Benaych-Georges Florent
email: florent.benaych-georges@parisdescartes.fr.

GUE MINORS, MAXIMAL BROWNIAN FUNCTIONALS AND LONGEST INCREASING SUBSEQUENCES IN RANDOM WORDS

Keywords: 2000 Mathematics Subject Classification. 15A52;60F05 Random matrices, Brownian motion, random words, longest increasing subsequences, RSK correspondence

We present equalities in law between the spectra of the principal minors of a GUE matrix and some maximal functionals of independent Brownian motions. In turn, these results allow to recover the limiting shape (properly centered and scaled) of the RSK Young diagrams associated with a random word as a function of the spectra of these minors. Since the length of the top row of the diagrams is the length of the longest increasing subsequence of the random word, the corresponding limiting law also follows.

Introduction

It is by now well known that there exist strong and interesting connections between directed percolation and random matrices. The precise results we have in mind have their origins in the identity in law, due to Baryshnikov [START_REF] Baryshnikov | GUEs and queues[END_REF] and Gravner, Tracy and Widom [START_REF] Gravner | Limit theorems for height fluctuations in a class of discrete space and time growth models[END_REF], between the maximal eigenvalue of an M × M element of the GUE and a certain maximal functional of standard M-dimensional Brownian motion originating in queuing theory, with Glynn and Whitt [START_REF] Glynn | Whitt Departures form many queues in series[END_REF]. This first result has seen many extensions and complements. For example, Bougerol and Jeulin [START_REF] Bougerol | Jeulin Paths in Weyl chambers and random matrices Probab[END_REF] as well as O'Connell and Yor [START_REF] O'connell | Yor A representation for non-colliding random walks Electron[END_REF] obtained identities in law between (different) multivariate Brownian functionals and the spectrum of the GUE whose equivalence is shown in Biane, Bougerol and O'Connell [START_REF] Biane | Connell Littelmann paths and Brownian paths[END_REF]. Various related representations have also been put forward and studied for instance in Doumerc [START_REF] Doumerc | A note on representations of classical Gaussian matrices Sémimaire de Probabilités XXXVII[END_REF], Johansson [START_REF] Johansson | Shape fluctuations and random matrices[END_REF][START_REF] Johansson | Non-intersecting paths, random tilings and random matrices[END_REF], O'Connell [START_REF] O'connell | A path transformation for random walks and the Robinson-Schensted correspondence Trans[END_REF] to name but a few authors and pieces of work.

Our interest in such representations comes from the identification, first obtained by Kerov [START_REF] Kerov | Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis[END_REF], of the limiting length (properly centered and scaled) of the longest increasing subsequence of a random word, as the maximal eigenvalue of a certain random matrix. For example, in the case of a word with i.i.d. uniformly distributed letters in an alphabet of size M, the limiting law is the maximal eigenvalue of the M × M traceless GUE. Moreover, [START_REF] Kerov | Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis[END_REF] showed that the whole normalized limiting shape of the RSK Young diagrams associated with the random word, is the spectrum of the M × M traceless GUE. Since the length of the top row of the diagrams is the length of the longest increasing subsequence of the random word, the maximal eigenvalue result is recovered. (The asymptotic length result was rediscovered by Tracy and Widom [START_REF] Tracy | On the distribution of the lengths of the longest increasing monotone subsequences in random words[END_REF] and the asymptotic shape one by Johansson [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF], who positively answered a conjecture of Tracy and Widom. Extensions to non-uniform letters were also obtained by Its, Tracy and Widom [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF][START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems[END_REF].)

Limiting laws expressed in terms of maximal Brownian functionals are also obtained in [START_REF] Houdré | On the longest increasing subsequence for finite and countable alphabets[END_REF]. These last representations involve dependent Brownian motions and do not clearly recover the results of [START_REF] Tracy | On the distribution of the lengths of the longest increasing monotone subsequences in random words[END_REF] or [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF][START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems[END_REF], which themselves are mainly derived by analytical techniques. To resolve this issue, we provide below an extension of Baryshnikov's result [START_REF] Baryshnikov | GUEs and queues[END_REF] on the identification of the multivariate law of the maximal eigenvalues of the principal minors of a GUE matrix, with some maximal functionals of a standard multidimensional Brownian motion. This allows us to circumvent the analytical approach and provides a mixed combinatorial/probabilistic methodology for the solutions of these finite alphabet longest increasing subsequence problems. Our hope is that Theorem 1, below, will also be helpful to fully identify eigenvalues of random matrices as the limiting laws in the corresponding Markov random word problems (see Conjecture 7 in Kuperberg [START_REF] Kuperberg | Random words, quantum statistics, central limits, random matrices[END_REF]). In the Markovian setting, the analytical methodology is lacking, in contrast to the probabilistic one, and to date the limiting laws are mainly only expressed as Brownian functionals (see [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF]). Indeed, the multivariate functional appearing in Theorem 1 is exactly the one giving the limiting law of the shape of the RSK image of a Markov random word in [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF], the only difference being that the Brownian motions in [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF] are correlated, say, with covariance Σ. This correlation issue in maximal functionals often amounts to adding a condition on the trace of the random matrix (as in [START_REF] Tracy | On the distribution of the lengths of the longest increasing monotone subsequences in random words[END_REF][START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF][START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF][START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems[END_REF]). However, for general Markov random words the full identification of these functionals via random matrices remains open. For Markov random words with, e.g., cyclic transition matrix, the length of the longest increasing subsequence will be asymptotically identified as the eigenvalue of some random matrices once one obtains a more general version of Theorem 1 below, with correlated Brownian motions. Our intuition is that to get such a generalization, one needs to consider the principal minors of more general random matrices, namely Gaussian Hermitian matrices with as diagonal, and up to scaling factor, a Gaussian vector with covariance Σ. Besides providing the final touch to an essentially probabilistic proof of the random word asymptotics problem, our results also allow us to shed new lights on the queuing framework by providing, for example, joint limiting laws involving departing times and service times of individual customers.

Statements and proofs of the results

Throughout, fix a positive integer M and consider:

• an M × M GUE matrix H = [h ij ], i.e.
, a standard Gaussian variable on the space of M × M Hermitian matrices endowed with the Euclidean scalar product given by X • Y = Tr XY ,

• an M-dimensional standard Brownian motion B = (B k (t)) t∈[0,1],k=1,...,M .
For each k = 1, . . . , M, denote by

(1)

µ k 1 ≥ • • • ≥ µ k k ,
0 dB π(t) (t) = M j=1 (B j (t j ) -B j (t j-1 )), where 0 = t 0 ≤ t 1 ≤ • • • ≤ t M = 1 are such that π(•) = M -1 j=1 j × ½ [t j-1 ,t j ) (•) + M × ½ [t M -1 ,t M ] (•).
To complete our notations, for π 1 , π 2 ∈ P, we write π 1 < π 2 whenever π 1 (t) < π 2 (t) for all t ∈ [0, 1]. Let us now state our first result which, in particular, when ℓ = 1 below, identifies the joint law of the maximal eigenvalue of the principal minors of H and therefore extends Theorem 0.7 of [START_REF] Baryshnikov | GUEs and queues[END_REF].

Theorem 1. The following equality in law holds true:

ℓ i=1 µ k i 1≤ℓ≤k≤M law = sup ℓ i=1 ∆ π i (B) ; π 1 , . . . , π ℓ ∈ P, π 1 < • • • < π ℓ ≤ k 1≤ℓ≤k≤M .
This theorem has also a process version where the matrix H is replaced by an Hermitian Brownian motion, where and B is taken up to time t and not to time 1.

In the forthcoming proof, and throughout, =⇒ indicates convergence in distribution.

Proof. Let w N,M := [w ij ] (i = 1, . . . , N, j = 1, . . . , M) be an array of i.i.d. geometric random variables with parameter q ∈ (0, 1), i.e., with law k≥0 q k (1 -q)δ k . Such variables have mean e := q/(1 -q) and variance v := q/(1 -q) 2 .

Applying the RSK correspondence to w N,M (see e.g., [START_REF] Van Moerbeke | Random and Integrable Models in Mathematics and Physics in Random Matrices, Random Processes and Integrable Systems[END_REF] for an introduction to the RSK correspondence applied to arrays of integers) gives a pair (P, Q) of Young diagrams with the same shape. Let us denote the shape of these Young diagrams by

λ M 1 ≥ • • • ≥ λ M M ,
where the exponent M is here to emphasize the dependence on the dimension M of the GUE matrix H (the dependence on N is implicit). In the same way, one can of course define, for each k = 1, . . . , M, the shape

λ k 1 ≥ • • • ≥ λ k k ,
of the Young diagrams obtained by applying the RSK correspondence to the array w N,k , which is the array w N,M where all but the first k columns have been removed. Note that

(2)       λ 1 1 λ 2 1 λ 2 2 λ 3 1 λ 3 2 λ 3 3 . . . . . . . . . λ M 1 . . . . . . λ M M      
, is a Gelfand-Tsetlin pattern, i.e., satisfies the interlacing inequalities

λ k i ≥ λ k-1 i ≥ λ k i+1 (1 ≤ i < k ≤ M).
This can be seen from the fact that if (P k , Q k ) denotes the pair associated to w N,k by the RSK correspondence, then P k-1 can be deduced from P k by removing all the boxes filled with the number k.

Let us now define the random variables (3)

ξ k i := λ k i -eN √ vN .
We then have the following lemma.

Lemma 2. As N tends to infinity,

      ξ 1 1 ξ 2 1 ξ 2 2 ξ 3 1 ξ 3 2 ξ 3 3 . . . . . . . . . ξ M 1 . . . . . . ξ M M       =⇒       µ 1 1 µ 2 1 µ 2 2 µ 3 1 µ 3 2 µ 3 3 . . . . . . . . . µ M 1 . . . . . . µ M M       ,
where the µ k i s are the ones introduced in (1).

Proof. This lemma is stated as Proposition 3.12 in [START_REF] Johansson | Nordenstam Eigenvalues of GUE minors[END_REF] (with slightly different notation). However, for the convenience of the reader, we shall say a few words about its proof:

By a well known property of the geometric law, conditionally on S := i,j w i,j , the law of w N,M is the uniform law on the set of N × M integer arrays summing up to S. But the RSK correspondence w → (P (w), Q(w)) is a bijection, when defined on this set, and the information carried out by the Gelfand-Tsetlin pattern λ of ( 2) is the same as the one carried out by P (w). It follows that conditionally on the event S = s, the law of the Gelfand-Tsetlin pattern λ of (2) can be recovered as follows:

First choose λ M 1 ≥ • • • ≥ λ M M
according to its particular conditional law, and then λ is uniformly distributed on the set of integer valued Gelfand-Tsetlin patterns with bottom row λ

M 1 ≥ • • • ≥ λ M M .
Therefore, by conditioning, it is clear that the unconditional law of λ can be recovered exactly in the same way.

Moreover, it can also be proved [START_REF] Baryshnikov | GUEs and queues[END_REF]Proposition 4.7] that conditionally on the value of its bottom row µ

M 1 ≥ • • • ≥ µ M M , the law of (µ k i ) 1≤i≤k≤M is the uniform law of real valued Gelfand-Tsetlin patterns with bottom row µ M 1 ≥ • • • ≥ µ M M .
Thus as M is fixed, up to standard analysis (see e.g. [2, Lem. 2.5]), it suffices to prove that the convergence in distribution stated in the lemma is true when restricted to the bottom rows:

(4) (ξ M 1 , . . . . . . , ξ M M ) =⇒ (µ M 1 , . . . . . . , µ M M ).
To prove (4), one only needs to notice that the joint distribution of its right hand side has density

(5) 1 normalizing constant 1≤i<j≤M (µ M j -µ M i ) 2 e -1 2 i (µ M i ) 2 ,
(this is a well known fact, see e.g. [1, Th. 2.5.2]), whereas it has been proved by Johansson in [START_REF] Johansson | Shape fluctuations and random matrices[END_REF] that the joint density of the left hand side of ( 4) with respect to the counting measure is

(6) 1 normalizing constant 1≤i<j≤M (ξ M i -ξ M j + j -i) 2 M i=1 (ξ M i + M)! (ξ M i + M -i)!
.

It can easily be seen that the quantities of ( 5) and (6) agree at the limit.

Let us now give a few more definitions. Fix some positive integers k, N.

• An up-right path with values in {1, . . . , N} × {1, . . . , k} is a map π defined on a set {1, . . . , p}, for a positive integer p, taking values in {1, . . . , N} × {1, . . . , k} and having two components π 1 , π 2 , such that for all i ∈ {2, . . . , p}, we have equality of the sets ) the path obtained by removing all points in π strictly before (resp. strictly after) the point P (these last definitions are unambiguous since, as defined, an up-right path is always one-to-one).

{π 1 (i) -π 1 (i -1), π 2 (i) -π 2 (i -1)} = {0,
• For π, π ′ up-right paths with values in {1, . . . , N}×{1, . . . , k} such that the starting point of π ′ is just to the right or just above the ending point of π, we denote by π ∪ π ′ their (naturally defined) concatenation. • For π, π ′ up-right paths with values in {1, . . . , N} × {1, . . . , k}, we write π < π ′ , if for any n ∈ {1, . . . , N}, the intersection of the support of π with {n} × {1, . . . , k} is located strictly below the intersection of the support of π ′ with {n} × {1, . . . , k}.

We will also need the following lemmas.

Lemma 3. Let 1 ≤ ℓ ≤ k and
∪ ℓ i=1 support(π i ) ⊂ ∪ ℓ i=1 support(π ′ i ).
Proof. Note first that if ℓ = k, the lemma is clearly true by choosing π ′ 1 , . . . , π ′ k to be the maximal horizontal lines with starting abscissas all equal to 1. So let us assume that ℓ ≤ k -1, and present a trivial claim which will be useful in the sequel: Claim : The hypothesis ℓ ≤ k-1 implies that at least one point of the set {1}×{1, . . . , k} is the starting point of no π i or that at least one π i has starting point in {1} × {1, . . . , k} and second point just above its starting point.

Next, let sa max denote the maximum of the starting abscissas of π 1 , . . . , π ℓ and ℓ sa max denote the number of i's such that π i has starting abscissa sa max . Let us prove the lemma by induction (for the lexical order) on (sa max , ℓ samax ). If sa max = 1, then there is nothing to do (as all the π i 's have starting abscissas equal to 1). This begins the induction. So let us now assume that sa max ≥ 2. Let i 0 be such that π i 0 has starting abscissa sa max . Let P be the point in {1, . . . , N} × {1, . . . , k} just to the left of the starting point of π i 0 .

• If P is on none of the supports of the π i 's, then one can extend π i 0 by adding P at its beginning and conclude by the induction hypothesis. • If P is on the support of a path π i 1 , then two cases can occur:

-If π i 1 does not end at P : then the point after P in π i 1 must be the point Q just above P , and replacing π i 0 by π

P ] i 1 ∪ π i 0 and π i 1 by π [Q i 1 allows
∪ ℓ i=1 support(π i ) ⊂ ∪ ℓ i=1 support(π ′ i ).
Proof. The proof is quite similar to the proof of the previous lemma. The difference lies in that the hypotheses require all the starting abscissas to be equal to one, while the conclusion requires all the ending abscissas to be equal to N.

Let ea min denote the minimum of the ending abscissas of π 1 , . . . , π ℓ and ℓ ea min denote the number of i's such that π i has ending abscissa ea min . Let us prove the lemma by induction (for the lexical order) on (ea min , ℓ ea min ). If ea min = N, then there is nothing to do (as all the π i 's have ending abscissas equal to N). This begins the induction. So let us next assume that ea min ≤ N -1. Let i 0 be such that π i 0 have ending abscissa ea min . Let P be the point in {1, . . . , N} × {1, . . . , k} just to the right of the ending point of π i 0 .

• If P is on none of the supports of the π i 's, then one can extend π i 0 by adding P to its end and conclude by the induction hypothesis. • If P is on the support of a path π i 1 , then P is not the starting point of π i 1 and the point preceding P in π i 1 is the point Q just below P . Hence replacing π i 0 by

π i 0 ∪ π [P i 1 and π i 1 by π Q]
i 1 allows to conclude by the induction hypothesis.

Lemma 5. Let 1 ≤ ℓ ≤ k and let π 1 , . . . , π ℓ be up-right paths with values in {1, . . . , N} × {1, . . . , k} having pairwise disjoint supports, starting abscissas all equal to 1 and ending abscissas all equal to N. Then, the collection (π i ) 1≤i≤ℓ can be re-indexed in such a way that

π 1 < • • • < π ℓ .
Proof. It suffices to re-index the collection (π i ) 1≤i≤ℓ according to the order of their starting abscissas and to use the definition of up-right paths combined with the disjoint supports.

Lemma 6. For each 1 ≤ ℓ ≤ k ≤ M, (7)
λ k 1 + • • • + λ k ℓ = max π 1 ,...,π ℓ ℓ r=1 (i,j)∈πr w ij ,
where the max is over collections 7) is true when the paths are only required to be pairwise disjoint, without any condition on the starting and ending points.

{π 1 < • • • < π ℓ } of up-
Then, the previous three lemmas allow to claim that any set of ℓ pairwise disjoint paths can be changed into a set of ℓ pairwise disjoint paths π 1 < • • • < π ℓ , with starting abscissas all equal to 1 and ending abscissas all equal to N, in such a way that the union of the supports of the news paths contains the union of the supports of the former ones. To finish the proof, it then suffices to notice that since the w ij 's are non-negative, enlarging the union of the supports of the paths never decreases the total weight.

To complete the proof of Theorem 1, note that any up-right path π r as described in the previous lemma is a concatenation of at most k paths with fixed second coordinate and has length between N and N + M. Moreover, by Donsker theorem (see [START_REF] Billingsley | Convergence of probability measures[END_REF][START_REF] Glynn | Whitt Departures form many queues in series[END_REF]), the

M-dimensional process   1 √ vN ⌊N t⌋ i=1 (w ik -e)   k=1,...,M
, converges in distribution (for the Skorohod topology) to the M-dimensional Brownian motion B. To finish the proof, apply both Lemma 2 and Lemma 6.

Applications of Theorem 1

At first, Let us present and prove some corollaries of Theorem 1 which motivated, in large part, the present study.

Let us briefly recall the framework of the works on random words cited in the Introduction. Let (X i ) i≥1 be a sequence of i.i.d. random variables on a totally ordered finite alphabet A of cardinality k. Denote the elements of A by α 1 , . . . , α k listed in such a way that if

p i := P(X 1 = α i ), i = 1, . . . , k, then p 1 ≥ • • • • • • ≥ p k (
therefore this indexing of the letters in A has nothing to do with the order used on A). Next, decompose the alphabet A into subsets A 1 , . . . , A n in such a way that α i and α j belong to the same A m , m = 1, . . . , n ≤ k, if and only if p i = p j . Finally, let LI N be the length of the longest (weakly) increasing subsequence of the random word

X 1 • • • • • • X N .
When combined with [START_REF] Houdré | On the longest increasing subsequence for finite and countable alphabets[END_REF], the following corollary provides an alternative approach to [START_REF] Kerov | Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis[END_REF]Chap. 3,Sec. 3.4,Th. 2] and [START_REF] Tracy | On the distribution of the lengths of the longest increasing monotone subsequences in random words[END_REF] for uniform letters, or to [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF] 1 . Corollary 7. Let p max := p 1 , k 1 := #A 1 and let H = [h ij ] be a k 1 × k 1 GUE matrix with largest eigenvalue µ max . Then, as N tends to infinity,

LI N -Np max √ Np max =⇒ √ 1 -k 1 p max -1 k 1 k 1 j=1 h jj + µ max . ( 8
)
Proof. From Theorem 1, with the notation introduced above, and if B is now a k 1dimensional standard Brownian motion,

(9) (µ j 1 ) 1≤j≤k 1 ∪(h jj ) 1≤j≤k 1 law = max 0=t 0 ≤•••≤t j =1 j i=1 (B i (t i ) -B i (t i-1 )) 1≤j≤k 1 ∪ (B j (1)) 1≤j≤k 1 ,
by also noticing that h 11 = µ 1 1 = B 1 [START_REF] Anderson | An Introduction to Random Matrices[END_REF] and that for all j = 2, . . . , k 1 ,

h jj = j i=1 µ j i - j-1 i=1 µ j-1 i law = j i=1 B i (1) -j-1 i=1 B i (1) = B j (1)
. Next, Corollary 3.3 in [START_REF] Houdré | On the longest increasing subsequence for finite and countable alphabets[END_REF] asserts that

(10) LI N -Np max √ Np max =⇒ √ 1 -k 1 p max -1 k 1 k 1 j=1 B j (1) + max 0=t 0 ≤•••≤t k 1 =1 k 1 i=1 (B i (t i ) -B i (t i-1 )) ,
combining ( 9) and ( 10) gives [START_REF] Doumerc | A note on representations of classical Gaussian matrices Sémimaire de Probabilités XXXVII[END_REF].

Denote by λ 1 ≥ • • • ≥ λ k the shape of the Young diagrams obtained by applying the RSK correspondence to the random word

X 1 • • • • • • X N ,
and let

ξ i = λ i -Np i √ Np i ,
1 ≤ i ≤ k, be the corresponding rescaled variables. Next, introduce independent GUE matrices H 1 , . . . , H n , where each H j has size k j := #A j , and let

H :=   H 1 . . . H n   and    H 1 . . . H n    := H -Tr(HJ)J, where J = diag( √ p 1 , . . . , √ p k ).
Remark 8. Note that J is a unit vector of the space H k 1 × • • • × H kn endowed with the Euclidean inner product structure, so H -Tr(HJ)J is the orthogonal projection onto J ⊥ , so that its law is the law of H conditioned to belong to J ⊥ .

The following corollary makes full use of Theorem 1 and, when combined with [START_REF] Houdré | On the limiting shape of Young diagrams associated with inhomogeneous random words[END_REF] or [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF], provides an alternative approach to [START_REF] Kerov | Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis[END_REF], [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF], [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF], [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems[END_REF] (see also, Bufetov [START_REF] Bufetov | A central limit theorem for the extremal characters of the infinite symmetric group[END_REF] and Méliot [START_REF] Méliot | Fluctuations of central measures on partitions[END_REF]).

Let us now define the random vector (µ 1 , . . . , µ k ) by (µ 1 , . . . , µ k ) := (ordered spectrum of H 1 , . . . . . . , ordered spectrum of H n ).

Corollary 9. As N → ∞, (ξ 1 , . . . , ξ k ) =⇒ (µ 1 , . . . , µ k ).
Remark 10. The limiting law of LI N , rescaled, is simply the law of µ 1 and is given by [START_REF] Gravner | Limit theorems for height fluctuations in a class of discrete space and time growth models[END_REF] λ max (H 1 ) -

p 1 Tr(H 1 ) -p 1 (1 -k 1 p 1 )Z,
where Z is a standard normal random variable, independent of H 1 . Note also that this law only depends on p 1 and k 1 .

Proof. First,

Tr(HJ) = n j=1 √ p (j) Tr H j ,
where for all j, p (j) := p ℓ , for ℓ ∈ A j . So for each i, we have

H i = H i - √ p i n j=1 √ p (j) Tr H j I,
where I is the corresponding identity matrix. Then, Theorem 3.1 and Remark 3.2 (iv) in [START_REF] Houdré | On the limiting shape of Young diagrams associated with inhomogeneous random words[END_REF] together with Theorem 1 allow to conclude.

In case the i.i.d. random variables generating the random word are replaced by an (irreducible, aperiodic) homogeneous Markov chain, with state space A of cardinality k, the corresponding limiting laws can also be given in terms of maximal Brownian functionals similar to those in Theorem 1 (see [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF]). However, an important difference is that now the standard Brownian motion B is replaced by a correlated one B with, say, covariance matrix Σ instead of I. The possible identification of (the law of) these functionals as (the law of) maximal eigenvalues (or spectra) of random matrices has not been fully accomplished yet, although various cases are done. In particular, for cyclic transition matrices P , in which case the stationary distribution is the uniform one, there is a curious dichotomy between alphabets of size at most three and of size four or more. Indeed for k ≤ 3, the cyclic hypothesis forces Σ to have a permutation-symmetric structure seen in the i.i.d. uniform case. For example, for k = 3, Σ is, a rescaled version of, [START_REF] Houdré | On the longest increasing subsequence for finite and countable alphabets[END_REF] Σ u :=

  1 -1/2 -1/2 -1/2 1 -1/2 -1/2 -1/2 1   ,
and so (up to a multiplicative constant) and with k = k 1 = 3, p max = p 1 = 1/3, (8) continues to hold for cyclic Markov chains. For k ≥ 4, the cyclicity constraint on P forces Σ to be cyclic but does no longer force the permutation-symmetric structure, and, say, for k = 4, Σ might differ from, a rescaled version of,

Σ u :=     1 -1/3 -1/3 -1/3 -1/3 1 -1/3 -1/3 -1/3 -1/3 1 -1/3 -1/3 -1/3 -1/3 1     . (13)
In fact, if

P =     p 1 p 2 p 3 p 4 p 4 p 1 p 2 p 3 p 3 p 4 p 1 p 2 p 2 p 3 p 4 p 1     , then Σ = (σ i,j ) 1≤i,j≤4 (14)
is a rescaled version of Σ u if and only if p 2 3 = p 2 p 4 . Nevertheless, see [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF], for k ≥ 2, and if σ = σ i,i , i = 1, . . . , k,

LI N -N/k σ √ N =⇒ sup ∆ π ( B); π ∈ P, π ≤ k = max 0=t 0 ≤•••≤t k =1 k j=1 Bj (t j ) -Bj (t j-1 ) . (15)
Assuming that in addition to be cyclic, P is also symmetric (for k = 2 the cyclic and symmetric assumptions are the same, and Chistyakov and Götze [START_REF] Chistyakov | Götze Distribution of the shape of Markovian random words[END_REF], see also [START_REF] Houdré | Litherland Asymptotics for the length of the longest increasing subsequence of a binary Markov random word[END_REF], showed that the corresponding limiting law is the maximal eigenvalue of the 2 × 2 traceless GUE) a diagonalization argument, combined with [START_REF] Houdré | On the limiting shape of Young diagrams associated with inhomogeneous random words[END_REF], leads to the following result.

Proposition 11. Let P := (p i,j ) 1≤i,j≤k be cyclic and symmetric, i.e., P = (p(j -i)) 1≤i,j≤k , where p is a k-periodic function defined on Z such that p(r) = p(-r), for all r ∈ Z. Let [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF] λ ℓ :=

k r=1 p(r) cos (2π(ℓ -1)r/k) (1 ≤ ℓ ≤ k),
and let (B j ) j=2,...,k be a (k -1)-dimensional standard Brownian motion on [0, 1]. Then,

LI N -N/k σ √ N =⇒ max 0=t 0 ≤•••≤t k =1      2 k k j=1 ⌊ k-1 2 ⌋ r=1 1 + λ r+1 1 -λ r+1 cos 2πjr k (B 2r (t j )- B 2r (t j-1 )) + sin 2πjr k (B 2r+1 (t j ) -B 2r+1 (t j-1 )) - 1 √ k 1 + λ k 2 +1 1 -λ k 2 +1 2 k j=1 (B k (t j ) -B k (t j-1 )) -B k (1)    , ( 17
)
where the last term, above, is only present for k even.

Proof. Since P is symmetric, it can be diagonalized as P = SΛS ⊤ , where Λ is the diagonal matrix formed with its eigenvalues (λ ℓ ) 1≤ℓ≤k (we will see below that these are the quantities defined at [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF]) and where S is a matrix formed by the orthonormal column eigenvectors (u ℓ ) 1≤ℓ≤k where u ⊤ 1 = (1/ √ k, . . . , 1/ √ k). Next, by Theorem 4.3 in [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF], Σ, the covariance matrix of the k-dimensional correlated Brownian motion B, is given by Σ = SΛ Σ S ⊤ , where Λ Σ is the diagonal matrix with diagonal entries 0, (1 + λ 2 )/(1 -λ 2 ), . . . , (1 + λ k )/(1 -λ k ). Therefore, B = S √ Λ Σ B, where now B is a standard k-dimensional Brownian motion. Next, the symmetric and cyclic structures imply that the eigenvalues of P are in fact the λ ℓ 's defined at [START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems. I. Random matrix models and their applications[END_REF]: λ ℓ = k r=1 p(r) cos (2π(ℓ -1)r/k), 1 ≤ ℓ ≤ k, (clearly they are not all simple since λ ℓ = λ k-ℓ+2 , ℓ = 2, . . . , k). The corresponding orthonormal column eigenvectors are

v ℓ := (v j,ℓ ) 1≤j≤k = ( √ 2 cos(2π(ℓ -1)j/k)/ √ k) 1≤j≤k , ℓ = 1, 2, . . . , ⌊k/2⌋ + 1,
and

w ℓ := (w j,ℓ ) 1≤j≤k = ( √ 2 sin(2π(ℓ -1)j/k)/ √ k) 1≤j≤k , ℓ = 2, 3, . . . , ⌊(k -1)/2⌋ + 1.
Clearly, v 1 = u 1 is an eigenvector corresponding to the simple eigenvalue 1, while if k is even,

v (k/2)+1 = (1/ √ k, -1/ √ k, . . . , 1/ √ k, -1/ √ k
) is an eigenvector corresponding to the simple eigenvalue k r=1 p(r) cos (2π(k/2 + 1 -1)r/k) = k r=1 (-1) r p(r). Moreover, for ℓ = 2, 3, . . . , ⌊(k -1)/2⌋ + 1, v ℓ and w ℓ share the same eigenvalue λ ℓ . Therefore,

S = v 1 , v 2 , w 2 , . . . , v ⌊ k-1 2 ⌋+1 , w ⌊ k-1 2 ⌋+1 , vk 2 +1 ,
where, above, the last column is only present if k is even. Next, from the transformation B = S √ Λ Σ B, and since

√ Λ Σ B ℓ = (1 + λ ⌊ℓ/2⌋+1 )/(1 -λ ⌊ℓ/2⌋+1 )B ℓ , ℓ = 2, . . . , k, and √ Λ Σ B 1 = 0, then for j = 1, . . . , k, Bj = k ℓ=2 u j,ℓ 1 + λ ⌊ ℓ 2 ⌋ +1 1 -λ ⌊ ℓ 2 ⌋+1 B ℓ , (18)
where u j,ℓ = v j,⌊ℓ/2⌋+1 or u j,ℓ = w j,⌊ℓ/2⌋+1 , for ℓ even or odd. Therefore, for j = 1, . . . , k,

Bj = 2 k ⌊ k-1 2 ⌋ r=1 1 + λ r+1 1 -λ r+1 cos 2πrj k B 2r + sin 2πrj k B 2r+1 (19) + (-1) j+1 √ k 1 + λ k 2 +1 1 -λ k 2 +1 B k ,
where the last term on the right of ( 19) is only present for k even. With [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF], the sum on the right hand side of (15) becomes:

2 k k j=1 ⌊ k-1 2 ⌋ r=1 1 + λ r+1 1 -λ r+1 cos 2πjr k B 2r (t j ) + sin 2πjr k B 2r+1 (t j ) (20) -cos 2πjr k B 2r (t j-1 ) -sin 2πjr k B 2r+1 (t j-1 ) + 1 √ k 1 + λ k 2 +1 1 -λ k 2 +1 k j=1 (-1) j+1 (B k (t j ) -B k (t j-1 )) ,
an expression only involving standard Brownian motions and where, again, the last term

1 √ k 1 + λ k 2 +1 1 -λ k 2 +1 k j=1 (-1) j+1 (B k (t j ) -B k (t j-1 )) (21) = -1 √ k 1 + λ k 2 +1 1 -λ k 2 +1 B k (1) + k-1 j=1 2(-1) j B k (t j ) = -1 √ k 1 + λ k 2 +1 1 -λ k 2 +1 2 k j=1 (B k (t j ) -B k (t j-1 )) -B k (1) , (22)
is only present if k is even.

Remark 12. Let us specialize the previous results to instances where further simplifications and identifications occur.

(i) For k = 3, and up to the factor 2(1 + λ 2 )/(k(1 -λ 2 )) = 2(1 + 3p 1 )/(3(3 -3p 1 )), the right-hand side of (17) becomes max

0=t 0 ≤t 1 ≤t 2 ≤t 3 =1 3 j=1 cos 2πj 3 (B 2 (t j ) -B 2 (t j-1 )) + sin 2πj 3 (B 3 (t j ) -B 3 (t j-1 )) = max 0≤t 1 ≤t 2 ≤1 B 2 (1) + √ 3B 3 (t 1 ) - √ 3 2 B 3 (t 2 ) - 3 2 B 2 (t 2 ) law = 2 3 max 0≤t 1 ≤t 2 ≤t 3 =1 2 j=1 - j j + 1 B j (t j+1 ) + j j + 1 B j (t j ) law = 2 3 max 0=t 0 ≤t 1 ≤t 2 ≤t 3 =1 3 j=1 (B j (t j ) -B j (t j-1 )) - 1 3 3 j=1 B j (1) ,
where the last equality, in law, follows either by using, in [START_REF] Houdré | On the limiting shape of Young diagrams associated with inhomogeneous random words[END_REF], the simple linear transformation

(23) Bj = 2(1 + λ 2 ) 3(1 -λ 2 ) 2 3 B j - 1 6 3 i=1,i =j B i , j = 1, 2, 3,
which, by comparing covariances, is easily verified from [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF]; or, still by comparing covariances, by arguments such as those in the proof of Theorem 3.2 in [START_REF] Houdré | On the longest increasing subsequence for finite and countable alphabets[END_REF]. Therefore, with the help of Theorem 1, and up to a scaling factor, the limiting law of LI N is that of the maximal eigenvalue of the 3 × 3 traceless GUE.

(ii) For k = 4, Σ, the covariance matrix of B = ( Bj ) j=1,...,4 is given, up to a scaling constant, by: [START_REF] Méliot | Fluctuations of central measures on partitions[END_REF] Σ :=

    2η 2 + η 3 -η 3 -2η 2 + η 3 -η 3 -η 3 2η 2 + η 3 -η 3 -2η 2 + η 3 -2η 2 + η 3 -η 3 2η 2 + η 3 -η 3 -η 3 -2η 2 + η 3 -η 3 2η 2 + η 3     .
where η 2 = (1 + λ 2 )/(1 -λ 2 ), λ 2 = p 1 -p 3 , and η 3 = (1 + λ 3 )/(1 -λ 3 ), λ 3 = p 1 -2p Then, and up to the multiplicative constant 4η 2 , Σ becomes:

Σ :=     1 -1/2 0 -1/2 -1/2 1 -1/2 0 0 -1/2 1 -1/2 -1/2 0 -1/2 1     ,
which is clearly different from, a rescaled version of, [START_REF] Houdré | On the limiting shape of Young diagrams associated with Markov random words[END_REF]. In fact, if Σ = Σ u , then clearly (iii) Finally, it is easy to see that the properties just described continue to hold for arbitrary dimension k ≥ 4. In arbitrary dimension, if Σ = Σ u (the k-dimensional version of the matrix defined at [START_REF] Houdré | On the longest increasing subsequence for finite and countable alphabets[END_REF] and ( 13)), then the linear transformation corresponding to [START_REF] Van Moerbeke | Random and Integrable Models in Mathematics and Physics in Random Matrices, Random Processes and Integrable Systems[END_REF] is given by

Bj = k -1 k B j - 1 k(k -1)

  the eigenvalues of the principal k × k minor of H. Next, introduce the set P := {π : [0, 1] → {1, . . . , M}, càdlàg, non-decreasing}, and for π ∈ P, let ∆ π (B) :=

1

 1

  1}. • The support of an up-right path π : {1, . . . , p} → {1, . . . , N} × {1, . . . , k} is the set {π(1), . . . , π(p)}. • The starting point (resp. ending point) of an up-right path π : {1, . . . , p} → {1, . . . , N} × {1, . . . , k} is π(1) (resp. π(p)). The starting abscissa (resp. ending abscissa) of π is the first coordinate of its starting point (resp. ending point). • For π an up-right path with values in {1, . . . , N} × {1, . . . , k} and P a point of the support of π, we denote by π [P (resp. π P ]

  right paths in the set {1, . . . , N} × {1, . . . , k} starting in the subset {1} × {1, . . . , k} and ending in the subset {N} × {1, . . . , k}. Proof. By [25, Th. 1.1.1] and [9, Chap. 3, Lemma 1], (

4 j=1( 4 j=1B

 44 clearly, for a linear transformation such asBj = α j B j -4 i=1,i =j β i B i , j = 1, 2, 3, 4,to lead to Σ, one needs Σ to be permutation-symmetric and, up to a multiplicative constant, the right-hand side of[START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems[END_REF] becomes equal in law to max 0=t 0 ≤t 1 ≤t 2 ≤t 3 ≤t 4 =1 B j (t j ) -B j (t j-1 )) -1 4 j (1), and corresponds to the matrix P in (14) with p 2 = p 3 = p 4 .

  k i=1,i =j B i , j = 1, . . . , k,Conversely, for a linear transformation such asBj = α j B j -k i=1,i =j β i B i , j = 1, 2, . . . , k,to lead to Σ, one needs Σ to be permutation-symmetric. In either instance, and up to a multiplicative constant, the right-hand side of (17) has the same law as max0=t 0 ≤•••≤t k =1 k j=1(B j (t j ) -B j (t j-1 ))turn, via Theorem 1, is the law of the maximal eigenvalue of an element of the k × k traceless GUE.

  let π 1 , . . . , π ℓ be up-right paths with values in {1, . . . , N} × {1, . . . , k} and having pairwise disjoint supports. Then, there exist up-right paths π ′ 1 , . . . , π ′ ℓ with values in {1, . . . , N} × {1, . . . , k}, having pairwise disjoint supports and starting abscissas all equal to 1 such that

  to conclude by the induction hypothesis.-If π i 1 ends at P : then replacing π i 0 by π i 1 ∪ π i 0 and π i 1 by a point of the set {1} × {1, . . . , k} allows to conclude (note that replacing π i 1 by a point of the set {1} × {1, . . . , k} is possible by the claim stated above: if a point of the set {1} × {1, . . . , k} is the starting point of no π i then this is trivial, while if a π i 2 has starting point in {1} × {1, . . . , k} and second point just above its starting point, one can remove its starting point from π i 2 to get back to the first case). Lemma 4. Let 1 ≤ ℓ ≤ k and let π 1 , . . . , π ℓ be up-right paths with values in {1, . . . , N} × {1, . . . , k}, pairwise disjoint supports and starting abscissas all equal to 1. Then there exist up-right paths π ′ 1 , . . . , π ′ ℓ with values in {1, . . . , N} × {1, . . . , k}, having pairwise disjoint supports, starting abscissas all equal to 1 ending abscissas all equal to N, and such that

  2 + p 3 . Clearly, Σ can differ from Σ u , e.g., let 2η 2 = η 3 , i.e., let

	P =	    	p 1 p 2 p 2 (1-2p 2 ) p 2 1+2p 2	p 2 p 1 1+2p 2 p 2 (1-2p 2 ) p 2	p 2 (1-2p 2 ) 1+2p 2 p 2 p 2 p 1	p 2 p 2 (1-2p 2 ) 1+2p 2 p 1 p 2	    	.

On the same year, the same authors published a second paper on the subject,[START_REF] Its | Widom Random words, Toeplitz determinants, and integrable systems[END_REF]. This paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a Riemann-Hilbert problem and an associated system of integrable partial differential equations. In particular, it shows that the Poissonization of the distribution function of this length can be identified as the Jimbo-Miwa-Ueno tau function.

This work was partially supported by the grant #246283 from the Simons Foundation and by a Simons Foundation Fellowship, grant #267336. Many thanks to the Laboratory MAS of École Centrale, Paris, and to the LPMA of the Université Pierre et Marie Curie for their hospitality while part of this research was carried out.


N Arcozzi

K Domelevo

S Petermichl

Sharp L p estimates for second order Riesz transforms on multiply-connected Lie groups

We study a class of combinations of second order Riesz transforms on Lie groups G = G x × G y that are multiply connected, composed of a discrete abelian component G x and a compact connected component G y . We prove sharp L p estimates for these operators, therefore generalizing previous results [13][4].

The proof uses stochastic integrals with jump components adapted to functions defined on the semi-discrete set G x × G y . The analysis shows that Itô integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local.

Introduction

Sharp L p inequalities for pairs of differentially subordinate martingales date back to the celebrated work of Burkholder [START_REF] Burkholder | Boundary value problems and sharp inequalities for martingale transforms[END_REF] in 1984 where the optimal constant is exhibited. See also from the same author [START_REF] Donald | Sharp inequalities for martingales and stochastic integrals[END_REF] [START_REF] Donald | Explorations in martingale theory and its applications[END_REF]. The relation between differentially subordinate martingales and Caldéron-Zygmund operators is known since Gundy-Varopoulos [START_REF] Gundy | Les transformations de Riesz et les intégrales stochastiques[END_REF]. Starting with a test function f , martingales are built using Brownian motion and Poisson extensions in the upper half space R + × R n .

It is shown that the martingale arising from R f , where R is a Riesz transform in R n , is a martinagle transform of that arising from f . The two form a pair of martingales with differential subordination and orthognality. Thus, in the case of Riesz tranforms, the optimal L p constants could be recovered using probabilistic methods. One derives martingale inequalities under hypotheses of strong differential subordination and orthogonality, see Banuelos-Wang [START_REF] Bañuelos | Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms[END_REF].

In the case of second order Riesz transforms, the use of heat extensions in the upper half space instead of Poisson extensions originated in Petermichl-Volberg [START_REF] Petermichl | Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular[END_REF] and was used to prove L p estimates for the second order Riesz transforms based on the results of Burkholder in Nazarov-Volberg [START_REF] Volberg | Heat extension of the Beurling operator and estimates for its norm[END_REF] as part of their best-at-time estimate for the Beurling-Ahlfors operator, whose real and imaginary parts themselves are second order Riesz transforms. The sharpness of the constant for the real part of the Beurling-Alfhors operator is proved using probabilistic methods in conjunction with a technique by Bourgain in Geiss-Montgomery-Saksman [START_REF] Geiss | On singular integral and martingale transforms[END_REF]. See also applications in Banuelos-Baudoin [START_REF] Bañuelos | Martingale Transforms and Their Projection Operators on Manifolds[END_REF] All these martingale inequalities use special functions found in Pichorides [START_REF] Pichorides | On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov[END_REF], Essén [START_REF] Essén | A superharmonic proof of the M. Riesz conjugate function theorem[END_REF], when orthogonality is present in addition to differential subordination or Burkholder [START_REF] Burkholder | Boundary value problems and sharp inequalities for martingale transforms[END_REF][8] [START_REF] Donald | Sharp inequalities for martingales and stochastic integrals[END_REF], when differential subordination is the only hypothesis.

Deterministic proofs of sharp L p estimates of Caldéron-Zygmund operators that use Burkholder's theorems are available in the literature. The technique of Bellman functions was used in Nazarov-Volberg [START_REF] Volberg | Heat extension of the Beurling operator and estimates for its norm[END_REF] for an L p estimate for certain second order Riesz transforms in the euclidean plane as well as in the recent version on discrete abelian groups [START_REF] Domelevo | Sharp L p estimates for discrete second order Riesz transforms[END_REF].

The aim of the present work is two-fold. On one hand we want to derive a stochastic proof of the latter result on discrete abelian groups. On the other hand, we want to generalize the estimate to second order Riesz transforms acting on multiply connected Lie groups, built as the cartesian product of a discrete abelian group with a connected compact Lie group. Previous works based on stochastic methods for the analysis of Riesz transforms on connected compact Lie groups are in Arcozzi [START_REF] Arcozzi | Riesz transforms on spheres and compact Lie groups[END_REF][2], and sharp L p estimates were proved in this setting in Banuelos-Baudoin [START_REF] Bañuelos | Martingale Transforms and Their Projection Operators on Manifolds[END_REF] for second order Riesz transforms. The novelty of this text is the generalization to the multiply connected setting. In this sense, it is also a generalization of [START_REF] Domelevo | Sharp L p estimates for discrete second order Riesz transforms[END_REF], by regarding each point in the discrete abelain group as a Lie group of dimension zero.

Differential operators and Riesz transforms.

First order derivatives and tangent planes. For a general n-dimensional Riemannian manifold M, the tangent plane T y M at a point y ∈ M is spanned by a local family of vectors (Y 1 (y), ..., Y n (y)). Identifying as usual the vector Y j with the first order derivative Y j = ∂/∂x j , we have for a real function f defined on M, the gradient written as a column vector of size n:

∇ M f (y) := (∂f /∂y 1 , ..., ∂f /∂y n )(y) = (Y 1 f , ..., Y n f )(y) = j =1 n Y j f (y) Y j (y) ∈ T y M.
Now for a Lie group G := G x × G y with G x a discrete group of order m and G y a connected Lie group of dimension n we define the tangent plane

T z G at a point z = (x, y) ∈ (G x × G y ) = G in three steps.
First, let G y be a compact Lie group endowed with a biinvariant Riemannian metric and denote by G y its Lie algebra. Since G y is n-dimensional, we can find a family (Y j ) j =1,...,n of n left invariant vector fields that form an orthonormal basis of G y . If f is a function defined on G y , then the partial derivative of f in the direction Y j at point y ∈ G y is

(∂f /∂y j )(y) := (Y j f )(y) := (∂ j f )(y)
and the gradient accounting for the infinitesimal variations of f about a point y ∈ G y is as before the n-column-vector

∇ y f (y) := (∂f /∂y 1 , ..., ∂f /∂y n )(y) = (Y 1 f , ..., Y n f )(y) = j =1 n Y j f (y) Y j (y) ∈ T y G y
Second, the discrete component G x is of order m and one has m generators G x = (g i ) i=1,...,m . At any point x ∈ G x , and given a direction i ∈ {1, ..., m}, one has two choices of discrete derivatives for each generator, namely the right and the left derivatives:

(∂ + f /∂x i )(z) := f (xg i , y) -f (x , y) := (∂ i + f )(z) (∂ -f /∂x i )(z) := f (xg i -1 , y) -f (x , y) := (∂ i -f )(z).
Comparing with the continuous component, this suggests that the tangent plane T ˆxG x at a point x of the discrete group G x is actually composed of the "right" tangent plane T x + G x and the "left" tangent plance T x -G x . We consequently define the augmented discrete gradient ∇ ˆxf (x) noted with a hat, as the 2m-vector of T ˆxG

x := T x + G x × T x - G
x accounting for all the local variations of the function f in the direct vicinity of x, that is the 2m-column-vector

∇ ˆxf (x) := (X 1 + f , X 2 + f , ..., X 1 -f , X 2 -f , ...)(x) = i=1 m τ =± X i τ f (x) X i τ ∈ T ˆxG x .
We noted the discrete derivatives X i ± f := ∂ i ± f and we noted the discrete 2m-vectors X i ± as the column vectors of Z 2m

X i + = (0, ..., 1, ..., 0) × 0 m , X i -= 0 m × (0, ..., 1, ..., 0),
where the 1's in X i ± are located at the i-th position of respectively the first or the second m-tuple. Notice that those vectors are independent of the point x.

Finally, for a function f defined on the cartesian product G := G x × G y , the (augmented) gradient ∇ ˆzf (z) at the point z = (x, y) is an element of the tangent plane T ˆzG := T ˆxG x × T y G y , that is a (2m + n)-column-vector

∇ ˆzf (z) := ∇ ˆxf (z), ∇ y f (z) = (X 1 + f , X 2 + f , ..., X 1 -f , X 2 -f , ..., Y 1 f , Y 2 f , ...)(z) = i=1 m τ =± X i τ f (z) X ˆiτ + j =1 n Y j f (z) Y ˆj(z)
where X ˆiτ and Y ˆj(z) are now column vectors of size (2m + n) with obvious definitions.

Riesz transforms. Following [START_REF] Arcozzi | Riesz transforms on spheres and compact Lie groups[END_REF][2], recall first that for a general Riemannian manifold M without boundary, one denotes by ∇ M , div M and ∆ M :=div M ∇ M the gradient, the divergence and the Laplacian associated with M, respectively. Then -∆ M is a positive operator and the vector Riesz transform is defined as the linear operator

R M := ∇ M • (-∆ M ) -1/2
acting on L 0 2 (M). It follows that if f is a function defined on M and y ∈ M then R M f (y) is a vector of the tangent plane T y M.

Similarily on M = G, we define ∇ M := ∇ ˆz as before, and then we define the divergence operator as the formal dual, that is -div M =div z := ∇ ˆz * , with respect to the following scalar product of

R (2m+n) : a ˆ, b ˆ L 2 (R 2m+n ) := 1 2 i=1 m ± a i + b i + + j =1 n a j b j
where

a ˆ:= (a 1 + , a 2 + , ..., a 1 - , a 2 - , ..., a 1 , a 2 , ...) ∈ R 2m+n ,
and a similar definition holds for b ^. The factor one-half is necessary so as to ensure a natural isometry between vectors of R 2m+n and vectors of R m+n . Thanks to the duality relations

(X i ± ) * = -X i ∓ and Y j * = -Y j , the divergence operator writes as the (2m + n)-row-vector ∇ ˆz * f (z) := 1 2 ∇ ˆx * f (z), ∇ y * f (z) = - 1 2 i=1 m τ =± X i -τ f (z) X ˆiτ , * - j =1 n Y j f (z) Y ˆj * (z)
where X ˆiτ , * and Y ˆj * (z) are now row vectors of size (2m + n) that are the transpose of X ˆiτ and Y ˆj(z) respectively. It follows that the Laplacian -∆ G is as expected

-∆ z f (z) := ∇ ˆz * f (z) • ∇ ˆzf (z) = 1 2 ∇ ˆx * f (z) • ∇ ˆxf (z) + ∇ y * f (z) • ∇ y f (z) = - i=1 m X i -X i + f (z) - j =1 n Y j 2 f (z) = - i=1 m X i 2 f (z) - j =1 n Y j 2 f (z) =: (-∆ x f )(z) + (-∆ y f )(z)
where we noted X

i 2 := X i + X i -= X i - X i + . The Riesz vector R ˆzf (z) is the (2m + n)-column-vector
of the tangent plane T ˆzG defined as the linear operator

R ˆzf := ∇ ˆzf • (-∆ z f ) -1/2
In particular, we have coordinatewise

R i ± = ∂ i ± • (-∆ z ) -1/2 and R j = ∂ j • (-∆ z ) -1/2 .

Main results.

In this text, we are concerned with second order Riesz transforms and combinations thereof. We first define the square Riesz transform in the (discrete) direction i to be

R i 2 := R i + R i -= R i - R i + .
Then, given α := ((α i x ) i=1...m , (α jk y ) j ,k=1...n ) ∈ C m × C n×n , we define R α 2 to be the following combination of second order Riesz transforms:

R α 2 := i=1 m α i x R i 2 + j ,k=1 n α jk y R j R k ,
where the first sum involves squares of discrete Riesz transforms as defined above, and the second sum involves products of continuous Riesz transforms. This combination is written in a condensed manner as the quadratic form

R α 2 = R ˆz, A α R ˆz where A α is the (2m + n) × (2m + n) block matrix A α := A α x 0 0 A α y with A α x = diag(α 1 x , ..., α m x , α 1 x , ..., α m x ) ∈ C 2m×2m , A α y = (α j k y ) j ,k=1...n ∈ C n×n .
When p and q are conjugate exponents, let p * = max {p, q }. Our main results are Theorem 1. Let G be a Lie group as defined before. Let R α 2 : L p (G, C) → L p (G, C) be a combination of second order Riesz transforms as defined above. This operator enjoys the estimate

R α 2 A α 2 (p * -1).

The estimate above is sharp when the group

G = G x × G y and dim (G y ) + dim ∞ (G x ) 2, where dim ∞ (G x ) denotes the number of infinite components of G x . Notice that A α 2 = max ( A α x 2 , A α y 2 ) = max (|α 1 x |, ..., |α m x |, A α y 2 ).
In the case where G = G x only consists of the discrete component, this whas proved in [START_REF] Domelevo | Sharp L p estimates for discrete second-order Riesz transforms[END_REF][13] using the deterministic Bellman function technique. In the case where G = G y is a connected compact Lie group, this was proved by Banuelos-Baudouin [START_REF] Bañuelos | Martingale Transforms and Their Projection Operators on Manifolds[END_REF] using Brownian motions defined on manifolds and projections of martingale transforms.

In the case where the function f is real valued, we obtain better estimates involving the Choi constants. Compare with Banuelos-Osekowski [START_REF] Banuelos | Martingales and sharp bounds for Fourier multipliers[END_REF] and with [START_REF] Domelevo | Sharp L p estimates for discrete second order Riesz transforms[END_REF].

Theorem 2. Assume that aI

A α bI in the sense of quadratic forms, where a, b are real numbers. Then R α

2 : L p (G, R) → L p (G, R) enjoys the norm estimate R α 2 p
C a,b,p , where these are the Choi constants.

The Choi constants (see [START_REF] Choi | A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in L p (0, 1)[END_REF]) are not explicit, except C -1,1,p = p * -1. An approximation of C 0,1,p is known and writes as

C 0,1,p = p 2 + 1 2 log 1 + e -2 2 + β2 p + ...., with β 2 = log 2 1 + e -2 2 + 1 2 log 1 + e -2 2
-2

e -2 1 + e -2 2 .

Plan of the paper.

In the next subsection, we recall the weak formulations characterizing the combinations of semidiscrete second order Riesz transforms we are interested in. Section 2 is devoted to the stochastic integrals for semi-discrete functions f defined on G, together with their martingale transforms. Finally, the proofs of the main results are given in Section 3.

Weak formulations.

Let f : G → C be given. The heat extension f ˜(t) of f is defined as f ˜(t) := e t∆z f =: P t f . We have therefore f ˜(0) = f . The aim of this section is to derive weak formulations for the second order Riesz transforms. We start with the weak formulation of the identity operator I.

Lemma 3. Assume f and g in L 0 2 (G), then (If , g) := (f , g) = 2 0 ∞ ∇ ˆzP t f , ∇ ˆzP t g L 2 (G;T G) dt = 2 0 ∞ z ∈G    i=1 m τ =± (X i τ P t f )(z) (X i τ P t g)(z) + j =1 n (Y j P t f )(z) (Y j P t g)(z)    dt
and the sums and integrals that arise converge absolutely.

Proof. This classical formula can be obtained by observing that d t P t = ∆ z P t and writing the ODE satisfied by φ(t) := (P t f , P t g).

In order to pass to the weak formulation for the squares of Riesz transforms, we need the following hypothesis and commutation properties.

Hypothesis. We assume everywhere in the sequel:

Y j • ∆ z = ∆ z • Y j X i τ • ∆ z = ∆ z • X i τ , τ ∈ {+, -, 0, 2} Proof. Since G = G x × G y is a cartesian product, we have [Y j , X i τ ] =
(R α 2 f , g) = -2 0 ∞ A α ∇ ˆPt f , ∇ ˆPt g L 2 G;T ˆG dt = -2 0 ∞ z ∈G 1 2 i=1 m τ =± α i x (X i τ P t f )(z) (X i τ P t g)(z) + j =1 n α j k y (Y j P t f )(z) (Y k P t g)(z)    dt
and the sums and integrals that arise converge absolutely.

Proof. We apply the previous Lemma to R α 2 f instead of f and we are left with integrands of the form

∇ ˆzP t R α 2 f , ∇ ˆzP t g = (-∆ z )P t R α 2 f , P t g = i,τ α i x (-∆ z )P t R i 2 f , P t g + j ,k α j ,k y (-∆ z )P t R j R k f , P t g = i,τ α i x (-∆ z )P t X i - (-∆ z ) -1/2 X i + (-∆ z ) -1/2 f , P t g + j ,k α j ,k y (-∆ z )P t X j (-∆ z ) -1/2 X k (-∆ z ) -1/2 f , P t g = i,τ α i x X i ± P t f , X i ± P t g + j ,k α j ,k y X j P t f , X k P t g
where we used successively the commutation properties of the Laplacian ∆ z with the vector fields and the commutation properties of the vector fields with P t = e t∆z . This yields the desired result.

Stochastic integrals and martingale transforms

In all what follows, we assume that we have a complete probability space (Ω, F , P) with a càdlàg (i.e. right continuous left limit) filtration (F t ) t 0 of sub-σ-algebras of F . We assume as usual that F 0 contains all events of probability zero. All random walks and martingales are adapted to this filtration.

We define below a semi-discrete random walk Z t := (X t , Y t ) ∈ G x × G y with generator L = ∆ z . The jump component X t is built thanks to compound Poisson jump processes on the discrete set G x wheras the continuous component Y t involves standard brownian motions on the manifold G y . Then, Itô's formula ensures that semi-discrete "harmonic" functions f : R + × G → C solving the backward heat equation (∂ t + ∆ z )f = 0 are actually martingales M t f := f (t, Z t ) for which we define a class of martingale transforms.

Stochastic integrals on Riemannian manifolds and Itô integral. Following Emery

[15] [START_REF] Emery | An Invitation to Second-Order Stochastic Differential Geometry[END_REF], see also Arcozzi [START_REF] Arcozzi | Riesz transforms on spheres and compact Lie groups[END_REF][2], we define the Brownian motion Y t on G y , a compact Riemannian manifold, as the process Y t : Ω → (0, T ) × G y such that for all smooth functions f :

G y → R, the quantity f (Y t ) -f (Y 0 ) - 1 2 0 t (∆ y f )(Y s ) ds =: (I df ) t (1)
is an R-valued continuous martingale. For any adapted continuous process Ψ with values in the cotangent space

T * G y of G y , if Ψ t (ω) ∈ T Yt(ω) *
G y for all t 0 and ω ∈ Ω, then one can define the continuous Itô integral I Ψ of Ψ as

(I Ψ ) t := 0 t Ψ s , dY s so that in particular (I df ) t = 0 t d y f (Y s ), dY s The integrand therefore involves the 1-form of T y * G y d y f (y) := j (∂ j f )(y) dy j = j (X j f )(y) X j *
Discrete random walks and jump processes. We define the discrete m-dimensional random walk X t on the discrete abelian group G x as a tuple of the form X t = (X t 1 , ..., X t m ) where each X t i , 1 i m is a compound jump process defined as follows: i. For any 1 i m, let N t i be a càdlàg Poisson process of parameter λ, that is

∀t, P(N t i = n) = (λt) n n! e -λt
The sequence of instants where jumps occur is noted (T k i ) k∈N , with the convention T 0 i = 0.

ii. Let (τ k ) k∈N be a sequence of independent Bernouilli variables

∀k, P(τ k = 1) = P(τ k = -1) = 1/2
We set

N t = i=1 m N t i
Almost surely the instants of jumps ((T k i ) k∈N ) i=1,...,m are disjoints. Let (T k ) k∈N = ∪ i=1 m (T k i ) k ∈N the sequence of instants of jumps of N t , and let i Nt (ω) be the index of the coordinate where the jumps occurs at time t,

dN t = i=1 m dN t i = dN t iN t
The random walk X t started at X 0 ∈ G x is the càdlàg compound Poisson process (see e.g. Protter [START_REF] Protter | Stochastic integration and differential equations[END_REF], Privault [START_REF] Privault | Notes on Stochastic Finance[END_REF]) defined as

X t (ω) := X 0 + k=1 Nt X ˆik(ω) τk(ω)
where we used an additive notation for the discrete abelian group. In differential form, we have ∀t, dX t = X ˆiN t τN t dN t Stochastic integrals on discrete groups. We recall for the convenience of the reader the derivation of stochastic integrals for jump processes. We will emphasize the fact that the corresponding Itô's formula involves the action of a discrete 1-form written in a well-chosen local coordinate system of the discrete augmented cotangent plane (see details below). Let 1 k N t and let (T k , i k , τ k ) be respectively the instant, the axis and the direction of the k-th jump. We set

T 0 = 0. Let f := f (t, x), t ∈ R + , x ∈ G x a function defined on R + × G x . Then f (t, X t ) = f (t, X t ) -f (t, X TN t ) + k=1 Nt {f (t, X Tk ) -f (t, X Tk-1 )} = f (t, X t ) -f (t, X TN t ) + k=1 Nt {f (t, X Tk ) -f (t, X Tk-) + f (t, X Tk-) -f (t, X Tk-1 )} = TN t t (∂ t f )(s, X s )ds + k=1 Nt f (t, X T k ) -f (t, X T k -) + Tk-1 Tk (∂ t f )(s, X s )ds = 0 t (∂ t f )(s, X s )ds + 0 t (f (s, X s ) -f (s, X s-)) dN s = 0 t (∂ t f )(s, X s )ds + i=1 m 0 t (f (s, X s ) -f (s, X s-)) dN s i
At an instant s = T k of jump, the integrand in the last term writes as

(f (s, X s ) -f (s, X s-)) dN s i = f s, X s-+ τ Ns X ˆi -f (s, X s-) dN s i = (X i τN s f )(X s-) dN s i = 1 2 {(X i 2 f )(X s-) + τ Ns (X i 0 f )(X s-)} dN s i
where we introduced, for all 1 i m,

X i 0 := (X i + + X i -)/2 X i 2 := (X i + -X i -)/2.
Notice that, for any given 1 i m, up to a normalisation factor, the system of coordinate

(X i 2 , X i 0 )
is obtained thanks to a rotation of π/4 of the canonical system of coordinate

(X i + , X i - ). Finally, f (t, X t ) = 0 t (∂ t f )(s, X s )ds + 1 2 i=1 m 0 t {(X i 2 f )(X s-) + τ Ns (X i 0 f )(X s-)} dN s i = 0 t (∂ t f )(s, X s ) + λ 2 (∆ x f )(s, X s ) ds + + 1 2 i=1 m 0 t (X i 2 f )(s, X s-) d(N s i -λs) + (X i 0 f )(s, X s-) dX s i = 0 t (∂ t f )(s, X s ) + λ 2 (∆ x f )(s, X s ) ds + 0 t d ˆf (s, X s-), dW ˆs =: 0 t (∂ t f )(s, X s ) + λ 2 (∆ x f )(s, X s ) ds + I d ˆf x t , (2)
where we set dX s i := τ Ns dN s i . Here and in the sequel, we take λ = 2.

Discrete Itô integral. The stochastic integral above shows that Itô formula (1) for continuous processes has a discrete counterpart involving stochastic integrals for jump processes, namely we have the discrete Itô integral

I d ˆf x t := 1 2 i=1 m 0 t (X i 2 f )(s, X s -) d(N s i -λs) + (X i 0 f )(s, X s -) dX s
This has a more intrinsic expression similar to the continuous Itô integral (1). If we regard the discrete component G x as a "discrete Riemannian" manifold, then this discrete Itô integral involves discrete vectors (resp. 1-forms) defined on the augmented discrete tangent (resp. cotangent) space T ˆxG x (resp. T ˆx * G x ) of dimension 2m defined as

T ˆxG x = span{X 1 + , X 2 + , ..., X 1 -, X 2 -, ...} = span{X 1 2 , X 2 2 , ..., X 1 0 , X 2 0 , ...} T ˆx * G x = span{(X 1 + ) * , (X 2 + ) * , ..., (X 1 -) * , (X 2 -) * , ...} = span{(X 1 2 ) * , (X 2 2
) * , ..., (X 1 0 ) * , (X 2 0 ) * , ...} Indeed, let dW ˆs ∈ T ˆX s G x be the vector and d ˆf ∈ T ˆX s * G x be the 1-form respectively defined as:

dW ˆs = d(N s 1 -λs) X 1 2 + ... + d(N s m -λs) X m 2 + dX s 1 X 1 0 + ... + dX s m X m 0 d ˆxf = X 1 2 f (X 1 2 ) * + ... + X m 2 f (X m 2 ) * + X 1 0 f (X 1 0 ) * + ... + X m 0 f (X m 0 ) *
We have with these notations

I d ˆf x t := d ˆxf , dW ˆs T ˆx * G x ×T ˆxG x
where the factor 1/2 is included in the pairing •, • T ˆx * Gx×T ˆxGx .

Semi-discrete stochastic integrals. Let finally Z t = (X t , Y t ) be a semi-discrete random walk on the cartesian product G = G x × G y , where X t is the random walk above defined on G x with generator ∆ x and where Y t is the Brownian motion defined on G y with generator ∆ y . For f := f (t, z) = f (t, x, y) defined from R + × G onto C, we have easily the stochastic integral involving both discrete and continuous parts:

f (t, Z t ) = 0 t {(∂ t f )(s, Z s ) + (∆ z f )(s, Z s )} ds + I d ˆzf t
where the semi-discrete Itô integral writes as Martingale transforms. We are interested in martingale transforms allowing us to represent second order Riesz transforms. Let f (t, z) be a solution to the heat equation ∂ t -∆ z = 0. Fix T > 0 and Z 0 ∈ G. Then define

I d ˆzf t := I d ˆxf t + (I d y f ) t := 0 t d ˆxf (s, Z s -), dW ˆs T ˆX s * G x ×T ˆX s G x + 0 t d y f (s, Z s -),
∀0 t T , M t T ,Z0,f = f (T -t, Z t ).
This is a martingale since f (Tt) solves the backward heat equation ∂ t + ∆ z = 0, and we have in terms of stochastic integrals

M t f ,T ,Z0 = f (T -t, Z t ) = f (T , Z 0 ) + 0 t d ˆzf (T -s, Z s -), dZ s
Given A α the C (2m+n)×(2m+n) matrix defined earlier, we note M t α,f ,T ,Z0 the martingale transform

A α * M t f ,T ,Z0 defined as M t α,f ,T ,Z0 := f (T , Z 0 ) + 0 t A α ∇ ˆf (s, Z s-), dZ s = f (T , Z 0 ) + 0 t d ˆzf (T -s, Z s-)A α * , dZ s
where the first integral involves the L2 scalar product on T ˆzG × T ˆzG and the second integral involves the duality T ˆz * G × T ˆzG. In differential form:

dM t α,f ,T ,Z0 = A α ∇ ˆf (s, Z s -), dZ s = i=1 m ± α i x {(X i 2 f )(T -t, Z t-) d(N t i -λt) + (X i 0 f )(t, Z t-) dX t i } + j =1 n α j ,k y (X j f )(T -t, Z t-) dY t k
Quadratic covariation and subordination. We have the quadratic covariations (see Protter [START_REF] Protter | Stochastic integration and differential equations[END_REF], Dellacherie-Meyer [START_REF] Dellacherie | Probabilities and potential. B[END_REF], or Privault [START_REF] Privault | Notes on Stochastic Finance[END_REF])

d[N i -λt, N i -λt] t = dN t i d[N i -λt, X i ] t = τ Nt dN t i d[X i , X i ] t = dN t i d[Y j , Y j ] t = dt,
the other quadratic covariations being zero. For any two martingales M t f and M t g defined as above thanks to their respective heat extensions P t f et P t g, we have the quadratic covariations

d[M f , M g ] t = i=1 m (X i 2 f )(T -t, Z t-) (X i 2 g)(T -t, Z t-) d[N i -λt, N i -λt] t + i=1 m (X i 0 f )(T -t, Z t-) (X i 0 g)(T -t, Z t-) d[X i , X i ] t + i=1 m (X i 2 f )(T -t, Z t -) (X i 0 g)(T -t, Z t -) d[N i -λt, X i ] t + i=1 m (X i 0 f )(T -t, Z t-) (X i 2 g)(T -t, Z t-) d[X i , N i -λt] t + j =1 n (X j f )(T -t, Z t-) (X j g)(T -t, Z t-) d[Y j , Y j ] t = i=1 m { (X i + f ) (X i + g)(T -t, Z t-) 1(τ Nt = 1) + (X i -f ) (X i -g)(T -t, Z t-) 1(τ Nt = -1) } dN t i + (∇ y f , ∇ y g)(T -t, Z t -)
d[M α,f , M α,f ] t = i=1 m |α i x | 2 { (X i + f ) 2 (T -t, Z t-) 1(τ Nt = 1) + (X i -f ) 2 (T -t, Z t-) 1(τ Nt = -1) } dN t i + (A α y ∇ y f , A α y ∇ y f )(T -t, Z t

Proofs of the main results

Proof. (of Theorem 1)

The proof uses the well-known connection between martingale transforms and singular operators, through the use of projection operators. We refer to Gundy-Varopoulos [START_REF] Gundy | Les transformations de Riesz et les intégrales stochastiques[END_REF] as well as [3][4]. Following the same strategy, the random trajectories (B t ) -T t 0 defined on the band [-T , 0] × G by B t := (-t, Z t ), B -∞ = (T , Z -T ), -T t 0, Z -T ∈ G are replaced by random trajectories (B t ) -∞ t 0 defined on the upper half space R + × G B t := (-t, Z t ), B -∞ = (∞, Z -∞ ), -∞ t 0, Z -∞ ∈ G.

The latter are therefore trajectories starting at time t = -∞ from a point Z -∞ ∈ G chosen at random uniformly, and finishing at time t = 0, when hitting the boundary G of the upper half space. If f (t) = P t f is as in the previous section, then M t f = f (B t ), -∞ t 0 is a martingale and M t α,f its martingale transform as defined previously. The Gundy-Varopoulos approach adapted to second order Riesz transforms -see also [START_REF] Bañuelos | Martingale Transforms and Their Projection Operators on Manifolds[END_REF] yields the projection operator T α defined as ∀z ∈ G, (T α f )(z) := M 0 α,f Z 0 = z , so that using quadratic covariations as above, one observes that ∀g, (T α f , g) = -2

0 ∞ A α ∇ ˆPt f , ∇ ˆPt g L 2 G;T ˆG dt.
Thanks to Lemma 5, this means T α = R α 2 . Thanks to Wang's result [START_REF] Wang | Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities[END_REF] (see Theorem 6 and Lemma 7 above) we have easily

R α 2 f p = T α f p A α 2 (p * -1) f p ,
which concludes the proof of Theorem 1.

The proof of Theorem 2 follows exactly the same procedure. Recall Choi's result [START_REF] Choi | A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in L p (0, 1)[END_REF] for discrete martingales.

Theorem 8. (Choi, 1992) Let (Ω, (F ) n∈N , P) a probability space and X n an adapted real valued martingale. Let (α n ) n∈N be a predictable sequence taking values in [0, 1]. Let Y := α * X be the martingale transform of X defined for almost all ω ∈ Ω as Y 0 (ω) = X 0 (ω), and (Y n+1 -Y n )(ω) = α n (X n+1 -X n )(ω).

1 .

 1 The discrete component G x of the Lie group G is an abelian group 2. The connected component G y of the Lie group G is a compact Lie group Lemma 4. (Commutation relations) Assume the Hypothesis above. Then, we have

Lemma 5 . 2 (

 52 0 and as a consequence [Y j , ∆ x ] = 0 and [X i τ , ∆ y ] = 0. Moreover [Y j , ∆ y ] = 0 thanks to the existence of a biinvariant metric on the compact Lie group G y from which we chose the vector fields Y j . This yields [Y j , ∆ z ] = 0. Finally, since G x is abelian we have [X i τ , ∆ x ] = 0 and therefore [X i τ , ∆ z ] = 0. Assume the Hypothesis and the Commutation lemma above. Assume f and g in L 0

2 . 1 .

 21 dY s T ˆYs * Gy ×T ˆYs Gy Martingale transforms and quadratic covariations.

dt

  Differential subordination. Following Wang[START_REF] Wang | Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities[END_REF], given two adapted càdlàg Hilbert valued martingales X t and Y t , we say that Y t is differentially subordinate by quadratic variation toX t if |Y 0 | H |X 0 | H and [Y , Y ] t -[X ,X] t is nondecreasing nonnegative for all t. As a consequence, the estimate

  -) dt A ˆαshows that the martingale transform Y t := M t α is differentially subordinate (by quadratic variation) to the martingale X t := A ˆα 2 M t f . The following result of Wang[START_REF] Wang | Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities[END_REF]: Let X t and Y t be two adapted càdlàg Hilbert-valued martingales such that Y t is differentially subordinate by quadratic covariation to X t For 1 < p < ∞, Y t p (p * -1) X t p and the constant p * -1 is best possible. Strict inequality holds when 0 < X p < ∞ and p = / 2,

	Theorem 6. (Wang, 1995) implies in our situation that
	Lemma 7. Let M t f and M t α,f as defined above. We have
	∀t,	M t α,f	p	A ˆα	2 (p * -1) M t f	p .

N. Arcozzi, K. Domelevo, S. Petermichl

d[M f , M f ] t (3)

Sharp L p estimates for second order Riesz transforms on multiply-connected Lie groups

Then there exists a constant C p depending only on p such that Y p C p X p and the estimate is best possible.

The previous result from Choi is only for discrete martingales. For continuous-in-time martingales, we invoke Theorem 1.6 from the paper [START_REF] Banuelos | Martingales and sharp bounds for Fourier multipliers[END_REF], namely Lemma 9. (Banuelos-Osekowski, 2012) Let X t and Y t be two real-valued martingales satisfying

t for all t 0. Then for all 1 < p < ∞, we have Y p C p X p .

Proof. (of Theorem 2)

The result is now a corollary of Lemma 9 above with X t = M t f and

. It is not difficult to prove that the difference of quadratic variations above writes in terms of a jump part and a continuous part as

which is nonpositive since we assumed aI A α bI. This proves the result.

Bibliography