = (h I ) I such that h Lp(N ;ℓ c 2 (D)) ≤ l ϕ and for f = (f I ) I ∈ T c p ′ , l ϕ (f ) = τ I∈D h * I f I |I| 1 2 1 I dx So we get ϕ, w I |I| 1 2 1 I = h I , thus, ϕ H c p = I∈D ϕ, w I * |I| 1 2 1 I 1 2 Lp(N ) = h I Lp(N ;ℓ c 2 (D)) ≤ l ϕ
Chapter 2. Wavelet approach to operator-valued Hardy spaces

On the other hand, by Theorem 3.1, we have

[H c q , H c s ] θ = H c p ,
where(and in the rest of the paper) θ denote the interpolation parameter. Then Wolff's interpolation theorem yields the result. Substep 22: p ≤ 2. Let s > 2, then by Substep 21, we have

[BMO c (R, M), H c p (R, M)] p s = H c s (R, M).
Then together with Lemma 3.1, Wolff's interpolation theorem yields the result.

Step 3: we prove the conclusion for 1 = q < p < ∞. Take s > max(p, 2). By Step 2 and duality [3, Theorem 4.3.1], we get

[H c 1 , H c s ] θ = H c p .
Then together with Step 2, Wolff's interpolation yields the conclusion.

Remark 3.3. If one can directly prove Lemma 3.1 for p 0 = 1, we can prove the above theorem without the help of L c p MO(R, M) for 2 < p < ∞ as carried out in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF], where one needs an auxiliary space. Theorem 3.4. For 1 < p < ∞, we have

H p (R, M) = L p (N )

with equivalent norms.

Proof. There are several ways to prove this result. One can prove it by the strategy in [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] together with Stein's inequality (2.5). Here, we just use the fact that L p (M) with 1 < p < ∞ is a UMD space and our (w I ) I is an complete orthonormal basis. So by Theorem 3.8 in [START_REF] Hytönen | Wavelet expansions for weighted vector-valued BMO functions[END_REF], we have

f Lp(N ) ≃ E I∈D ε I f, w I |I| 1 2 1 I p Lp(N ) 1 p .
Then we complete the proof for 2 ≤ p < ∞ by Khintchine's inequalities. Now, let us prove the case 1 < p < 2. Let f ∈ H p (R, M), then for any ǫ > 0, by the definition of H p (R, M), there exists a decomposition f = f c + f r such that

f c H c p (R,M) + f r H r p (R,M) ≤ f Hp(R,M) + ǫ. Take any g ∈ L p ′ (N )
, by the results for p ′ > 2, the operator-valued Calderón identity (2.1.5) yields

|τ gf * | = | I∈D τ g, w I |I| 1 2 1 I • f * , w I |I| 1 2 1 I | ≤ | I∈D τ g, w I |I| 1 2 1 I • f * c , w I |I| 1 2 1 I | + | I∈D τ g, w I |I| 1 2 1 I • f * r , w I |I| 1 2 1 I | ≤ S c (g) L p ′ (N ) S c (f c ) Lp(N ) + |S r (g) L p ′ (N ) S r (f r ) Lp(N ) 2.3. Interpolation 75 ≤ c p ′ g L p ′ ( f Hp(R,M) + ǫ).
Taking sup and let ǫ → 0, we get the required result. Finally, we prove the inverse inequality. Let f ∈ L p (N ), by duality, we can find two sequences of functions (F c,I ) I ∈ L p (N ; ℓ c 2 (D)) and (F r,I )

I ∈ L p (N ; ℓ r 2 (D)) such that F c,I + F r,I = f, w I |I| -1 2 1 I and (F c,I ) I Lp(N ;ℓ c 2 (D)) + (F r,I ) I Lp(N ;ℓ r 2 (D)) ≤ f Lp(N ) . Let f c = Ψ((F c,I ) I ) and f r = Ψ((F r,I ) I )
, by identity (2.1.5), we have f = f c + f r . On the other hand, by the Stein inequality (2.5), we have

f c H c p (R,M) ≤ (F c,I ) I Lp(N ;ℓ c 2 (D)) and f r H r p (R,M) ≤ (F r,I ) I Lp(N ;ℓ r 2 (D)
) . So we have found the desired decomposition of f . Theorem 3.5. The following results hold with equivalent norms:

(i) Let 1 ≤ q < p < ∞, we have [BMO(R, M), L q (N )] q p = L p (N ). (2.3.7)
(ii) Let 1 < q < p ≤ ∞, we have [H 1 (R, M), L p (N )] p ′ q ′ = L q (N ). (2.3.8)
(iii) Let 1 < p < ∞, we have [BMO(R, M), H 1 (R, M)] 1 p = L p (N ). (2.3.9)
In order to prove this theorem, we need the following result from the theory of interpolation. We formulate it here without proof. Lemma 3.6. Let A 0 , B 0 , A 1 , B 1 be four Banach spaces satisfying the property needed for interpolation. Then

[A 0 + B 0 , A 1 + B 1 ] θ ⊃ [A 0 , A 1 ] θ + [B 0 , B 1 ] θ and [A 0 ∩ B 0 , A 1 ∩ B 1 ] θ ⊂ [A 0 , A 1 ] θ ∩ [B 0 , B 1 ] θ .
Proof. (i) We also exploit the similar but different strategy with that in the proof of Theorem 3.2.

Step 1: we prove the results for 2 ≤ q < p < ∞. By Theorem 3.4, Theorem 3.2 and the lemma, we have

[BMO(R, M), L q (N )] q p ⊂ L p (N ).
The inverse direction follows from L ∞ (N ) ⊂ BMO(R, M),

L p (N ) = [L ∞ (N ), L q (N )] q p ⊂ [BMO(R, M), L q (N )] q p
Step 2: we prove the results for 1 ≤ q < 2 ≤ p < ∞. By Step 1, we have

[BMO(R, M), L 2 (N )] 2 p = L p (N ).
(

) D 2 := {J ∈ D; 2|J| ≤ |I|, 2J ∩ 2I = ∅}, 2
D 3 := {J ∈ D; 2|J| ≤ |I|, 2J ∩ 2I = ∅}. (3)
Let a J = ϕ, ω J , then we have a priori formal series

ϕ 1 (x) = J∈D 1 a J [ω J (x) -ω J (c I )], ϕ i (x) = J∈D i a J ω J (x), i = 2, 3,
where c I is the center of the interval I. Denote ϕ I = ϕ 1 + ϕ 2 + ϕ 3 , by a similar discussion in [START_REF] Hytönen | Wavelet expansions for weighted vector-valued BMO functions[END_REF], we only need to prove:

1 |I| I |ϕ I (x)| 2 dx M < ∞.
By scaling we can assume:

sup I 1 |I| J⊂I |a J | 2 = 1.
Then we have the obvious bound for individual terms a J ≤ |J| 1 2 . Estimates for ϕ 1 :

1 |I| I |ϕ 1 (x)| 2 dx ≤ 1 |I| ( J∈D 1 a J |ω J (x) -ω J (c I )|) 2 dx ≤ c 1 |I| I [ J∈D 1 |J| 1 2 |I||J| -3 2 (1 + dist(I, J) |J| ) -2 ] 2 dx = c[ ∞ j=0 |J|∈(2 j-1 ,2 j ]|I| |I||J| -1 (1 + dist(I, J) |J| ) -2 ] 2 < ∞.
Estimates for ϕ 2 :

1 |I| I |ϕ 2 (x)| 2 dx ≤ 1 |I| I D 2 a J ω J (x) 2 dx ≤ 1 |I| I ( D 2 a J |ω J (x)|) 2 dx ≤ c 1 |I| I [ D 2 |J| 1 2 |J| -1 2 ( dist(I, J) |J| ) -2 ] 2 dx = c[ ∞ j=1 |J|∈(2 -j-1 ,2 -j )|I|,dist(I,J)>2 -1 |I| ( dist(I, J) |J| ) -2 ] 2 < ∞.
Estimates for ϕ 3 :

1 |I| I |ϕ 3 (x)| 2 dx ≤ 1 |I| J∈D 3 |a J | 2 ≤ 1 |I| J⊂4I |a J | 2 < ∞
Hence we deduce that:

I |ϕ I (x)| 2 dx M ≤ c 3 i=1 I |ϕ i (x)| 2 dx M ≤ c|I|
proved that H 1 (R) does not have complete unconditional basis. However, in noncommutative analysis, one can introduce another natural operator space structure on H 1 (R) as follows:

S 1 (H 1 (R)) = H 1 (R, B(ℓ 2 ))
, where S 1 is the trace class on ℓ 2 . Then we have the following result. Note that Ricard [START_REF] Ricard | Décomposition de H 1 , Multiplicateurs de Schur et Espaces d'Operateurs[END_REF] obtained a similar result using Hilbert space techniques.

Corollary 4.3. The complete orthogonal systems (w I ) I∈D of L 2 (R) is a completely unconditional basis for H 1 (R) if we define the operator space structure imposed on H 1 (R) by

S 1 (H 1 (R)) = H 1 (R, B(ℓ 2 )).
Proof. Fix a finite subset I ⊂ D. Let T ε f . = I∈I ε I f, w I w I , where ε I = ±1. By the definition of H c 1 (R, M), the orthogonality of (w I ) I∈D yields immediately that

T ε f H c 1 = I∈I | f, w I | 2 |I| 1 I (x) 1 2 L 1 (N ) ≤ I∈D | f, w I | 2 |I| 1 I (x) 1 2 L 1 (N ) = f H c 1
Similarly, the above inequality holds for H r 1 (R, M). Now, let f ∈ H 1 (R, M), then for any ǫ > 0, there exists a decomposition f = g + h such that

g H c 1 (R,M) + h H r 1 (R,M) ≤ f H 1 (R,M) + ǫ. Therefore T ε f H 1 (R,M) ≤ T ε g H c 1 (R,M) + T ε h H c 1 (R,M)

Introduction

A semicommutative CZO has the formal expression

T f (x) ∼ R n k(x, y)(f (y)) dy,
where the kernel acts linearly on the matrix-valued function f = (f ij ) and satisfies standard size/smoothness Calderón-Zygmund type conditions. This is the operator model for quite a number of problems which have attracted some attention in recent years, including matrixvalued paraproducts, operator-valued Calderón-Zygmund theory or Fourier multipliers on group von Neumann algebras, see [START_REF] Junge | Noncommutative Riesz transforms-A probabilistic approach[END_REF][START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF][START_REF] Mei | Notes on matrix valued paraproducts[END_REF][START_REF] Nazarov | Sharp Estimates in Vector Carleson Imbedding Theorem and for Vector Paraproducts[END_REF][START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF] and the references therein. To be more precise, let B(ℓ 2 ) stand for the matrix algebra of bounded linear operators on ℓ 2 . Consider the algebra formed by essentially bounded functions f : R n → B(ℓ 2 ). Its weak operator closure is a von Neumann algebra A and as such we may construct noncommutative L p spaces over it. Let us highlight a few significant examples:

• Scalar kernels. k(x, y) ∈ C and k(x, y)(f (y)) = k(x, y)f ij (y) .
• Schur product actions. k(x, y) ∈ B(ℓ 2 ) and

k(x, y)(f (y)) = k ij (x, y)f ij (y) .
• Fully noncommutative model. k(x, y) ∈ B(ℓ 2 ) ⊗B(ℓ 2 ) and

k(x, y)(f (y)) = m tr k ′′ m (y)f (y) k ′ m (x) ij .
Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

• Partial traces, noncommuting kernels. k(x, y) ∈ B(ℓ 2 ) and

k(x, y)(f (y)) =    s k is (x, y)f sj (y) , s f is (y)k sj (x, y) .
Scalar kernels required in [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF] a matrix-valued Calderón-Zygmund decomposition in terms of noncommutative martingales and a pseudo-localization principle to control the tails of Tf in the L 2 -metric. Hilbert space valued kernels were later considered in [START_REF] Mei | Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities[END_REF], see also [START_REF] Mei | Operator valued Hardy spaces[END_REF][START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF][START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] for previous related results. The second case refers to the Schur matrix product k(x, y) • f (y), considered for the first time in [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] to analyze cross product extensions of classical CZO's. It is instrumental for Hörmander-Mihlin type theorems on Fourier multipliers associated to discrete groups and for Schur multipliers with a Calderón-Zygmund behavior [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF][START_REF] Junge | Transference in noncommutative harmonic analysis[END_REF]. In the fully noncommutative model, we approximate k(x, y) by a sum of elementary tensors m k ′ m (x) ⊗ k ′′ m (y) and the action is given by

T f (x) ∼ R n (id ⊗ tr) k(x, y) 1 ⊗ f (y) dy.
In this case, we regard the space L p (A) = L p (R n ; L p (B(ℓ 2 ))) as a whole. In other words, the noncommutative nature of L p (A) predominates and the presence of a Euclidean subspace is ignored. That is what happens for purely noncommutative CZO's [START_REF] Junge | Algebraic Calderón-Zygmund theory[END_REF] and justifies the presence of id ⊗ tr, to integrate over the full algebra A and not just over the Euclidean part. The last case refers to matrix-valued kernels acting on f by left/right multiplication, k(x, y)f (y) and f (y)k(x, y). Matrix-valued paraproducts are prominent examples [START_REF] Katz | Matrix valued paraproducts[END_REF][START_REF] Mei | Notes on matrix valued paraproducts[END_REF][START_REF] Mei | An extrapolation of operator-valued dyadic paraproducts[END_REF][START_REF] Nazarov | Sharp Estimates in Vector Carleson Imbedding Theorem and for Vector Paraproducts[END_REF][START_REF] Pott | Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg theorem[END_REF]. This is the only case in which the kernel does not commute with f , since the Schur product is abelian and we find (id⊗tr)[k(x, y)(1⊗f (y))] = (id⊗tr)[(1⊗f (y))k(x, y)] by traciality. Our main goal is to obtain endpoint estimates for CZO's with noncommuting kernels, motivated by a recent estimate from [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] for semicommutative CZO's. If k(x, y) acts linearly on B(ℓ 2 ) and satisfies the Hörmander smoothness condition in the norm of bounded linear maps on B(ℓ 2 ), the content of [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF]Lemma 1.3] can be summarized as follows

• If T is L ∞ (B(ℓ 2 ); L r 2 (R n ))-bounded, then T : L ∞ (A) → BMO r (A), • If T is L ∞ (B(ℓ 2 ); L c 2 (R n ))-bounded, then T : L ∞ (A) → BMO c (A). Here, the L ∞ (L c
2 )-boundedness assumption refers to

R n T f (x) * T f (x) dx 1 2 B(ℓ 2 ) R n f (x) * f (x) dx 1 2 B(ℓ 2 )
, while the column-BMO norm of a matrix-valued function g is given by

sup Q cube - Q g(x) -g Q * g(x) -g Q dx 1 2 B(ℓ 2 )
.

Taking adjoints -so that the * switches everywhere from left to right-we find L ∞ (L r 2 )boundedness and the row-BMO norm. The noncommutative BMO space BMO(A) = BMO r (A) ∩ BMO c (A) was introduced in [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF]. According to [START_REF] Musat | Interpolation Between Non-commutative BMO and Non-commutative Lpspaces[END_REF] it has the expected interpolation behavior in the L p scale. Thus, standard interpolation and duality arguments show that T : L p (A) → L p (A) for 1 < p < ∞ provided the kernel is smooth enough in both variables and T is a normal self-adjoint map satisfying the L ∞ (L r 2 ) and L ∞ (L c 2 ) boundedness assumptions. In other words, the row/column boundedness conditions essentially play the role of the L 2 -boundedness assumption in classical Calderón-Zygmund theory.

Although this certainly works for non-scalar kernels -Schur product actions were used e.g. in [20, Theorem B]-the boundedness assumptions impose nearly commuting conditions on the kernel which are too strong for CZO's associated to noncommuting kernels. Namely, given k : R 2n \ ∆ → B(ℓ 2 ) smooth and given x / ∈ supp R n f , let us set formally the row/column CZO's

T c f (x) = R n k(x, y)f (y) dy and T r f (x) = R n f (y)k(x, y) dy.
It is not difficult to construct noncommuting kernels with i) T r and T c are L 2 (A)-bounded, ii) T r and T c are not L p (A)-bounded for 1 < p = 2 < ∞, see e.g. [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF]Section 6.1] for specific examples. Therefore, the L ∞ (L r

2 ) and L ∞ (L c 2 ) boundedness assumption is in general too restrictive when kernel and function do not commute. Assume for what follows that T r and T c are L 2 (A)-bounded. We are interested in weakened forms of L p boundedness and endpoint estimates for these CZO's. A dyadic noncommuting CZO will be a L 2 (A)-bounded pair (T r , T c ) associated to a noncommuting kernel satisfying one of the following conditions: a) Perfect dyadic kernels k(x, y)k(z, y) B(ℓ 2 ) + k(y, x)k(y, z) B(ℓ 2 ) = 0 whenever x, z ∈ Q and y ∈ R for some disjoint dyadic cubes Q, R.

b) Cancellative Haar shift operators

k(x, y) = Q dyadic R,S dyadic ⊂ Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(S) α Q RS h R (x)h S (y),
for some fixed r, s ∈ Z + where the

α Q RS ∈ B(ℓ 2 ) with α Q RS B(ℓ 2 ) ≤ √ |R||S| |Q| .
Here h Q refers to any of the 2 n -1 Haar functions related to the cube Q.

Perfect dyadic kernels were introduced in [START_REF] Auscher | Carleson measures tress, extrapolation, and T(b) theorems[END_REF] and include Haar multipliers, as well as paraproducts and their adjoints. If J -and J + denote the left/right halves of a dyadic interval in R, the standard model for Haar shifts is the dyadic Hilbert transform with kernel J (h J -(y)h J + (y))h J (x). It appeared after Petermichl's crucial result [START_REF] Petermichl | Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol[END_REF], showing the classical Hilbert transform as a certain average of dyadic Hilbert transforms. Hytönen's representation theorem [START_REF] Hytönen | The sharp weighted bound for general Calderón-Zygmund operators[END_REF] extends this result to arbitrary CZO's. We will write generic noncommuting CZO for L 2 (A)-bounded pairs (T r , T c ) with a noncommuting kernel satisfying the standard smoothness. Our first significant result is the following.

Theorem A. The following inequalities hold

: i) Dyadic noncommuting CZO's. Given f ∈ L 1 (A) inf f =fr+fc T r f r 1,∞ + T c f c 1,∞ f 1 .
Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels ii) Generic noncommuting CZO's.

Given f ∈ H 1 (A) inf f =fr+fc T r f r 1 + T c f c 1 f H 1 (A) .
The noncommutative forms of L 1,∞ and the Hardy space H 1 are well-known in the subject. Nevertheless, they will also be properly defined in the body of the chapter. Our main result is the inequality given in Theorem A i) and their noncommutative generalizations in Theorem C below. As we shall explain in the Appendix, the left/right modular nature of T r /T c is essential for the weak type (1, 1) estimates, see also Remark 2.5. The following result easily follows from Theorem A by interpolation/duality and it can also be derived from [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF]. Nevertheless, it is worth mentioning the L p inequalities that we find.

Theorem B. The following inequalities hold for generic noncommuting CZO's

: i) If 1 < p < 2 and f ∈ L p (A) inf f =fr+fc T r f r p + T c f c p f p . ii) If 2 < p < ∞ and f ∈ L p (A) T r f H r p (A) + T c f H c p (A) f p . iii) Given f ∈ L ∞ (A), we also have T r f BMOr(A) + T c f BMOc(A) f ∞ .
Theorems A and B also hold for other operator-valued functions, replacing B(ℓ 2 ) by any semifinite von Neumann algebra M. Our proof will be written in this framework. Let us now consider a weak- * dense filtration Σ A = (A n ) n≥1 of von Neumann subalgebras of an arbitrary semifinite von Neumann algebra A. In the following result, we will consider two kind of operators in L p (A): a) Noncommuting martingale transforms

M r ξ f = k≥1 ∆ k (f )ξ k-1 and M c ξ f = k≥1 ξ k-1 ∆ k (f ). b) Paraproducts with noncommuting symbol Π r ρ (f ) = k≥1 E k-1 (f )∆ k (ρ) and Π c ρ (f ) = k≥1 ∆ k (ρ)E k-1 (f ).
Here ∆ k denotes the martingale difference operator E k -E k-1 and ξ k ∈ A k is an adapted sequence. Of course, the symbols ξ and ρ do not necessarily commute with the function. Randrianantoanina considered in [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF] noncommutative martingale transforms with commuting coefficients. As for paraproducts with noncommuting symbols, Mei studied the L p -boundedness for p > 2 and regular filtrations in [START_REF] Mei | Notes on matrix valued paraproducts[END_REF] and also analyzed in [START_REF] Mei | An extrapolation of operator-valued dyadic paraproducts[END_REF] the case p < 2 in the dyadic matrix-valued case under a strong BMO condition of the symbol. Our theorem below goes beyond these results, see also [START_REF] Mei | Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities[END_REF] for related results.

Theorem C. Consider the pairs : If Σ A is regular, we obtain weak type (1, 1) inequalities like in Theorem Ai) for martingale transforms and paraproducts . The estimates in Theorems Aii) and B also hold for both families and for arbitrary filtrations Σ A . Moreover, the martingale paraproducts Π r ρ and Π c ρ are L p -bounded for 2 < p < ∞ and L ∞ → BMO.

i) Martingale transforms (M r ξ , M c ξ ), with sup k ξ k M < ∞.
In the case of martingale transforms, there are also examples of noncommuting kernels failing L p -boundedness for p = 2. Hence, our results recover those in [START_REF] Randrianantoanina | Non-commutative martingale transform[END_REF][START_REF] Randrianantoanina | A weak type inequality for non-commutative martingales and applications[END_REF] and are in some sense sharp, providing appropriate substitutes for noncommuting coefficients. Our result for paraproducts goes beyond [START_REF] Mei | Notes on matrix valued paraproducts[END_REF]Theorem 1.2] in two aspects. First, our estimates for p > 2 hold for arbitrary martingales, not just for regular ones. Second, we give a partial answer to Mei's question in [START_REF] Mei | Notes on matrix valued paraproducts[END_REF] after the proof of Theorem 1.2 for the case p < 2 and also for the weak type (1, 1) estimates. The chapter is organized following the order in the Introduction. We include an Appendix at the end with further comments and open problems. Along the chapter we shall assume some familiarity with basic notions from noncommutative integration. The content of [43, Section 1] is enough for our purposes, more can be found in [START_REF] Kadison | Fundamentals of the Theory of Operator Algebras I and II[END_REF][START_REF] Pisier | Handbook of the Geometry of Banach Spaces[END_REF][START_REF] Takesaki | Theory of operator algebras I, II and III[END_REF].

Calderón-Zygmund decomposition

Let M be a semifinite von Neumann algebra equipped with a normal semifinite faithful trace τ . Consider the algebra of essentially bounded functions R n → M equipped with the n.s.f. trace

ϕ(f ) = R n τ (f (x)) dx.
Its weak-operator closure is a von Neumann algebra A. If 1 ≤ p ≤ ∞, we write L p (M) and L p (A) for the noncommutative L p spaces associated to the pairs (M, τ ) and (A, ϕ). The lattices of projections are written M π and A π , while 1 M and 1 A stand for the unit elements. The set of dyadic cubes in R n is denoted by Q and we use Q k for the k-th generation, formed by cubes Q with side length ℓ

(Q) = 2 -k . If f : R n → M is integrable on Q ∈ Q, we set the average f Q = 1 |Q| Q f (y) dy.
Let us write (E k ) k∈Z for the family of conditional expectations associated to the classical dyadic filtration on R n . E k will also stand for the tensor product

E k ⊗ id M acting on A. If 1 ≤ p ≤ ∞ and f ∈ L p (A) E k (f ) = f k = Q∈Q k f Q 1 Q , ∆ k (f ) = df k = Q∈Q k f Q -f Q 1 Q ,
where Q denotes the dyadic parent of Q. We will write (A k ) k∈Z for the filtration A k = E k (A). The noncommutative weak L 1 -space, denoted by L 1,∞ (A), is the set of all ϕmeasurable operators f for which f 1,∞ = sup λ>0 λϕ{|f | > λ} < ∞, see [START_REF] Fack | Generalized s-numbers of τ -measurable operators[END_REF] for a more in depth discussion. In this case, we write ϕ{|f | > λ} to denote the trace of the spectral projection of |f | associated to the interval (λ, ∞). We find this terminology more intuitive, Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels since it is reminiscent of the classical one. The space L 1,∞ (A) is a quasi-Banach space and satisfies the quasi-triangle inequality below which will be used with no further reference

λ ϕ |f 1 + f 2 | > λ ≤ λ ϕ |f 1 | > λ/2 + λ ϕ |f 2 | > λ/2 .
Let us consider the dense subspace

A c,+ = L 1 (A) ∩ f : R n → M f ∈ A + , supp R n f is compact ⊂ L + 1 (A).
Here supp R n means the support of f as a vector-valued function in R n . In other words, we have supp R n f = supp f M . We employ this terminology to distinguish from supp f , the support of f as an operator in A. Any function f ∈ A c,+ gives rise to a martingale (f k ) k∈Z with respect to the dyadic filtration. Moreover, it is clear that given f ∈ A c,+ and λ > 0, there must exist m λ (f ) ∈ Z so that 0 ≤ f k ≤ λ for all k ≤ m λ (f ). The noncommutative analogue of the weak type (1, 1) boundedness of Doob's maximal function is due to Cuculescu. Here we state it in the context of operator-valued functions from A.

Cuculescu's construction [START_REF] Cuculescu | Martingales on von Neumann algebras[END_REF]. Let f ∈ A c,+ and consider the corresponding martingale (f k ) k∈Z relative to the filtration (A k ) k∈Z . Given λ ∈ R + , there exists a decreasing sequence of projections

(q k (λ)) k∈Z in A satisfying i) q k (λ) commutes with q k-1 (λ)f k q k-1 (λ) for each k, ii) q k (λ) belongs to A k for each k and q k (λ)f k q k (λ) ≤ λq k (λ),
iii) The following estimate holds

ϕ 1 A - k∈Z q k (λ) ≤ 1 λ sup k∈Z f k 1 = 1 λ f 1 .
Explicitly, take q k (λ) = χ (0,λ] (q k-1 (λ)f k q k-1 (λ)) with q k (λ) = 1 A for k ≤ m λ (f ).

Given f ∈ A c,+ , consider the Cuculescu's sequence (q k (λ)) k∈Z associated to (f, λ) for a given λ > 0. Since λ will be fixed most of the time, we will shorten the notation by q k and only write q k (λ) when needed. Define the sequence (p k ) k∈Z of disjoint projections p k = q k-1q k , so that

k∈Z p k = 1 A -q with q = k∈Z q k .
Calderón-Zygmund decomposition [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF] . Given f ∈ A c,+ and λ > 0, we may decompose f = g d + g off + b d + b off as the sum of four operators defined in terms of the Cuculescu's construction as follows

g d = qf q + k∈Z p k f k p k , b d = k∈Z p k (f -f k )p k , b off = i =j p i (f -f i∨j )p j , g off = i =j p i f i∨j p j + qf (1 A -q) + (1 A -q)f q.
3.2. Proof of Theorems A and B 87 Moreover, we have the diagonal estimates

qf q + k∈Z p k f k p k 2 2 ≤ 2 n λ f 1 and k∈Z p k (f -f k )p k 1 ≤ 2 f 1 .
The expression below for g off will be also instrumental

g off = ∞ s=1 ∞ k=m λ +1 p k df k+s q k+s-1 + q k+s-1 df k+s p k = ∞ s=1 ∞ k=m λ +1 g k,s = ∞ s=1 g (s) .

Proof of Theorems A and B

The key result of this chapter is Theorem A, since the remaining theorems follow from it or by using analog ideas. We begin with the proof of the weak type estimates for perfect dyadic CZO's and then make the necessary adjustments to make it work for Haar shift operators. The proof of Theorem Aii) will require to recall some recent results on square function and atomic Hardy spaces.

Perfect dyadic CZO's

To the best of our knowledge, the notion of perfect dyadic Calderón-Zygmund operator was rigorously defined for the first time in [START_REF] Auscher | Carleson measures tress, extrapolation, and T(b) theorems[END_REF] by Auscher, Hofmann, Muscalu, Tao and Thiele. Accordingly, we define a perfect dyadic CZO with noncommuting kernel as a pair (T r , T c ) formally given by

T r f (x) ∼ R n k(x, y)f (y) dy, T c f (x) ∼ R n f (y)k(x, y) dy,
with an M-valued kernel satisfying the perfect dyadic conditions k(x, y)k(z, y) M + k(y, x)k(y, z) M = 0 whenever x, z ∈ Q and y ∈ R for some disjoint dyadic cubes Q, R. Alternatively, we may think of perfect dyadic kernels k : R 2n \ ∆ → M as those which are constant on 2ncubes of the form Q × R, where Q, R are distinct dyadic cubes in R n with the same side length and sharing the same dyadic parent. Classical perfect dyadic CZO's include Haar multipliers/martingale transforms and dyadic paraproducts. In other words, operators of the following form

H ξ f (x) = R n Q∈Q ξ( Q) |Q| 1 Q (x)(1 Q -2 -n 1 Q )(y) f (y) dy, Π ρ f (x) = R n Q∈Q 1 |Q| (ρ Q -ρ Q )1 Q (x)2 -n 1 Q (y) f (y) dy,
with sup Q |ξ(Q)| < ∞ and ρ : R n → C in dyadic BMO. Adjoints of paraproducts are also perfect dyadic. In the noncommuting setting, the coefficients ξ(Q) and the symbol ρ become operators in M and an M-valued function respectively which do not commute a priori with f ∈ L p (A). Nevertheless, the perfect dyadic condition for the kernel is still satisfied in these cases.

Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

Proof of Theorem Ai) -Perfect dyadic CZO's. Splitting f as a sum of four positive operators and by density of A c,+ in the positive cone of L 1 (A), we may clearly assume that f ∈ A c,+ . A well-known lack of Cuculescu's construction is that we do not necessarily have q k (λ 1 ) ≤ q k (λ 2 ) for λ 1 ≤ λ 2 . This is typically solved restricting our attention to lacunary values for λ. Define

π j,k = s≥j q k (2 s ) - s≥j-1
q k (2 s ) for j, k ∈ Z.

We have j π j,k SOT = 1 Aψ k , where

ψ k = s∈Z q k (2 s ).
Observe that ψ k df k = df k ψ k = 0 for k ∈ Z. Indeed, we have

ψ k df k A ≤ ψ k f 1 2 k A f k 1 2 A + ψ k f 1 2 k-1 A f k-1 1 2 A = ψ k f k ψ k 1 2 A f k 1 2 A + ψ k f k-1 ψ k 1 2 A f k-1 1 2 A ≤ lim s→-∞ 2 1+ s 2 f 1 2
A .

In particular, we find

f = k (1 A -ψ k-1 )df k (1 A -ψ k-1 ) and set f = f r + f c with f r = k∈Z LT k-1 (df k ) = k∈Z i>j π i,k-1 df k π j,k-1 , f c = k∈Z UT k-1 (df k ) = k∈Z i≤j π i,k-1 df k π j,k-1 .
This is the decomposition we will use for any perfect dyadic CZO. Given such an operator T = (T r , T c ) and λ > 0, the goal is to show that there exists an absolute constant c 0 so that λϕ{|T r f r | > λ} + λϕ{|T c f c | > λ} ≤ c 0 f 1 for any f ∈ A c,+ and any λ > 0. By symmetry in the argument, we will just prove the inequality for T c f c . Moreover, replacing c 0 by 2c 0 we may also assume that λ = 2 ℓ for some ℓ ∈ Z. Having fixed the value of λ, we may consider the Calderón-Zygmund decomposition f = g d + g off + b d + b off and set

g c d = k∈Z UT k-1 ∆ k (g d ) , g c off = k∈Z UT k-1 ∆ k (g off ) , b c d = k∈Z UT k-1 ∆ k (b d ) , b c off = k∈Z UT k-1 ∆ k (b off ) .
By the quasi-triangle inequality it suffices to show

λ ϕ |T c g c d | > λ + ϕ |T c b c d | > λ + ϕ |T c g c off | > λ + ϕ |T c b c off | > λ f 1 .
The first term is first estimated by Chebychev's inequality in A

λϕ |T c g c d | > λ ≤ 1 λ T c g c d 2 2 1 λ g c d 2
2 .

We use that UT k-1 ∆ k (g d ) are in fact martingale differences, so that

1 λ g c d 2 2 = 1 λ k∈Z UT k-1 ∆ k (g d ) 2 2 ≤ 1 λ k∈Z ∆ k (g d ) 2 = 1 λ k∈Z ∆ k (g d ) 2 2 = 1 λ qf q + k∈Z p k f k p k 2 2 ≤ 2 n f 1 .
Indeed, the first inequality above follows from the fact that triangular truncations are contractive in L 2 (A) while the last inequality arise from the diagonal estimates in the noncommutative CZ decomposition stated above. To handle the remaining terms, we introduce the projection

q = s≥ℓ q(2 s ) = s≥ℓ k∈Z q k (2 s ).
According to Cuculescu's construction, we find

ϕ 1 A -q ≤ s≥ℓ ϕ 1 A -q(2 s ) ≤ s≥ℓ 1 2 s f 1 = 2 λ f 1 .
This reduces our problem to show that

λ ϕ T c (b c d ) q > λ + ϕ T c (g c off ) q > λ + ϕ T c (b c off ) q > λ f 1 .
The perfect dyadic nature of T c comes now into scene. Indeed, we claim that the three terms T c (b c d ) q, T c (g c off ) q, T c (b c off ) q vanish whenever T c is perfect dyadic. This will be enough to conclude the proof.

If Q k (x) is the only cube in Q k containing x, we find a.e. x T c (b c d )(x) q(x) = k∈Z T c UT k-1 (∆ k (b d )) (x) q(x) = k∈Z T c UT k-1 (∆ k (b d ))1 Q k-1 (x) (x) q(x) + k∈Z Q∈Q k-1 x / ∈Q Q k(x, y)UT k-1 (∆ k (b d ))(y) dy q(x).
The last term on the right vanishes since the term

UT k-1 (∆ k (b d )) has mean 0 in any Q ∈ Q k-1
, so that we may replace k(x, y) by k(x, y)k(x, c Q ), which is 0 when x / ∈ Q by the perfect dyadic cancellation of the kernel. On the other hand, if we define the projection

q k-1 = s≥ℓ q k-1 (2 s ),
we see that q(x) = q k-1 (x) q(x) = q k-1 (y) q(x) for any y ∈ Q k-1 (x). This gives

T c (b c d )(x) q(x) = k T c UT k-1 (∆ k (b d )) q k-1 1 Q k-1 (x) (x) q(x).
The exact same argument applies for g c off and b c off , so that it suffices to prove

UT k-1 (∆ k (b d )) q k-1 = 0, UT k-1 (∆ k (g off )) q k-1 = 0, UT k-1 (∆ k (b off )) q k-1 = 0,
for all k ∈ Z. In all these cases we will be using the following two key identities

• q k-1 π i,k-1 = π j,k-1 q k-1 = 0 for i, j > ℓ and k ∈ Z, Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

• π i,k-1 p k-s = p k-s π j,k-1 = 0 for s ≥ 1, i, j ≤ ℓ and k ∈ Z.
The proof is straightforward and left to the reader. It only requires to apply the monotonicity properties of s≥j q k (2 s ), which increases in j and decreases in k. If we apply the first identity to UT k-1 (∆ k (γ)) q k-1 for any γ, we get

UT k-1 (∆ k (γ)) q k-1 = i≤j≤ℓ π i,k-1 dγ k π j,k-1 q k-1 .
Therefore, if we know that dγ k = A k +B k where the left support of A k and the right support of B k are dominated by s≥1 p k-s = 1 A -q k-1 , then we deduce that UT k-1 (∆ k (γ)) q k-1 = 0. In other words, it suffices to prove that

q k-1 ∆ k (γ)q k-1 = 0 for γ = b d , g off , b off .
We have

∆ k (b d ) = j ∆ k p j (f -f j )p j = j<k p j (f k -f j )p j - j<k-1 p j (f k-1 -f j )p j = j≤k-1 p j df k p j = (1 A -q k-1 )∆ k (b d )(1 A -q k-1 ).
To calculate the martingale differences for g off , we invoke the formula

g off = ∞ s=1 j∈Z
p j df j+s q j+s-1 + q j+s-1 df j+s p j given in the statement of the Calderón-Zygmund decomposition. Then we find

∆ k (g off ) = ∞ s=1 p k-s df k q k-1 + q k-1 df k p k-s = (1 A -q k-1 )df k q k-1 + q k-1 df k (1 A -q k-1 ).
Finally, it remains to consider the martingale differences of b off

∆ k (b off ) = ∞ s=1 j∈Z ∆ k p j (f -f j+s )p j+s + p j+s (f -f j+s )p j = ∞ s=1 j<k-s p j (f k -f j+s )p j+s + p j+s (f k -f j+s )p j - ∞ s=1 j<k-s-1 p j (f k-1 -f j+s )p j+s + p j+s (f k-1 -f j+s )p j = ∞ s=1 j<k-s p j df k p j+s + ∞ s=1 j<k-s p j+s df k p j = A k + B k . So q k-1 A k = B k q k-1 = 0 and q k-1 ∆ k (γ)q k-1 = 0 for γ = b d , g off , b off as desired.

Haar shift operators

The Haar system has the form

h ε Q (x) = 1 |Q| n j=1 1 I - j (x j ) + ε j 1 I + j (x j )
where

Q = I 1 × I 2 × • • • × I n ∈ Q and ε = (ε 1 , ε 2 , . . . , ε n ) = (1, 1, . . . , 1
) with ε j ∈ ±1. We are using I - j and I + j for the left/right halves of the intervals I j . It yields an orthonormal system in L 2 (R n ) composed of mean zero functions. If we write h Q for any Haar function of the form h ε Q , a noncommuting dyadic shift with complexity (r, s) has the form

X α f (x) = Q∈Q A Q f = Q∈Q R,S dyadic ⊂Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(Q) α Q RS f, h S h R (x), where f, h S = f h S and α Q RS are operators in M satisfying α Q RS M ≤ √ |R||S| |Q| . Lemma 2.1. We have X α f 2 ≤ f 2 .
Proof. The argument is standard, observe that

X α f 2 2 = Q,Q ′ R,R ′ S,S ′ τ f, h S * α Q * RS α Q ′ R ′ S ′ f, h S ′ R n h R (y)h R ′ (y) dy.
The integral on the right imposes R = R ′ , which in turn gives Q = Q ′ since Q is the unique r-th ancestor of R and the same happens for (R ′ , Q ′ ). Once we know that Q = Q ′ , we may write

X α f 2 2 = Q∈Q A Q f 2 2 = Q∈Q A Q S⊂Q ℓ(S)=2 -s ℓ(Q) f, h S h S 2 2 .
It is worth mentioning that the double use above of h S always refers to the same choice of h ε S in both instances. On the other hand, it is easily seen that A Q is a contractive operator on L 2 (A). Indeed, we have

A Q g 2 2 ≤ R n R,S α Q R,S M 1 |S| S g(y) L 2 (M) dy 1 |R| 1 R (x) 2 dx ≤ Q - Q g(y) L 2 (M) dy 2 dx ≤ - Q g 2 L 2 (A) dx = g 2 L 2 (A) .
This yields

X α f 2 2 ≤ Q∈Q S⊂Q ℓ(S)=2 -s ℓ(Q) f, h S h S 2 2 = Q∈Q f, h Q h Q 2 2 = f 2 2 .
The next lemma is crucial to analyze Haar shifts and general Calderón-Zygmund operators with noncommuting kernels. We take here the opportunity to slightly modify the argument in [43, Lemma 4.2], which was not entirely correct. Lemma 2.2. Given s ∈ Z + , there exists ζ ∈ A π such that : Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

i) λϕ(1 A -ζ) ≤ 2 sn f 1 , ii) If Q 0 ∈ Q k 0 and x ∈ Q s 0 , then ζ(x) ≤ q k 0 (y) for all y ∈ Q 0 .
In the second property, we write Q s 0 for the unique s-th dyadic ancestor of Q 0 .

Proof. We have

1 A -q k = j≤k q j-1 -q j = j≤k Q∈Q j ρ Q ⊗ 1 Q = Q∈Q k R⊃Q ρ R ⊗ 1 Q for some family of projections ρ Q ∈ M π . Define ζ = k∈Z ζ k with ζ k = 1 A - j≤k Q∈Q j ρ Q 1 Q s .
It is clear that the ζ k 's are decreasing in k and we find

λϕ(1 A -ζ) = λ lim k→∞ ϕ(1 A -ζ k ) ≤ λ lim k→∞ j≤k Q∈Q j τ (ρ Q )| Q s | = 2 sn lim k→∞ λ j≤k Q∈Q j ϕ(ρ Q ⊗ 1 Q ) = 2 sn λ ϕ 1 A -q = 2 sn λ m≥ℓ ϕ 1 A -q(2 m ) 2 sn f 1 .
To prove the second property, it will be useful to observe that Q 1 Q 2 implies that ρ Q 1 ⊥ ρ Q 2 are orthogonal projections. Indeed, according to the definition of ρ Q above, we have ρ

Q 1 ρ Q 2 1 Q 1 = ( q j 1 -1 -q j 1 )( q j 2 -1 -q j 2 )1 Q 1 = 0 for ℓ(Q 1 ) = 2 -j 1 and ℓ(Q 2 ) = 2 -j 2 . Then, we find ζ(x) ≤ ζ k 0 (x) = 1 M - j≤k 0 Q∈Q j ρ Q 1 Q s (x) ≤ 1 M - R⊃Q 0 ρ R = 1 M - R⊃Q 0 ρ R = 1 A - Q∈Q k 0 R⊃Q ρ R ⊗ 1 Q (y) = q k 0 (y).
Proof of Theorem Ai) -Haar shift operators. As in the perfect dyadic case, we assume f ∈ A c,+ and decompose f = f r + f c in the same way. Once more the argument is row/column symmetric, and we just consider the column part. After fixing λ = 2 ℓ for some ℓ ∈ Z, we construct the corresponding Calderón-Zygmund decomposition for

f c = g c d + g c off + b c d + b c off .
According to Lemma 2.1, we may control the term X α (g c d ) in the usual way. Given γ ∈ {b d , g off , b off }, the other terms can be decomposed as follows

X α (γ c ) = k∈Z X α UT k-1 (∆ k (γ)) 3.2. Proof of Theorems A and B 93 = k∈Z Q∈Q R,S⊂Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(Q) α Q RS R n UT k-1 (∆ k (γ))h S dy h R (x) = k∈Z Q∈Q ℓ(Q)≤2 -k+1 + Q∈Q ℓ(Q)>2 -k+1 ℓ(Q)≤2 s-k+1 + Q∈Q ℓ(Q)>2 s-k+1 = A γ + B γ + C γ .
We claim that C γ = 0. Namely, we have ℓ(S) = 2 -s ℓ(Q) > 2 -k+1 . This means that E k-1 (h S ) = h S since the Haar functions h S are constant in the dyadic children of S, whose length sides are greater or equal than 2 -(k-1) . This yields

R n UT k-1 (∆ k (γ))h S dy = R n E k-1 UT k-1 (∆ k (γ))h S dy = R n E k-1 UT k-1 (∆ k (γ)) h S dy = R n UT k-1 (E k-1 ∆ k (γ))h S dy = 0.
To deal with the remaining terms A γ and B γ , we invoke the identity q k-1 ∆ k (γ)q k-1 = 0 which was already justified in the perfect dyadic case whenever γ

= b d , g off , b off . Namely, since π i,k-1 (1 A -q k-1 ) = (1 A -q k-1 )π j,k-1 = 0 for i, j ≤ ℓ, we find UT k-1 (∆ k (γ)) = i≤j π i,k-1 ∆ k (γ)π j,k-1 = i≤j j>ℓ π i,k-1 ∆ k (γ)π j,k-1 .
Let us now consider the term A γ , we have

λ ϕ |A γ | > λ ≤ λ ϕ 1 A -q + λ ϕ A γ q > λ 2 .
We already know that the first term on the right is dominated by f 1 and

A γ q = k∈Z Q∈Q ℓ(Q)≤2 -k+1 R,S⊂Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(Q) α Q RS R n UT k-1 (∆ k (γ))h S dy h R (x) q(x). Given Q ∈ Q with ℓ(Q) ≤ 2 -k+1 let k Q ≥ k -1 determined by ℓ(Q) = 2 -k Q . It is clear that q(x) = q k Q (x) q(x) = q k Q (y) q(x) = q k-1 (
y) q(x) whenever x, y belong to Q. However, the presence of h R (x), h S (y) implies (unless the corresponding term is 0) that the pair (x, y) ∈ R × S ⊂ Q × Q so that we may write

A γ q = k∈Z Q∈Q ℓ(Q)≤2 -k+1 R,S⊂Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(Q) α Q RS R n UT k-1 (∆ k (γ)) q k-1 h S dy h R (x) q(x).
Therefore, we conclude

UT k-1 (∆ k (γ)) q k-1 = i≤j j>ℓ π i,k-1 ∆ k (γ)π j,k-1 q k-1 = 0
Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels since π j,k-1 q k-1 = 0 when j > ℓ. This shows that A γ q = 0. Let us finally consider the term B γ . We will follow a similar argument with the projection ζ from Lemma 2.2 instead. Namely, we have

λ ϕ |B γ | > λ ≤ λ ϕ 1 A -ζ + λ ϕ B γ ζ > λ 2 .
According to property i) of Lemma 2.2, it suffices to show that B γ ζ = 0. Now we know that ℓ(Q) ≤ 2 s-k+1 , so that k Q ≥ ks -1. Let us now consider the 2 ns dyadic cubes T j having Q as their s-th dyadic ancestor. This gives rise to the identities

ζ(x) = ζ k Q +s (x)ζ(x) = ζ k Q +s (y)ζ(x) = q k Q +s (z)ζ(x) = q k-1 (z)ζ(x)
for (x, y, z) ∈ Q×Q×T j . Indeed, the second identity follows from the fact that

E k Q (ζ k Q +s ) = ζ k Q +s
, the third one from the second property in Lemma 2.2 and the last one from the inequality k Q ≥ ks -1. Hence, given y ∈ S ⊂ Q we pick the unique j for which S = T j and deduce that ζ(x) = q k-1 (y)ζ(x). Then it yields the identity

B γ ζ = k∈Z Q∈Q ℓ(Q)>2 -k+1 ℓ(Q)≤2 s-k+1 R,S⊂Q ℓ(R)=2 -r ℓ(Q) ℓ(S)=2 -s ℓ(Q) α Q RS R n UT k-1 (∆ k (γ)) q k-1 h S dy h R (x) ζ(x).
The integrand UT k-1 (∆ k (γ)) q k-1 vanishes for the same reason as it did above.

Remark 2.3. Our constants are ∼ 2 sn and seem far to be sharp. Unfortunately, the classical argument leading to constants ∼ s encounters a major obstacle due to the presence -in the noncommutative setting-of triangular truncations, which are not bounded in L 1 . The Appendix below contains more details on this topic.

Noncommuting CZO's

The proofs of Theorems Aii), B and C arise from a careful combination of recent results in the theory of noncommutative Hardy spaces. Let us begin introducing Mei's notion [START_REF] Mei | Operator valued Hardy spaces[END_REF] of row and column Hardy spaces for our algebra of operator-valued functions A. In order to distinguish from order Hardy spaces to be introduced below, let us follows Mei's notation and define

H 1 (R n ; M) = H r 1 (R n ; M) + H c 1 (R n ; M) as the space of functions f ∈ L 1 (A) for which we have f H 1 (R n ;M) = inf f =g+h g H r 1 (R n ;M) + h H c 1 (R n ;M) < ∞,
where the row/column norms are given by

g H r 1 (R n ;M) = Γ ∂ g ∂t ∂ g * ∂t + j ∂ g ∂x j ∂ g * ∂x j (x + •, t) dxdt t n-1 1 2 1 , h H c 1 (R n ;M) = Γ ∂ h * ∂t ∂ h ∂t + j ∂ h * ∂x j ∂ h ∂x j (x + •, t) dxdt t n-1 1 2 1 , with Γ = {(x, t) ∈ R n+1 + | |x| < y} and f (x, t) = P t f (x)
for the Poisson semigroup (P t ) t≥0 . In other words, operator-valued forms of Lusin's square function. We say that a

∈ L 1 (M; L c 2 (R n )) is a column atom if there exists a cube Q so that • supp R n a = Q, • Q a(y) dy = 0, • a L 1 (M;L c 2 (R n )) = τ Q |a(y)| 2 dy 1 2 ≤ 1 |Q| .
According to [34, Theorem 2.8], we have

f H c 1 (R n ;M) ∼ inf k |λ k | f = k λ k a k with a k column atoms .
On the other hand, we have already settled a dyadic filtration (A k ) k∈Z for our algebra of operator-valued functions A. Then, we may follow [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] to define the corresponding noncommutative Hardy space H 1 (A) as the completion of the space of finite martingales in L 1 (A) with respect to the norm

f H 1 (A) = inf f =g+h g,h martingales k∈Z dg k dg * k 1 2 1 + k∈Z dh * k dh k 1 2 1 .
In other words,

H 1 (A) = H r 1 (A) + H c 1 (A)
where the spaces on the right are the completions of the spaces of finite L 1 -martingales with respect to the norms in L 1 of the corresponding row/column square functions given above. By the use of a dyadic covering [START_REF] Conde | On dyadic coverings and nondoubling Calderón-Zygmund theory[END_REF][START_REF] Mei | Operator valued Hardy spaces[END_REF], it can be shown that there exists n + 1 dyadic filtrations

Σ j A (0 ≤ j ≤ n) in R n so that H 1 (R n ; M) ≃ n j=0 H 1 (A, Σ j A ),
where the latter spaces are defined as H 1 (A) after replacing the standard filtration Σ 0 A by any other dyadic filtration in our family. Moreover, this isomorphism also holds independently for row/column Hardy spaces.

Proof of Theorem Aii). It suffices to show

T r : H r 1 (A) → L 1 (A) and T c : H c 1 (A) → L 1 (A),
for any generic noncommuting CZO (T r , T c ). Indeed, in that case we decompose f

= f r + f c ∈ H 1 (A), so that f H 1 (A) ∼ f r H r 1 (A) + f c H c 1 (A) and we deduce that T r f r 1 + T c f c 1 f r H r 1 (A) + f c H c 1 (A) ∼ f H 1 (A)
. According to our observation above, H 1 (A) embeds isomorphically into H 1 (R n ; M) by means of a suitably choice of dyadic coverings of R n , and the same holds for row and column spaces isolatedly. Therefore, it also suffices to show that

T r : H r 1 (R n ; M) → L 1 (A), T c : H c 1 (R n ; M) → L 1 (A).
Both estimates are identical, let us prove the column case. According to the atomic decomposition of H c 1 (R n ; M) we just find a uniform upper estimate for the L 1 norm of T c (a) valid for an arbitrary column atom

T c (a) 1 ≤ T c (a)1 2Q 1 + T c (a)1 R n \2Q 1 .
Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

The second term is dominated by

T c (a)1 R n \2Q 1 = τ R n \2Q Q k(x, y)a(y) dy dx ≤ Q R n \2Q k(x, y) -k(x, c Q ) M dx τ |a(y)| dy τ Q |a(y)| dy ≤ |Q|τ Q |a(y)| 2 dy 1 2 ≤ 1,
where the next to last estimate follows from Hansen's inequality or as a consequence of the operator-convexity of the function a → |a| 2 . As for the first term, it suffices to show that

T c : L 1 (M; L c 2 (R n )) → L 1 (M; L c 2 (R n ))
, since then we find again

T c (a)1 2Q 1 = τ 2Q |T c (a)(x)| dx ≤ |2Q| τ 2Q |T c (a)(x)| 2 dx 1 2 |2Q| τ Q |a(x)| 2 dx 1 2 1. The L 1 (M; L c 2 (R n ))-boundedness of T c follows from anti-linear duality T c (f ) L 1 (M;L c 2 (R n )) ≤ sup g L∞(L c 2 ) ≤1 T * c (g) L∞(M;L c 2 (R n )) f L 1 (M;L c 2 (R n )) .
It is easily checked that the adjoint T * c (g) has the form T * c g(x) ∼ R n k(y, x) * g(y) dy when we construct it with respect to the anti-linear bracket f, g = ϕ(f * g). This means in particular that T * c is still an L 2 -bounded column CZO associated to a kernel satisfying Hörmander smoothness. This gives rise to

T * c (g) L∞(M;L c 2 (R n )) = R n |T * c (g)(x)| 2 dx 1 2 M = sup u L 2 (M) ≤1 R n |T * c (g)(x)| 2 u, u L 2 (M) dx 1 2 = sup u L 2 (M) ≤1 R n T * c (gu)(x) 2 L 2 (M) dx 1 2 sup u L 2 (M) ≤1 R n g(x)u 2 L 2 (M) dx 1 2 = R n |g(x)| 2 dx 1 2 M .
The third identity above uses the right M-module nature of column CZO's.

Remark 2.4. Theorem Aii) could have also been derived from the L ∞ → BMO type estimates in [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF]. We have preferred to include this alternative argument using atomic decompositions. Still a third approach is possible using more recent atomic decompositions from [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF][START_REF] Hong | John-Nirenberg inequality and atomic decomposition for noncommutative martingales[END_REF]. This will be needed below for martingale transforms and paraproducts. The proof goes in fact a little further than the statement, since it emphasizes row/column H 1 → L 1 type estimates for T r /T c respectively. This also works for arbitrary semicommutative CZO's under suitable assumptions, see [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] for details.

Remark 2.5. The proof above also shows that L 1 (L † 2 ) and L ∞ (L † 2 ) boundedness of T † for † ∈ {r, c} follow from the corresponding L 2 boundedness of the same operator. As noticed in [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF], this is very specific of CZO's with noncommuting kernels since other semicommutative CZO's fail to satisfy this implication. The key property here is left/right M-modularity, so that uT r (f ) = T r (uf ) and T c (f )u = T c (f u).

This also explains our approach through weak type estimates, see the Appendix.

Row/column L p estimates

Theorem B follows as an easy consequence of Theorem A after applying suitable interpolation/duality results. Thus, we will only outline the definition of the involved spaces and the necessary results to deduce Theorem B from Theorem A. Given 1 < p < ∞, the noncommutative Hardy space H p (A) is defined as

H p (A) = H r p (A) + H c p (A) if 1 < p ≤ 2, H r p (A) ∩ H c p (A) if 2 ≤ p < ∞,
where the corresponding row/column Hardy spaces arise as the completion of the subspace of finite martingales in L p (A) with respect to the norms given by the row and column square functions

f H r p (A) = k∈Z df k df * k 1 2 p , f H c p (A) = k∈Z df * k df k 1 2 p .
Pisier/Xu obtained in [START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF] the noncommutative Burkholder-Gundy inequalities which can be formulated as L p (A) ≃ H p (A) for 1 < p < ∞. On the other hand, we know from [START_REF] Junge | Doob's Inequality for Non-commutative Martingales[END_REF][START_REF] Junge | Non-commutative Burkholder/Rosenthal Inequalities[END_REF] that H † p (A) * ≃ H † p ′ (A) for † ∈ {r, c} and 1 < p < ∞. Regarding interpolation, we know from Musat [?] that

H † p (A) ≃ H † p 0 (A), H † p 1 (A) θ
, where † ∈ {r, c} and 1 p = 1-θ p 0 + θ p 1 . The proof of Theorem B is now straightforward. Proof of Theorem B. We know that

T r : H r 1 (A) → L 1 (A) and T c : H c 1 (A) → L 1 (A). If 1 < p < 2, we find T r : H r p (A) → L p (A) and T c : H c p (A) → L p (A) by interpolation with L 2 (A) = H r 2 (A) = H c 2 (A). Hence, taking a decomposition f = f r + f c satisfying f p ∼ f Hp(A) ∼ f r H r p (A) + f c H c p (A) we get T r f r p + T c f c p f p . Now if 2 < p < ∞, recalling that T *
r , T * c are again row/column CZO's with the same properties, duality gives T r : L p (A) → H r p (A) and T c : L p (A) → H c p (A). This immediately yields the inequality in Theorem Bii). The L ∞ → BMO type estimates were originally proved in [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF], these also follows by duality from Theorem A. Remark 2.6. Alternatively, it can be proved that the row/column L p estimates in Theorem Bi) for 1 < p < 2 also follow by real interpolation from the weak type estimates in Theorem Ai). Moreover, since Mei's spaces H p (R n ; M) also behave well for interpolation and duality, the statement of Theorem B could have been done in terms of these other Hardy spaces.

Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

Proof of Theorem C

In this section we turn our attention to noncommutative martingale transforms and paraproducts. In particular, the former pair (A, ϕ) The martingale coefficients ξ k ∈ A k form an adapted sequence and it is easy to show that L 2 -boundedness of M r ξ and M c ξ holds iff the ξ k 's are uniformly bounded in the norm of A. On the other hand, the classical characterization Π ρ : L 2 → L 2 iff ρ ∈ BMO was disproved by Nazarov, Pisier, Treil and Volberg [START_REF] Nazarov | Sharp Estimates in Vector Carleson Imbedding Theorem and for Vector Paraproducts[END_REF], see also Mei's paper [START_REF] Mei | Notes on matrix valued paraproducts[END_REF]. Hence, the L 2 -boundedness of Π r ρ and Π c ρ will be simply assumed in what follows. Regarding Cuculescu's construction and CZ decomposition, no essential changes are needed. Namely, given f ∈ L + 1 (A) (the former space A c,+ is unnecessary since our filtration starts now at k = 1) and λ ∈ R + , Cuculescu's construction is verbatim the same. The only difference is on the diagonal estimate

qf q + ∞ k=1 p k f k p k 2 2 λ f 1 .
This inequality requires to work with regular filtrations, which are defined through the additional condition E k (f ) ≤ cE k-1 (f ) for some absolute constant c > 0 and every pair (f, k) ∈ A + × Z + . Of course, the reader might think that it is more appropriate to use in this case the noncommutative form of Gundy's decomposition [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF], which does not require any regularity assumption on the martingale. This leads unfortunately to some problems related to our triangular truncations which will be explained in the Appendix below. Thus, arguing as in the proof of Theorem A it suffices to show that q M r ξ (γ r ) = M c ξ (γ c ) q = q Π r ρ (γ r ) = Π c ρ (γ c ) q = 0 for any γ ∈ {g off , b d , b off }. As usual, we just consider the column case by symmetry. Let us begin with martingale transforms. Since γ c = j UT j-1 (∆ j (γ)) and the triangular truncation UT j-1 is built with j-predictable projections, we see that UT j-1 (∆ j (γ)) is a j-th martingale difference, so that

Proof of

∆ k (γ c ) = UT k-1 (∆ k (γ)).
By the proof of Theorem A, we know UT k-1 (∆ k (γ)) q k-1 = 0 and

M c ξ (γ c ) q = ∞ k=1 ξ k-1 ∆ k (γ c ) q = ∞ k=1 ξ k-1 UT k-1 (∆ k (γ)) q k-1 q = 0.
For martingale paraproducts, we observe that E k-1 (γ c ) = j<k UT j-1 (∆ j (γ)) and

Π c ρ (γ c ) q = ∞ k=1 ∆ k (ρ)
j<k UT j-1 (∆ j (γ)) q j-1 q = 0.

Remark 3.1. Adjoints of martingale paraproducts have the form

Π c ρ * f = k≥1 E k-1 ∆ k (ρ * )∆ k (f ) and Π r ρ * f = k≥1 E k-1 ∆ k (f )∆ k (ρ * )
when using the anti-linear duality bracket. It is easy to adapt the argument above for these maps, to obtain weak type inequalities for adjoints of noncommutative paraproducts associated to regular filtrations

inf f =fr+fc Π r ρ * f r 1,∞ + Π c ρ * f c 1,∞ ≤ f 1 .
We defined above the noncommutative Hardy spaces H 1 (A). Alternatively, we may also consider the noncommutative form h 1 (A) = h r 1 (A) + h c 1 (A) + h d 1 (A) of the conditional Hardy space h 1 , where the norms are given by

f h r 1 (A) = k≥1 E k-1 df k df * k 1 2 1 , f h c 1 (A) = k≥1 E k-1 df * k df k 1 2 1 , f h d 1 (A) = k≥1 |df k | 1 = k≥1 df k 1 .
The space h 1 (A) was studied in [START_REF] Junge | Noncommutative Riesz transforms-A probabilistic approach[END_REF][START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF], it was independently proved that

H r 1 (A) ≃ h r 1 (A) + h d 1 (A), H c 1 (A) ≃ h c 1 (A) + h d 1 (A).
In conjunction, these isomorphisms could be regarded as a noncommutative form of Davis' decomposition for martingales. Shortly after, it was found in [START_REF] Bekjan | Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales[END_REF] an atomic decomposition for the spaces h r 1 (A) and h c 1 (A). More precisely, an element a in L 1 (A) ∩ L 2 (A) is called a column atom with respect to the filtration (A k ) k≥1 if there exists k 0 ∈ Z + and a finite projection e ∈ A k 0 such that • a = ae, Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels

• E k 0 (a) = 0,
• a 2 ≤ ϕ(e) -1 2 .

An element a ∈ L 1 (A) is called a catom if it is a column atom or a ∈ A 1 with a 1 ≤ 1.

Row atoms are defined to satisfy a = ea instead and ratoms are defined similarly. We also refer to [START_REF] Hong | John-Nirenberg inequality and atomic decomposition for noncommutative martingales[END_REF] for q-analogs of these notions. In the following result, we collect some norm equivalences coming from atomic decompositions and John-Nirenberg type inequalities. Recall that

f BMOc(A) = sup k≥1 E k (f -f k-1 ) * (f -f k-1 ) 1 2 A , f bmoc(A) = max E 1 (f ) 1 , sup k≥1 E k (f -f k ) * (f -f k ) 1 2 A .
As usual, the corresponding row norms of f arise as the column norms of f * . If we also define f bmo d (A) = sup k df k A , then we can define the spaces BMO(A) and bmo(A) as follows f BMO(A) = max f BMOr(A) , f BMOc(A) , f bmo(A) = max f bmor(A) , f bmoc(A) , f bmo d (A) .

The isomorphism BMO(A) ≃ bmo(A) was independently proved in [START_REF] Junge | Noncommutative Riesz transforms-A probabilistic approach[END_REF][START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF].

Atoms and

β∈A k β 1 ≤1 β(f -f k ) 1 ∨ sup β∈A k β 1 ≤1 (f -f k )β 1 .
The last equivalence is a John-Nirenberg type inequality, which differs from [START_REF] Junge | Non-commutative John-Nirenberg theorem[END_REF].

Proof of Theorem C -H p /L p type inequalities. Let us begin with H 1 → L 1 type inequalities. As pointed out in the proof of Theorem Aii), it suffices to show that T † : H † 1 (A) → L 1 (A) with † ∈ {r, c} and for both martingale transforms and paraproducts. Since we have H † 1 (A) ≃ h † 1 (A) + h d 1 (A), it suffices to show that T † : X → L 1 (A) with X any of the two spaces appearing on the right. Once more, the argument is row/column symmetric and we just consider columns. To see that T c : h c 1 (A) → L 1 (A) we may use the atomic decomposition above, so that it suffices to find a uniform upper bound for T c (a) 1 with a being a catom. If a ∈ A 1 with a 1 ≤ 1, then we see that M c ξ (a) = ξ 0 a 1 and Π c ρ (a) = ba = Π c ρ (u|a|

M c ξ (f ) 1 ≤ k≥1 ξ k ∞ ∆ k (f ) 1 ≤ sup k≥1 ξ k ∞ f h d 1 (A)
As for the paraproduct, we use the John-Nirenberg inequality above

Π c ρ (f ) 1 = k≥1 ∆ k (ρ) j<k ∆ j (f ) 1 = k≥1 ρ -ρ k ∆ k (f ) 1 ρ bmo(A) f h d 1 (A) .
According to [START_REF] Junge | Noncommutative Riesz transforms-A probabilistic approach[END_REF][START_REF] Perrin | A noncommutative Davis' decomposition for martingales[END_REF] and [START_REF] Mei | Notes on matrix valued paraproducts[END_REF][START_REF] Nazarov | Sharp Estimates in Vector Carleson Imbedding Theorem and for Vector Paraproducts[END_REF], we have

ρ bmo(A) ∼ ρ BMO(A) max Π r ρ : L 2 → L 2 , Π c ρ : L 2 → L 2 .
All together gives that M c ξ and Π c ρ take H c 1 (A) into L 1 (A) as we claimed. In fact slight modifications of the given argument yield the same result for [Π c ρ ] * , details are left to he reader. This is all what is needed to produce analog inequalities in this setting to those in Theorems A and B, we just need to follow the arguments verbatim. It remains to show that Π c ρ : L p (A) → L p (A) for p > 2, for which it will be enough to prove L ∞ → BMO boundedness and use interpolation. The L ∞ → BMO c boundedness follows by duality from the H c 1 → L 1 boundedness of [Π c ρ ] * . On the other hand, the L ∞ → BMO r boundedness is very simple

Π c ρ f BMOr(A) = sup k≥1 E k j≥k ∆ j (Π c ρ (f ))∆ j (Π c ρ (f )) * 1 2 A = sup k≥1 E k j≥k ∆ j (ρ)E j-1 (f )E j-1 (f ) * ∆ j (ρ) * 1 2 A ≤ sup k≥1 E k j≥k ∆ j (ρ)∆ j (ρ) * 1 2 A f ∞ ≤ ρ BMOr(A) f ∞ .
Now we majorize ρ BMOr(A) by the L 2 → L 2 norm of Π ρ as we did above.

Observe that we have not needed to assume regularity of our martingale filtration and we find that [Π r ρ ] * , [Π c ρ ] * take H 1 → L 1 and L p → L p for 1 < p < 2 by duality. In some sense, row/column noncommutative paraproducts present a similar behavior as row/column square functions in the noncommutative Burkholder-Gundy and Khintchine inequalities [START_REF] Lust-Piquard | Inégalités de Khintchine dans C p (1 < p < ∞)[END_REF][START_REF] Lust-Piquard | Non-commutative Khintchine and Paley inequalities[END_REF][START_REF] Pisier | Non-commutative Martingale Inequalities[END_REF]. On the other hand, [START_REF] Randrianantoanina | A weak type inequality for non-commutative martingales and applications[END_REF]Theorem 5.7] where p k = q k-1q k . The column decomposition just requires to put p k and q on the right. The advantage of this approach is that we do not find off-diagonal terms which were much harder to deal in [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF]. Moreover, it is not very difficult to show that max g r λ f 1 as expected. Problem 1 would be solved if we knew that

k∈Z p k f r -E k (f r ) 1 + f c -E k (f c ) p k 1 f 1 .
It is perhaps too optimistic to expect that the inequality above holds, since the triangular truncations LT k and UT k appear to be incomparable for different values of k. We wonder whether some noncommutative form of Davis' decomposition in the sense of [START_REF] Randrianantoanina | Conditioned square functions for noncommutative martingales[END_REF] could be useful to modify our row/column decomposition f = f r + f c before performing the CZ decomposition, see also [START_REF] Parcet | Weak type estimates associated to Burkholder's martingale inequality[END_REF] for related ideas. Note that such a row/column CZ decomposition would provide in particular a much simpler proof of the main result in [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF], since off diagonal terms would disappear.

Problem 2. Find a row/column CZ decomposition of f in the line explained above.

A.2. CZO's on general von Neumann algebras

As explained in [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF], a key ingredient for a successful application of the noncommutative CZ decomposition is to use it on M-bimoludar maps. In this paper, our decomposition f = f r + f c has allowed us to make it work for either left or right M-module maps. There are however many other semicommutative CZO's, some of which were mentioned in the Introduction. We know from [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] that a semicommutative CZO satisfying L ∞ (L r 2 ) and L ∞ (L c

2 ) boundedness also satisfies T : L ∞ (A) → BMO(A). Problem 3. Do we have T : L 1 (A) → L 1,∞ (A) under the same assumptions?

According to [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF], solving Problem 3 for CZO's associated to a kernel acting by Schur multiplication would provide weak type (1, 1) inequalities for crossed product extensions of classical CZO's T f (x) ∼ g∈G R n k(x, y)f g (y) ⋊ γ λ(g) dy 3.4. Appendix. Open problems 103 on A = L ∞ (R n ) ⋊ γ G. This in turn is closely related to weak type estimates for Fourier multipliers on group von Neumann algebras, see [START_REF] Junge | Smooth Fourier multipliers on group von Neumann algebras[END_REF] for further details. On the other hand, consider CZO's of the form

T f (x) ∼ R n
(id ⊗ tr) k(x, y) 1 ⊗ f (y) dy.

As we have seen along this paper and in [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF], weak type inequalities require to find vanishing products q 1 (y)q 2 (x) with q 1 , q 2 certain projections in A, see e.g. Lemma 2.2. However, we find T (f q 1 )(x)q 2 (x) ∼ R n (id⊗tr)[k(x, y)(q 2 (x)⊗f q 1 (y))]dy in the model above and no interaction between q 1 and q 2 takes place. This is due to the lack of right M-modularity for T . In fact, solving Problem 3 for this kind of CZO's is very much related to the CZ theory for von Neumann algebras developed in [START_REF] Junge | Algebraic Calderón-Zygmund theory[END_REF]. Namely, the projection in Lemma 2.2 is a dyadic dilation on R n of q not affecting its M 'structure' because the CZO is given as a partial trace on R n , but not on M. The idea in the model above is to dilate both in R n and M. Dilating in M has to do with finding a suitable 'metric' in M to work with. This is what is done in [START_REF] Junge | Algebraic Calderón-Zygmund theory[END_REF] in terms of diffusion semigroups on the given algebra. Under this point of view, we could relate CZO's on (A, ϕ) with those in [START_REF] Parcet | Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory[END_REF] when ϕ is tracial and with the ones considered in this paper when ϕ is a nontracial weight. Problem 4. Prove a CZ decomposition/weak type inequalities for CZO's in [START_REF] Junge | Algebraic Calderón-Zygmund theory[END_REF].

A.3. Gundy's decomposition vs triangular truncations

It is a little bit unsatisfactory to require regular filtrations to provide weak type inequalities for martingales transforms/paraproducts with noncommuting coefficients/symbols. It is well-known that these estimates hold in the classical setting for any filtration by means of Gundy's decomposition. The noncommutative extension of Gundy's decomposition was constructed in [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF]. Given a positive martingale f = (f 1 , f 2 , . . .) in L 1 (A), we may decompose it as f = α + β + γ with dα k = q k df k q k -E k-1 q k df k q k , dβ k = q k-1 df k q k-1q k df k q k + E k-1 q k df k q k , dγ k = df kq k-1 df k q k-1 .

It was proved in [START_REF] Parcet | Gundy's Decomposition for Non-Commutative Martingales and Applications[END_REF] that

max 1 λ α 2 2 , k≥1 dβ k 1 , λϕ k≥1 supp * dγ k f 1 ,
where supp * a = 1 A -q with q the greatest projection satisfying qaq = 0. If we try to prove Theorem C using Gundy's decomposition instead of Calderón-Zygmund decomposition, we will not find any trouble controlling the terms associated to α and γ. The term β presents however a significant difficulty due to the presence of triangular truncations LT k and UT k in L 1 (A). This difficulty can be summarized as follows.

  Notons que bmod (M) = h d ∞ (M). Définissons l'espace bmo(M) = bmo c (M) ∩ bmo r (M) ∩ bmo d (M),muni de la normex bmo = max{ x bmo c , x bmo r , x bmo d }.0.1. Chapitre 1

x

  BMO ≤ B p (x) ≤ cp x BMO , où B p (x) = max{ sup n sup b∈Mn, b p≤1 (xx n-1 )b p , sup n sup b∈Mn, b p≤1, b(xx n-1 ) p }.

Théorème 0. 1 . 1 . 1 px

 111 Pour tout 0 < p < ∞, α -bmo c ≤ x bmo c p,pr ≤ β p x bmo c , où x bmo c p,pr = max E 1 (x) ∞ , sup n sup e∈P(Mn) 1 (τ (e)) 1/p (xx n )e h c p .

  Il a aussi obtenu les résultats d'interpolation désirés et les inégalités de Littlewood-Paley. Mei a adapté l'approche classique dans sa recherche et réduit plusieurs problèmes au cas des martingales. Il a inventé une technique très intéressante et puissante : l'espace BMO sur R (aussi bien dans les cas classique et non commutatif) est l'intersection de deux espaces BMO dyadiques. D'un autre côté, en analyse harmonique classique, il est bien connu que H 1 (R) défini par la fonction carrée et H 1 (R) défini par ondelettes coïncident puisque'ils admettent la même décomposition atomique. Comme expliqué dans le chapitre 1, il est très difficile d'obtenir la décomposition atomique pour les espaces de Hardy non commutatifs par la construction explicite comme dans le cadre classique. On ne peut donc pas comparer ces deux espaces en utilisant la méthode classique. Dans ce chapitre, on définit directement H p (R, M) et BMO(R, M) via ondelette. Les définitions sont similaires à celles du cas des martingales, mais associées à une base d'ondelettes fixée (w I ) I∈D . Par souci de simplicité, on notera N l'espace L ∞ (R) ⊗M dans ce chapitre. Pour f ∈ S N , les fonctions carrées sont définies par

  Fix n, let b ∈ L p (M n ) with b p ≤ 1 and S = {z ∈ C : 0 ≤ Rez ≤ 1}. Then by interpolation between L p spaces L p = (L 2 , L ∞ ) θ , there exists an operator-valued function B which is continuous on S and analytic in the interior of S such that B(θ) = b and

Corollary 2 . 13 . 1 px

 2131 For 0 < p < ∞, we haveα -BMO ≤ B p (x) ≤ β p x BMO , where B p (x) = max{ sup n sup b∈Mn, b p≤1 (xx n-1 )b p , sup n sup b∈Mn, b p≤1 b(xx n-1 ) p }.The constant α p and β p have the same orders as those in Theorem 2.3.Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales

Corollary 2 . 19 .

 219 bmo , where Pb p (x) = max{ sup n (dx n ) n ∞ , sup n sup e∈Mn (xx n ) e (τ (e)) 1/p p , sup n sup e∈Mn e (τ (e)) 1/p (xx n ) p }. The constants α p and β p have the same orders as those in Theorem 2.3. For 0 < p < ∞, we have α -1 p x BMO ≤ PB p (x) ≤ β p x BMO , where PB p (x) = max{ sup n sup e∈Mn (xx n-1 ) e (τ (e)) 1/p p , sup n sup e∈Mn e (τ (e)) 1/p (xx n-1 ) p }. The constant α p and β p have the same orders as those in Theorem 2.3.

c 1 ,Lemma 3 . 13 . 2 e f e h c 1 g e bmo cτ

 13132 pr (M) isomorphically into Z via the following map π(y) = ((yy ne )e) e . Set Y = π(bmo c 1,pr (M)). Chapter 1. John-Nirenberg inequality and atomic decomposition for noncommutative martingales With the notation above we have (i) Z is a subspace of X * with equivalent norms, so is Y . (ii) Y is w*-closed in X * . Proof. (i). Let (f e ) e ∈ Z, for any (g e ) e ∈ X, we have | (f e ) e , (g e ) e | = | e τ ((f e ) * g e )| ≤ √ (e) g e bmo c = √ 2 (f e ) e Z (g e ) e X .

e g e h c 1 ≤

 1 e τ (e) g e bmo c = (g e ) e X . 1.3. Atomic decomposition 55 Therefore, for any (g e ) e ∈ X, we have π(y α ), (g e ) e = e τ (((y α ey α ne )e) * g e ) = τ ((y α ) * e g e ) Combining 1.3.2 and 1.3.3, we deduce that ξ = π(y) ∈ Y , as desired.

  e ∈ X : τ (y * e g e ) = 0, ∀y ∈ bmo c 1,pr (M)} = {(g e ) e ∈ X : e g e = 0 in h c 1 (M)}. Then for (g e ) e ∈ X/Y ⊥ , let g = e∈P g e . Then (g e ) e X/Y ⊥ = inf{ e τ (e) (g ′ e ) e bmo c : g = e g ′ e , (g ′ e ) e ∈ X} = inf{ e |λ e | : g = e λ e a e , (λ e a e ) e ∈ X, a e bmo c ≤ 1 τ (e) } = g h c 1,at∞,pr . Consequently, for any x ∈ h c 1,at∞,pr (M) and any decomposition x = e λ e a e , x h c 1,at∞,pr = (λ e a e ) e X/Y ⊥ = (λ e a e ) e Y * = sup y∈bmo c 1,pr , π(y) Y ≤1 | (λ e a e ), π(y) | ≤ sup y bmo c ≤c |τ (( e λ e a e ) * y)| ≤ c x h c 1 .

  ) e : g e e = g e or eg e = g e , E ne g e = 0, (g e ) e L P 1 (L∞) < ∞ , L P ∞ (L 1 ) = (f e ) e : f e e = f e or ef e = f e , E ne f e = 0, (f e ) e L P ∞ (L 1 ) < ∞ , where (g e ) e L P 1 (L∞) = e τ (e) g e ∞ , (f e ) e L P ∞ (L 1 ) = max sup e 1 τ (e) f e e 1 , sup e 1 τ (e) ef e 1 .

2 p . 1 . 4 .

 214 for some p ≥ 3. Then c p (n) ≥ c(log(n + 1)) An open question of Junge and Musat 59 Proof. Fix a selfadjoint M n -valued function b. By the operator Jensen inequality and Lemma 4.2, for p ≥ 3,

2. 2 .

 2 Duality 71 for all f ∈ S N . Therefore, we complete the proof about H c p (R, M) and L c p ′ MO(R, M).

3. 1 .

 1 Calderón-Zygmund decomposition 85 ii) Martingale paraproducts (Π r ρ , Π c ρ ), with Π r/c ρ L 2 (A)-bounded.

  will refer in what follows to an arbitrary semifinite von Neumann algebra equipped with a normal faithful semifinite trace. Our filtration Σ A = (A k ) k≥1 will be any increasing family of von Neumann subalgebras, whose union is weak- * dense in A. The operators E k and ∆ k still denote the corresponding conditional expectations and martingale difference operators. As mentioned in the Introduction, we will deal with a) Noncommuting martingale transformsM r ξ f = k≥1 ∆ k (f )ξ k-1 and M c ξ f = k≥1 ξ k-1 ∆ k (f ). b) Paraproducts with noncommuting symbol Π r ρ (f ) = k≥1 E k-1 (f )∆ k (ρ) and Π c ρ (f ) = k≥1 ∆ k (ρ)E k-1 (f ).

2 2 , g c d 2 2 ≤ g d 2 2 = qf q + k≥1 p k f k p k 2 2 λ f 1 .

 22221 Theorem C -Weak type inequalities. The argument is essentially the same as in the perfect dyadic case. Given f ∈ L + 1 (A), we construct the same decomposition f = f r + f c via the projections π j,k and fix λ = 2 ℓ for some ℓ ∈ Z. A further CZ decomposition gives f c = g c d + g c off + b c d + b c off as usual. According to our regularity assumption, we still have max g r d

John-Nirenberg inequality [ 2 , 12 ]. We have f h r 1 ∼ inf k |λ k | f = k λ k a k and a k r -atom , f h c 1 ∼

 21211 inf k |λ k | f = k λ k a k and a k catom , f bmo(A) ∼ sup k≥1 df k ∞ ∨ sup

1 2 for∆

 12 a = u|a|.In particular, M c ξ (a)1 + Π c ρ (a) 1 a 1 ≤ 1.If a is a column atom, we findM c ξ (a) = k>k 0 ξ k-1 ∆ k (a) = k>k 0 ξ k-1 ∆ k (a)e = M c ξ (a)e, k (ρ)E k-1 (a) = k>k 0 +1 ∆ k (ρ)E k-1 (a)e = Π c ρ (a)e.This gives rise to T c (a) 1 = T c (a)e 1 ≤ T c (a) 2 e 2 a 2 e 2 ≤ 1 for both martingale transforms and paraproducts. We have already justified the h c 1 → L 1 boundedness. Let us now look at h d

1

 1

Problem 1 .

 1 yields L log L → L 1 type estimates for a finite von Neumann algebra A with (T r , T c ) a martingale transform/paraproduct with noncommuting coefficients/symbolinf f =fr+fc T r f r 1 + T c f c 1 f L log L(A) .Chapter 3. Calderón-Zygmund operators associated to matrix-valued kernels3.4 Appendix. Open problems A.1. CZO's with noncommuting kernelsOur proof of Theorem Ai) is not entirely satisfactory, since it does not include arbitrary CZO's with noncommuting kernels. In the general case, we can not expect to annihilate the terms associated to g off , b d , b off . If the reader considers the simplest term b d , a difficulty with triangular truncations in L 1 will be immediately recognized. In fact, our proof for Haar shifts operators does not provide sharp constants for the same reason. Extend Theorem Ai) to arbitrary CZO's with noncommuting kernels.Here is a possible alternative argument. Once we have f = f r + f c , the same decomposition constructed in the proof of the perfect dyadic case, we could consider a left CZ decomposition for f r and a right CZ decomposition for f c as follows. Given λ ∈ R + we let f r = g r + b r and f c = g c + b c withg r = q f r + k∈Z p k E k (f r ) and b r = k∈Z p k f r -E k (f r ) ,

α k β k 1 ?

 1 Consider a family Tr k of upper triangular truncations and assume that(α k , β k ) ∈ L ∞ (A) × L 1 (A), do we have k≥1 α k Tr k (β k ) 1 k≥1Or at leastk≥1 α k Tr k (β k ) 1 sup k≥1 α k ∞ k≥1 β k 1 ?

  pour le semi-groupe de Poisson (P t ) t≥0 . Ensuite, H r p (R, M) et H p (R, M) sont définis d'une façon similaire aux espaces analogues dans le cas des martingales. Le résultat remarquable prouvé par Mei est que H c 1 (R, M) est le prédual de l'espace BMO apparu dans l'analyse harmonique à valeurs matricielles noté BM O c (R, M). Cet espace est constitué de tous les ϕ

  1 fonctions de Haar relativement au cube Q. On écrit OCZ générique non commutant pour des paires (T r , T c ) qui sont bornées dans L 2 (A) et dont les noyaux ne commutent pas avec les fonctions et vérifiant les conditions de finesse standard. Notre premier résultat est le suivant.

	Theorem 0.3.1. Les inégalités suivantes sont vraies :

  Si Σ A est régulière, on obtient les inégalités de type faible (1, 1) comme dans le Théorème 0.3.1i) pour les tranformées et paraproduits de martingales. Les estimations dans les Théorèmes 0.3.1ii) et 0.3.2 sont aussi vraies pour les deux familles et les Σ A arbitraires. De plus, les paraproduits de martingales Π r ρ et Π c ρ sont bornés dans L p pour 2 < p < ∞ et L ∞ → BMO.

			Introduction
	ii) Paraproduits de Martingales (Π r ρ , Π c ρ ), avec Π	r/c ρ	L 2 (A)-borné.

une séquence adaptée. Bien sûr, les symboles ξ et ρ ne commutent pas nécessairement avec les fonctions. Theorem 0.3.3. Considérons les paires : i) Transformées de martingales (M r ξ , M c ξ ), avec sup k ξ k M < ∞.

  1/p h c p .

	Similarly,
	bmo r p,pr (M) = {x : x * ∈ bmo c p,pr (M)} with x bmo r p,pr = x *	bmo c p,pr .
	Finally,
	bmo

p,pr (M) = bmo c p,pr (M) ∩ bmo r p,pr (M) ∩ bmo d (M) equipped with

x bmop,pr = max{ x bmo c p,pr , x bmo r p,pr , x bmo d }.

  at∞ (M) with equivalent norms due to Lemma 3.2. Thus Theorem 3.4 is completely proved.

	Chapter 1. John-Nirenberg inequality and atomic decomposition for
			noncommutative martingales
	Definition 3.14. We define
	h 1,atq,pr (M) = h c 1,atq,pr (M) + h r 1,atq,pr (M) + h d 1 (M)
	equipped with the sum norm
	x h 1,atq,pr =	inf x=xc+xr+x d	{ x c h c 1,atq,pr + x r h r 1,atq,pr + x d h d 1	}.

Remerciements

Interpolation

This section is devoted to the interpolation of our wavelet Hardy spaces. The interpolation results below will be needed in the next section to compare our Hardy spaces with those of Mei.

Lemma 3.1. Let 1 < p 0 < p < p 1 < ∞, we have

with equivalent norms, where θ satisfies 1 p = 1-θ p 0 + θ p 1 .

Proof. The embedding map Φ yields

On the other hand, it is the boundedness of the projection map Ψ from L p (N; ℓ c 2 (D)) to H c p (R, M) stated in Corollary 2.7 that yields the inverse direction.

Theorem 3.2. Let 1 ≤ q < p < ∞, we have

with equivalent norms.

Proof. We will prove the theorem by a general strategy as appeared in [START_REF] Musat | Interpolation Between Non-commutative BMO and Non-commutative Lpspaces[END_REF].

Step 1: We prove the conclusion for 2 < q < p < ∞:

The identity can be seen easily from the following two inclusions. On one hand, the operator Φ which in (2.1.10), together with (2.1.2) yields

Then by duality and Corollary 2.8, we have

On the other hand, the operator T identifies the space L c p MO(R, M) as a subspace of

together with Lemma 1.4 yields

Step 2: we prove the conclusion for 1 < q < p < ∞. This step can be divided into two substeps.

Substep 21: p > 2. Let p < s < ∞. By Step 1, we have

Chapter 2. Wavelet approach to operator-valued Hardy spaces

Wolff's interpolation yields the conclusion.

Step 3: we prove the results for 1 ≤ q < p < 2. By Step 2, we have

Together with

Wolff's interpolation yields the conclusion.

(ii) The results for 1 < q < p < ∞ can be immediately proved by duality and the partial results in (i). For p = ∞, take q < s < ∞, then by Wolff's argument, we get the conclusion.

(iii) First, we prove conclusion for p < 2. Then by (i) and (ii), we have

. Therefore, we end with Wolff's argument. Second, the proof for p > 2 is the same. At last, when p = 2, we can take s > 2, by the results for p = 2 and reiteration theorem in [3, Theorem 4.6.1], we get

Comparison with Mei's results

We denote the column Hardy space defined in [START_REF] Mei | Operator valued Hardy spaces[END_REF] through operator-valued Lusin square function by H c p (R, M) and the column bounded mean oscillation space appeared in the matrix-valued harmonic analysis by BM O c (R, M) (see e.g. [START_REF] Mei | Operator valued Hardy spaces[END_REF]). We have the following result.

Theorem 4.1. We have

with equivalent norms. Similar results holds for the row spaces. Consequently, BMO(R, M) = BMO(R, M) with equivalent norms.

The theorem can be easily seen from the corresponding BM O(R, H)-spaces. However, we can exploit the idea of [START_REF] Hytönen | Wavelet expansions for weighted vector-valued BMO functions[END_REF] to prove our BMO c (R, M) also coincide with that defined by the mean oscillation BM O(R, H).

As in the beginning of the proof of Theorem 1.2 in [START_REF] Hytönen | Wavelet expansions for weighted vector-valued BMO functions[END_REF], fix a finite interval I ⊂ R, and consider the collections of dyadic intervals

Chapter 2. Wavelet approach to operator-valued Hardy spaces Now we turn to the proof of inverse direction

The proof is very similar to that of Lemma 4.1 in Mei's work [START_REF] Mei | Operator valued Hardy spaces[END_REF]. For any dyadic interval

Estimates for ϕ 1 :

Estimates for ϕ 2 :

Similar results hold for H r p and H r p , and H p and H p .

also has the following characterization:

where H is the Hilbert transform. For any f ∈ H 1 (R),

Thus H 1 (R) can be viewed as a subspace of L 1 (R) ⊕ 1 L 1 (R). The latter direct sum has its natural operator structure as an L 1 space. This induces an operator space structure on H 1 (R). Although (w I ) I∈D is a unconditional basis of H 1 (R), Ricard [START_REF] Ricard | L'espace H 1 n'a pas de base complètement inconditionnelle[END_REF] (see also [START_REF] Ricard | Décomposition de H 1 , Multiplicateurs de Schur et Espaces d'Operateurs[END_REF])