Yumeng Ou
email: yumeng_ou@brown.edu

Stefanie Petermichl
email: stefanie.petermichl@gmail.com

Elizabeth Strouse
email: elizabeth.strouse@math.u-bordeaux1.fr

Elizabeth Strouse Mixed

Bmo Hal

Mixed commutators and little product BMO

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A classical result of Nehari [START_REF] Nehari | On bounded bilinear forms[END_REF] shows that a Hankel operator with anti-analytic symbol b mapping analytic functions into the space of anti-analytic functions by f Þ Ñ P ´bf is bounded with respect to an L 2 norm if and only if the symbol belongs to BMO. This theorem has an equivalent formulation in terms of the boundedness of the commutator of the multiplication operator with symbol function b and the Hilbert transform rH, bs " Hb ´bH. To see this correspondence one rewrites the commutator as a sum of Hankel operators with orthogonal ranges.

Let H 2 pT 2 q denote the Banach space of analytic functions in L 2 pT 2 q. In [START_REF] Ferguson | Characterisations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures[END_REF], Ferguson and Sadosky study the symbols of bounded 'big' and 'little' Hankel operators on the bidisk. Big Hankel operators are those which project on to a 'big' subspace of L 2 pT 2 q -the orthogonal complement of H 2 pT 2 q; while little Hankel operators project onto the smaller subspace of complex conjugates of functions in H 2 pT 2 q -or anti-analytic functions. The corresponding commutators are rH 1 H 2 , bs, and rH 1 , rH 2 , bss where b " bpx 1 , x 2 q and H k are the Hilbert transforms acting in the k th variable. Ferguson and Sadosky show that the first commutator is bounded if and only if the symbol b belongs to the so called little BMO class, consisting of those functions that are uniformly in BMO in each variable separately. They also show that if b belongs to the product BMO space, as identified by Chang and Fefferman [START_REF] Chang | Some recent developments in Fourier analysis and H p -theory on product domains[END_REF], [START_REF]A continuous version of duality of H 1 with BMO on the bidisc[END_REF] then the second commutator is bounded. The fact that boundedness of the second commutator implies that b is in product BMO was shown in the groundbreaking paper of Ferguson and Lacey [START_REF] Ferguson | A characterisation of product BMO by commutators[END_REF]. The techniques to tackle this question in several parameters are very different and have brought valuable new insight and use to existing theories, for example in the interpretation of Journé's lemma [START_REF]A covering lemma for product spaces[END_REF] in combination with Carleson's example [START_REF] Carleson | A counterexample for measures bounded on H p spaces for the bidisk[END_REF]. Lacey and Terwilliger extended this result to an arbitrary number of iterates in [START_REF] Lacey | Hankel operators in several complex variables and product BMO[END_REF], requiring thus, among others, a refinement of Pipher's iterated multi-parameter version [START_REF] Pipher | Journé's covering lemma and its extension to higher dimensions[END_REF] of Journé's lemma.

When leaving the notion of Hankel operators behind, their interpretation as commutators allow for natural generalizations. Through the use of completely different real variable methods, Coifman, Rochberg and Weiss [START_REF] Coifman | Factorization theorems for Hardy spaces in several variables[END_REF] extended Nehari's one-parameter theory to real analysis in the sense that the Hilbert transforms were replaced by Riesz transforms. The missing features of the Riesz transforms include analytic projection on one hand as well as strong factorisation theorems of H 1 pDq on the other.

The authors in [START_REF] Coifman | Factorization theorems for Hardy spaces in several variables[END_REF] obtained sufficiency, i.e. that a BMO symbol b yields an L 2 pR d q bounded commutator for certain more general, convolution type singular integral operators. For necessity, they showed that the collection of Riesz transforms was representative enough:

}b} BMO À sup 1ďjďd }rR j , bs} 2Ñ2 .
Notably this lower bound was obtained somewhat indirectly through use of spherical harmonics in combination with the mean oscillation characterisation of BMO in one parameter.

These one-parameter results in [START_REF] Coifman | Factorization theorems for Hardy spaces in several variables[END_REF] were extended to the multi-parameter setting in [START_REF] Lacey | Multiparameter Riesz commutators[END_REF]. Both the upper and lower estimate have proofs very different from those in one parameter. For the lower estimate, the methods in [START_REF] Ferguson | A characterisation of product BMO by commutators[END_REF] or [START_REF] Lacey | Hankel operators in several complex variables and product BMO[END_REF] find an extension to real variables through operators closer to the Hilbert transform than the Riesz transforms and an indirect passage on the Fourier transform side.

In a recent paper [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF] it is shown that iterated commutators formed with any arbitrary Calderón-Zygmund operators are bounded if the symbol belongs to product BMO.

The first part of the present note is concerned with mixed Hankel operators or commutators such as rH 1 , rH 2 H 3 , bss.

We classify boundedness of these commutators by a mixed BMO class (little product BMO): those functions b " bpx 1 , x 2 , x 3 q so that bp¨, x 2 , ¨q and bp¨, ¨, x 3 q are uniformly in product BMO.

Similar results can be obtained for any finite iteration of any finite tensor product of Hilbert transforms.

The second part is concerned with a real variable analog or commutators of the form rR 1,j 1 , rR 2,j 2 R 3,j 3 , bss, where R k,j k are Riesz transforms of direction j k acting in the k th variable. We show necessity and sufficiency of the little product BMO condition when the R k,j k are allowed to run through all Riesz transforms by means of a two-sided estimate. Our argument works for all higher iterates and tensor products.

After reading the formal definition of little product BMO, Definition 2.2, the main twosided estimates are stated in Theorem 5.3.

It is a general fact that two-sided commutator estimates have an equivalent formulation in terms of weak factorization. We find the pre-duals of our little product BMO spaces and prove a corresponding weak factorization result.

Much like discussed in the base cases of our results [START_REF] Coifman | Factorization theorems for Hardy spaces in several variables[END_REF], [START_REF] Lacey | Multiparameter Riesz commutators[END_REF], boundedness of commutators involving Hilbert or Riesz transforms are a testing condition. If these commutators are bounded, the symbol necessarily belongs to a little product BMO. We then show that iterated commutators using a much more general class than that of tensor products of Riesz transforms are also bounded: commutators with paraproduct-free Journé operators.

We make some remarks about the strategy of the proof. In the Hilbert transform case, Toeplitz operators with operator symbol arise naturally. While Riesz transforms in R d are a good generalisation of the Hilbert transform, there is absence of analytic structure and tools relying on analytic projection or orthogonal spaces are not readily available. We overcome this difficulty through a first intermediate passage via tensor products of Calderón-Zygmund operators whose Fourier multiplier symbols are adapted to cones. This idea is inspired by [START_REF] Lacey | Multiparameter Riesz commutators[END_REF]. A class of operators of this type classifies little product BMO through two-sided commutator estimates, but it does not allow the passage to a classification through iterated commutators with tensor products of Riesz transforms. In a second step, we find it necessary to consider upper and lower commutator estimates using a well-chosen family of Journé operators that are not of tensor product type. These operators are constructed to resemble the multiple Hilbert transform. A two-sided estimate of iterated commutators involving operators of this family facilitates a passage to iterated commutators with tensor products of Riesz transforms. There is an increase in difficulty when the arising tensor products involve more than two Riesz transforms and when the dimension is greater than two.

The actual passage to the Riesz transforms requires for us to prove a stability estimate in commutator norms for certain multi-parameter singular integrals in terms of the mixed BMO class. In this context, we prove a qualitative upper estimate for iterated commutators using paraproduct free Journé operators. We make use of recent versions of T p1q theorems in this setting. These recent advances are different from the corresponding theorem of Journé [START_REF] Journé | Calderón-Zygmund operators on product spaces[END_REF]. The results we allude to have the additional feature to provide a convenient representation formula for bi-parameter in [START_REF] Martikainen | Representation of bi-parameter singular integrals by dyadic operators[END_REF] and even multi-parameter in [START_REF]Multi-parameter singular integral operators and representation theorem[END_REF] Calderón-Zygmund operators by dyadic shifts.

Aspects of Multi-Parameter Theory

This section contains some review on Hardy spaces in several parameters as well as some new definitions and lemmas relevant to us.

2.1. Chang-Fefferman BMO. We describe the elements of product Hardy space theory, as developed by Chang and Fefferman as well as Journé. By this we mean the Hardy spaces associated with domains like the poly-disk or R d :" Â t s"1 R ds for d " pd 1 , . . . , d t q. While doing so, we typically do not distinguish whether we are working on R d or T d . In higher dimensions, the Hilbert transform is usually replaced by the collections of Riesz transforms.

The (real) one-parameter Hardy space H 1 Re pR d q denotes the class of functions with the norm

d ÿ j"0 }R j f } 1
where R j denotes the j th Riesz transform or the Hilbert transform if the dimension is one.

Here and below we adopt the convention that R 0 , the 0 th Riesz transform, is the identity. This space is invariant under the one-parameter family of isotropic dilations, while the product Hardy space H 1

Re pR d q is invariant under dilations of each coordinate separately. That is, it is invariant under a t parameter family of dilations, hence the terminology 'multi-parameter' theory. One way to define a norm on

H 1 Re pR d q is }f } H 1 " ÿ 0ďj l ďd l } t â l"1 R l,j l f } 1 .
R l,j l is the Riesz transform in the j th l direction of the l th variable, and the 0 th Riesz transform is the identity operator.

The dual of the real Hardy space H 1 Re pR d q ˚is BMOpR d q, the t-fold product BMO space. It is a theorem of S.-Y. Chang and R. Fefferman [START_REF] Chang | Some recent developments in Fourier analysis and H p -theory on product domains[END_REF], [START_REF]A continuous version of duality of H 1 with BMO on the bidisc[END_REF] that this space has a characterization in terms of a product Carleson measure.

Define (1)

b BMOpR d q :" sup

U ĂR d ¨|U | ´1 ÿ RĂU ÿ εPsig d |xb, w ε R y| 2 '1{2 .
Here the supremum is taken over all open subsets U Ă R d with finite measure, and we use a wavelet basis w ε R adapted to rectangles R " Q 1 ˆ¨¨¨ˆQ t , where each Q l is a cube. The superscript ε reflects the fact that multiple wavelets are associated to any dyadic cube, see [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] for details. The fact that the supremum admits all open sets of finite measure cannot be omitted, as Carleson's example shows [START_REF] Carleson | A counterexample for measures bounded on H p spaces for the bidisk[END_REF]. This fact is responsible for some of the difficulties encountered when working with this space.

Theorem 2.1 (Chang, Fefferman). We have the equivalence of norms

}b} pH 1 Re pR d qq ˚" }b} BMOpR d q .
That is, BMOpR d q is the dual to H 1 Re pR d q.

This BMO norm is invariant under a t-parameter family of dilations. Here the dilations are isotropic in each parameter separately. See also [START_REF] Fefferman | Bounded mean oscillation on the polydisk[END_REF] and [START_REF]Harmonic analysis on product spaces[END_REF].

2.2. Little BMO. Following [START_REF] Cotlar | The Helson-Szegö theorem in L p of the bidimensional torus[END_REF] and [START_REF] Ferguson | Characterisations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures[END_REF], we review the space little BMO, often written as 'bmo', and its predual. A locally integrable function b : R d " R d 1 ˆ. . . ˆRds Ñ C is in bmo if and only if

}b} bmo " sup Q"Q 1 ˆ¨¨¨ˆQs | Q| ´1 ż Q |bp xq ´b Q | ă 8
Here the Q k are d k -dimensional cubes and b Q denotes the average of b over Q.

It is easy to see that this space consists of all functions that are uniformly in BMO in each variable separately. Let x v " px 1 , . . . ., x v´1 , ¨, x v`1 , . . . , x s q. Then bp x vq is a function in x v only with the other variables fixed. Its BMO norm in x v is

}bp x vq} BMO " sup Qv |Q v | ´1 ż Qv |bp xq ´bp x vq Qv |dx v
and the little BMO norm becomes

}b} bmo " max v tsup x v }bp x vq} BMO u. On the bi-disk, this becomes }b} bmo " maxtsup x 1 }bpx 1 , ¨q} BMO , sup x 2 }bp¨, x 2 q} BMO u,
the space discussed in [START_REF] Ferguson | Characterisations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures[END_REF]. Here, the pre-dual is the space H 1 pTq b L 1 pTq `L1 pTq b H 1 pTq. All other cases are an obvious generalisation, at the cost of notational inconvenience.

Little product BMO.

In this section we define a BMO space which is in between little BMO and product BMO. As mentioned in the introduction, we aim at characterising BMO spaces consisting for example of those functions bpx 1 , x 2 , x 3 q such that bpx 1 , ¨, ¨q and bp¨, ¨, x 3 q are uniformly in product BMO in the remaining two variables. Definition 2.2. Let b : R d Ñ C with d " pd 1 , ¨¨¨, d t q. Take a partition I " tI s : 1 ď s ď lu of t1, 2, ..., tu so that 9 Y 1ďsďl I s " t1, 2, ..., tu. We say that b P BMO I pR d q if for any choices v " pv s q, v s P I s , b is uniformly in product BMO in the variables indexed by v s . We call a BMO space of this type a 'little product BMO'. If for any x " px 1 , ..., x t q P R d , we define x v by removing those variables indexed by v s , the little product BMO norm becomes

}b} BMO I " max v tsup x v }bp x v q} BMO u
where the BMO norm is product BMO in the variables indexed by v s .

For example, when d " p1, 1, 1q " 1, when t " 3 and l " 2 with I 1 " p13q and I 2 " p2q, writing I " p13qp2q the space BMO p13qp2q pT 1 q arises, which consists of those functions that are uniformly in product BMO in the variables p1, 2q and p3, 2q respectively, as described above. Some other examples of little product BMOs are BMO p12qp34q and BMO p234qp1qp5q . The first space consists of those functions which are uniformly in product BMO in the variables p1, 3q, p1, 4q, p2, 3q and p2, 4q, and the second one consists of functions that are uniformly in product BMO in the variables p2, 1, 5q, p3, 1, 5q and p4, 1, 5q. Moreover, as degenerate cases, it is easy to see that BMO p12...tq and BMO p1qp2q...ptq are exactly little BMO and product BMO respectively, the spaces we are familiar with.

The little product BMO spaces are closely related to classes of iterated commutators of tensors of operators, a fact we will elaborate on throughout the paper.

Little product BMO spaces on T d can be defined in the same way. Now we find the predual of BMO p13qp2q , which is a good model for other cases. We choose the order of variables most convenient for us.

Theorem 2.3. The pre-dual of the space BMO p13qp2q pT 1 q is equal to the space

H 1
Re pT p1,1q q b L 1 pTq `L1 pTq b H 1 Re pT p1,1q q :" tf `g : f P H 1 Re pT p1,1q q b L 1 pTq and g P L 1 pTq b H 1 Re pT p1,1q qu. Proof. The space

H 1 Re pT p1,1q q b L 1 pTq " tf P L 1 pT 3 q : H 1 f, H 2 f, H 1 H 2 f P L 1 pT 3 qu equipped with the norm }f } " }f } 1 `}H 1 f } 1 `}H 2 f } 1 `}H 1 H 2 f } 1 is a Banach space.
Let W 1 " L 1 pT 3 q ˆL1 pT 3 q ˆL1 pT 3 q ˆL1 pT 3 q equipped with the norm

}pf 1 , f 2 , f 3 , f 4 q} W 1 " }f 1 } 1 `}f 2 } 1 `}f 3 } 1 `}f 4 } 1 .
Then we see that H 1 Re pT p1,1q q b L 1 pTq is isomorphically isometric to the closed subspace V " tpf, H 1 pf q, H 2 pf q, H 1 H 2 pf qq : f P H 1 pT p1,1q q b L 1 pTqu of W 1 . Now, the dual of W 1 is equal to W 8 " L 8 pT 3 q ˆL8 pT 3 q ˆL8 pT 3 q ˆL8 pT 3 q equipped with the norm }pg 1 , g 2 , g 3 , g 4 q} 8 " maxt}g i } 8 pi " 1, ¨¨¨4qu so the dual space of V is equal to the quotient of W 8 by the annihilator U of the subspace V in W 8 . But, using the fact that the Hilbert transforms are self-adjoint up to a sign change, we see that U " tpg 1 , g 2 , g 3 , g 4 q : g 1 `H1 g 2 `H2 g 3 `H1 H 2 g 4 " 0u

and so:

V ˚-W 8 {U -Impθq
where θpg 1 , g 2 , g 3 , g 4 q " g 1 `H1 g 2 `H2 g 3 `H1 H 2 g 4 since U " kerpθq. But

Impθq " L 8 pT 3 q `H1 pL 8 pT 3 qq `H2 pL 8 pT 3 qq `H1 pH 2 pL 8 pT 3 qqq is equal to the functions that are uniformly in product BMO in variables 1 and 2.

Using the same reasoning we see that the dual of L 1 pTq b H 1 Re pT p1,1q q is equal to L 8 pT 3 q H2 pL 8 pT 3 qq `H3 pL 8 pT 3 qq `H2 H 3 pL 8 pT 3 qq, which is equal to the space of functions that are uniformly in product BMO in variable 2 and 3. Now, we consider the 'L 1 sum' of the spaces H 1 Re pT p1,1q q b L 1 pTq and L 1 pTq b H 1 Re pT p1,1q q; that is M p13qp2q " tpf, gq : f P H 1 Re pT p1,1q q b L 1 pTq; g P L 1 pTq b H 1 Re pT p1,1q qu equipped with the norm }pf, gq} " }f } H 1 Re pT p1,1q qbL 1 pTq `}g} L 1 pTqbH 1 Re pT p1,1q q . We see that, if φ : M p13qp2q Ñ L 1 ppT 3 q is defined by φpf, gq " f `g, then the image of φ is isometrically isomorphic to the quotient of M p13qp2q by the space N " tpf, gq P M p13qp2q : f `g " 0u " tpf, ´f q : f P H 1 Re pT p1,1q qbL 1 pTqXL 1 pTqbH 1 Re pT p1,1q qu. Now, recall that the dual of the quotient M {N is equal to the annihilator of N. It is easy to see that the annihilator of N is equal to the set of ordered pairs pφ, φq with φ in the intersection of the duals of the two spaces. Thus the dual of the image of θ is equal to BMO p13qp2q . The norm of an element in the predual is equal to its norm as an element of the double dual which is easily computed.

Following this example, the reader may easily find the correct formulation for the predual of other little product BMO spaces as well those in several variables, replacing the Hilbert transform by all choices of Riesz transforms. For instance, one can prove that the predual of the space BMO p13qp2q pR d q is equal to H 1 Re pR pd 1 ,d 2 q q b L 1 pR d 3 q `L1 pR d 1 q b H 1 Re pR pd 2 ,d 3 q q.

The Hilbert transform case

In this section, we characterise the boundedness of commutators of the form rH 2 , rH 3 H 1 , bss as operators on L 2 pT 3 q. In the case of the Hilbert transform, this case is representative for the general case and provides a starting point that is easier to read because of the simplicity of the expression of products and sums of projection onto orthogonal subspaces. Its general form can be found at the beginning of Section 5.

Now let b P L 1 pT n q and let P and Q denote orthogonal projections onto subspaces of L 2 pT n q. We shall describe relationships between functions in the little product BMOs and several types of projection-multiplication operators. These will be Hilbert transform-type operators of the form P ´P K ; and iterated Hankel or Toeplitz type operators of the form Q K bQ (Hankel), P bP (Toeplitz),P Q K bQP (mixed), where b means the (not a priori bounded) multiplication operator M b on L 2 pT n q.

We shall use the following simple observation concerning Hilbert transform type operators again and again: Remark 3.1. If H " P ´P K and T : L 2 pT n q Ñ L 2 pT n q is a linear operator then rH, T s " 2P T P K ´2P K T P and H is bounded if and only if P T P K and P K T P are.

Proof.

pP ´P K qT ´T pP ´P K q " pP ´P K qT pP `P K q ´pP `P K qT pP ´P K q " 2P T P K ´2P K T P.

The observation on boundedness comes from the fact that H is expressed as the sum of operators with orthogonal ranges.

We state the main result of this section. Theorem 3.2. Let b P L 1 pT 3 q. Then the following are equivalent with linear dependence on the respective norms (1) b P BMO p13qp2q (2) The commutators rH 2 , rH 1 , bss and rH 2 , rH 3 , bss are bounded on L 2 pT 3 q (3) The commutator rH 2 , rH 3 H 1 , bss is bounded on L 2 pT 3 q. Corollary 3.3. We have the following two-sided estimate }b} BMO p13qp2q À }rH 2 , rH 3 H 1 , bss} L 2 pT 3 qÑL 2 pT 3 q À }b} BMO p13qp2q .

Before we proceed with the proof of Theorem 3.2, we are going to need some notation and some lemmas. It will be useful to denote by Q 13 orthogonal projection on the subspace of functions which are either analytic or anti-analytic in the first and third variables; Q 13 " P 1 P 3 `P K 1 P K 3 . Then the projection Q K 13 onto the orthogonal of this subspace is defined by Q K 13 " P K 1 P 3 `P1 P K 3 . We reformulate properties (2) and (3) in the statement of Theorem 3.2 in terms of Hankel Toeplitz type operators. Lemma 3.4. We have the following algebraic facts on commutators and projection operators.

(1) The commutators rH 2 , rH 1 , bss and rH 2 , rH 3 , bss are bounded on L 2 pT 3 q if and only if Proof. Using Remark 3.1 it is easy to see that

P i P 2 bP K i P K 2 ,
rH 2 , rH 1 , bss " 4 `pP 2 P 1 bP K 1 P K 2 ´P2 P K 1 bP 1 P K 2 q ´pP K 2 P 1 bP K 1 P 2 ´P K 2 P K 1 bP 1 P 2 q
ȃnd that the corresponding equation for rH 2 , rH 3 , bss is also true. This, along with the observation that the ranges of all arising summands are mutually orthogonal, so that these operators are bounded if and only if each one of their summands is, gives assertion (1). To prove (2) we just notice that H 1 H 3 " Q 13 ´QK 13 is one of our Hilbert transform type operators which permits us to calculate:

rH 2 , rH 3 H 1 , bss " 4 `pP 2 Q 13 bQ K 13 P K 2 ´P2 Q K 13 bQ 13 P K 2 q ´pP K 2 Q 13 bQ K 13 P 2 ´P K 2 Q 13 bQ K 13 P 2 q
ȃnd conclude that the commutator is bounded if and only if the iterated Hankels are.

The other tools that we need in order to prove p3 q ñ p2 q. in Theorem 3.2 are Lemmas 3.5 and 3.6, Toeplitz type Lemmas which use the fact that norms of Toeplitz operators are determined by the L 8 norms of their symbols. Lemma 3.5. Let W 3 be the operator of multiplication by z 3 , W 3 pf q " z 3 f , acting on L 2 pT 3 q. If we define B " P K 1 P K 2 bP 1 P 2 as well as

A n " W ˚n 3 pP 3 P K 1 P K 2 bP 1 P 2 P 3 qW n 3 and C n " W n 3 pP K 3 P K 1 P K 2 bP 1 P 2 P K 3 qW ˚n 3
as operators acting on L 2 pT 3 q then the sequences A n and C n converge to B in the strong operator topology.

The proof below is a generalization of the Brown-Halmos classical proof that, if S is the unilateral shift and T b is the Toeplitz operator with symbol b on L 2 pTq, then S ˚nT b S n converges in the strong operator topology to M b the multiplication operator with symbol b [START_REF] Brown | Algebraic properties of Toeplitz operators[END_REF]. We sketch the argument adapted to our situation.

Proof. The two assertions are symmetric, we prove the fact for A n . It is easy to see that W 3 , W 3 ; and P 3 commute with P 1 , P 2 , P K 1 and P K 2 . The multiplier b satisfies the equation W ˚n 3 bW n 3 " b and W n 3 W ˚n 3 " Id. So we see that

A n " P K 1 P K 2 pW ˚n 3 P 3 W n 3 qbP 1 P 2 pW ˚n 3 P 3 W n 3 q.
But if f P L 2 pT 3 q, then, since W n 3 is a unitary operator:

}W ˚n 3 P 3 W n 3 pf q ´f } " }P 3 W n 3 pf q ´W n 3 pf q} " }pP 3 ´IqpW n 3 qpf q} Ñ 0 pn Ñ 8q,
as tail of a convergent Fourier series. This means that

W ˚n 3 P 3 W n 3 : L 2 pT 3 q Ñ L 2 pT 3 q
converges to the identity in the strong operator topology. Thus, for each f P L 2 pT 3 q we have

}pA n ´Bqpf q} ď }pP K 1 P K 2 pW ˚n 3 P 3 W n 3 ´IdqbP 1 P 2 W ˚n 3 P 3 W n 3 qpf q} `}pP K 1 P K 2 bP 1 P 2 pW ˚n 3 P 3 W n 3 ´Idqqpf q} Ñ 0 pn Ñ 8q Lemma 3.6. }P 3 P K 1 P K 2 bP 1 P 2 P 3 } L 2 ÑL 2 " }P K 1 P K 2 bP 1 P 2 } L 2 ÑL 2 .
Proof. The inequality ď is trivial, since P 3 is a projection which commutes with P K 1 and P K 2 . By Lemma 3.5,

}P K 1 P K 2 bP 1 P 2 } ď sup nPN }W ˚n 3 pP 3 P K 1 P K 2 bP 1 P 2 P 3 qW n 3 } ď }P 3 P K 1 P K 2 bP 1 P 2 P 3 },
using again the fact that the operators W 3 and W 3 are isometries on L 2 pT 3 q. Now, we are ready to proceed with the proof of the main theorem of this section.

Proof. (of Theorem 3.2) We show p1 q ô p2 q and p2 q ô p3 q p1 q ô p2 q Consider f " f px 1 , x 2 q and g " gpx 3 q. Observe that rH 2 , rH 1 , bsspf gq " g rH 2 , rH 1 , bsspf q. So }rH 2 , rH 1 , bsspf gq} 2 L 2 pT 3 q " }F g} 2 L 2 pTq where F px 2 q " }rH 2 , rH 1 , bsspf q} L 2 pT 2 q . The map g Þ Ñ F g has L 2 pTq operator norm }F } 8 . Now change the roles of x 1 and x 3 . The Ferguson-Lacey equivalences }rH 2 , rH i , bss} " }b} BMO give the desired result.

p2 q ñ p3 q We show that boundedness of the commutators rH 2 , rH 1 , bss and rH 2 , rH 3 , bss implies the boundedness of the mixed commutator rH 2 , rH 1 H 3 , bss. Indeed, the boundedness of the mixed commutator is equivalent to the boundedness of the four operators listed in Lemma 3.4. By direct calculation we can transform, for example, the first term P 2 Q 13 bQ K 13 P K into the sum of four terms with orthogonal range:

P 2 Q 13 bQ K 13 P K 2 "P 3 pP 1 P 2 bP K 1 P K 2 qP 3 `P K 3 pP K 1 P 2 bP 1 P K 2 qP K 3 `P1 pP 2 P 3 bP K 2 P K 3 qP 1 `P K 1 pP 2 P K 3 bP K 2 P 3 qP K 1 .
Notice that each term of this sum is either of the form P i T P i or of the form P K i T P K i where T is one of the eight operators from Lemma 3.4 which are bounded because we assume (2). Clearly }P i T P i } ď }T } and }P K i T P K i } ď }T } and so P 2 Q 13 bQ K 13 P K 2 is bounded. Analogous equations hold for the other four operators of (3) and so rH 2 , rH 3 H 1 , bss is bounded.

p3 q ñ p2 q This part relies on Lemma 3.6, which in turn relies on Lemma 3.5. We wish to conclude from the boundedness of rH 2 , rH 3 H 1 , bss the boundedness of rH 2 , rH 1 , bss and rH 2 , rH 3 , bss. To see boundedness of rH 2 , rH 1 , bss, let us look at one of the Hankels from Lemma 3.4. Lemma 3.6 shows that P K 2 P K 1 bP 2 P 1 is bounded if and only if the operator

P 3 P K 1 P K 2 bP 1 P 2 P 3 is.
And the latter is an operator found in the list from part (2) of Lemma 3.4. The analogous reasoning shows that all eight Hankels in 3.4 are bounded and so 2 is proved.

Real variables: upper bounds

In this section, we are interested in upper bounds for commutator norms by means of little product BMO norms of the symbol. In the case of the Hilbert transform, we have seen that these estimates, even in the iterated case, are straightforward. Other streamlined proofs exist for Hilbert or Riesz transforms when considering dyadic shifts of complexity one, see [START_REF] Petermichl | Dyadic Shifts and a Logarithmic Estimate for Hankel Operators with Matrix Symbol[END_REF], [START_REF] Petermichl | Why the Riesz transforms are averages of the dyadic shifts?[END_REF] and [START_REF]Iterated Riesz commutators: a simple proof of boundedness[END_REF]. When considering more general Calderón-Zygmund operators, the arguments required are more difficult, in each case. The first classical upper bound goes back to [START_REF] Coifman | Factorization theorems for Hardy spaces in several variables[END_REF], where an estimate for one-parameter commutators with convolution type Calderón-Zygmund operators is given. Next, the text [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] includes a technical estimate for the multi-parameter case for such Calderón-Zygmund operators with a high enough degree of smoothness. This smoothness assumption was removed in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF] thanks to an approach using the representation formula for Calderón-Zygmund operators by means of infinite complexity dyadic shifts [START_REF] Hytönen | The sharp weighted bound for general Calderón-Zygmund operators[END_REF]. This last proof also gives a control on the norm of the commutators which dependends on the Calderón-Zygmund norm of the operators themselves, a fact we will employ later. Below, we prove two generalizations to the mixed commutator setting. The first theorem is weaker and targets the tensor product case only. It will be relevant to the L p case in Section 7. The second theorem gives an estimate by little product BMO when the Calderón-Zygmund operators are of Journé type and cannot be written as a tensor product. While this estimate is interesting in its own right, it is also essential for our characterisation result, the lower estimate, in section 5. We start with the tensor product case, which is an almost direct consequence of [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF].

Theorem 4.1. Let d " pd 1 , . . . , d t q and I " pI s q 1ďsďl a partition of t1, . . . , tu. For each 1 ď s ď l let us consider the tensor product of Calderón-Zygmund operators T s " Â kPIs T k where T k acts on functions defined on the k th variable. Let the symbol b belong to the little product BMO space BMO I pR d q. Then,

}r T 1 , . . . r T l , bs . . .s} L 2 pR d qý À }b} BMO I pR d q ,
where the underlying constant depends only on dimensions and }T k } CZ .

The main ingredient of the proof is the upper bound theorem in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF]. Moreover, the oneparameter result (without iteration) can be similarly obtained, which we omit.

Proof. For convenience of notation, we provide the proof in the case R pd 1 ,d 2 ,d 3 q with partition I consisting of I 1 " p1q and I 2 " p23q and the commutator r T 1 , r T 2 , bss " rT 1 , rT 2 T 3 , bss only. The argument is simple and generalizes easily. By the fact that rT 2 T 3 , bs " T 2 rT 3 , bs`rT 2 , bsT 3 , one has

rT 1 , rT 2 T 3 , bss " rT 1 , T 2 rT 3 , bss `rT 1 , rT 2 , bsT 3 s " T 2 rT 1 , rT 3 , bss `rT 1 , rT 2 , bssT 3 .
We consider the first term only, the second term can be dealt with in the same way.

}T 2 rT 1 , rT 3 , bssf } 2 L 2 pR d q À }rT 1 , rT 3 , bssf } 2 L 2 pR d q " ż R d 2 ˆżR d 1 ˆRd 3 |rT 1 , rT 3 , bssf | 2 dx 1 dx 3 ˙dx 2 ď ż R d 2 }rT 1 , rT 3 , bp¨, x 2 , ¨qss} 2 L 2 pR d 1 ˆRd 3 qý }f p¨, x 2 ¨q} 2 L 2 pR d 1 ˆRd 3 q dx 2 À ż R d 2 }bp¨, x 2 , ¨q} 2 BMOpR d 1 ˆRd 3 q }f p¨, x 2 , ¨q} 2 L 2 pR d 1 ˆRd 3 q dx 2 ď }b} 2 BMO I pR d q }f } 2 L 2 pR d q .
The implied constants depend upon the L 2 norm of T 2 in the first inequality and the Calderón-Zygmund norms of T 1 and T 3 in the fourth line. Indeed, for any fixed x 2 , by the main theorem of [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF], }rT 1 , rT 3 , bp¨, x 2 , ¨qss} L 2 pR d 1 ˆRd 3 qý À }bp¨, x 2 , ¨q} BMOpR d 1 ˆRd 3 q with the implied constant depending only on the dimensions and }T 1 } CZ , }T 3 } CZ . The last inequality follows from }bp¨, x 2 , ¨q} BMOpR d 1 ˆRd 3 q ď }b} BMO I pR d q for all x 2 .

The first generation of multi-parameter singular integrals that are not of tensor product type goes back to Fefferman [START_REF]Singular integrals on product domains[END_REF] and was generalised by Journé in [START_REF] Journé | Calderón-Zygmund operators on product spaces[END_REF] to the non-convolution type in the framework of his T p1q theorem in this setting. Much later, Journé's T p1q theorem was revisited, for example in [START_REF] Pott | A T(1) theorem on product spaces[END_REF], [START_REF] Martikainen | Representation of bi-parameter singular integrals by dyadic operators[END_REF], [START_REF] Ou | A T pbq theorem on product spaces[END_REF], [START_REF]Multi-parameter singular integral operators and representation theorem[END_REF]. See also [START_REF] Martikainen | Some obstacles in characterising the boundedness of biparameter singular integrals[END_REF] for some difficulties related to this subject. The references [START_REF] Martikainen | Representation of bi-parameter singular integrals by dyadic operators[END_REF] in the bi-parameter case and [START_REF]Multi-parameter singular integral operators and representation theorem[END_REF] in the general multiparameter case include a representation formula by means of adapted, infinite complexity dyadic shifts. While these representation formulae look complicated, they have a feature very useful to us. Locally' they look as if they were of tensor product type, a feature we will exploit in the argument below. We start with the simplest bi-parameter case with no iterations and make comments about the generalization at the end of this section.

The class of bi-parameter singular integral operators treated in this section is that of any paraproduct free Journé type operator (not necessarily a tensor product and not necessarily of convolution type) satisfying a certain weak boundedness property, which we define as follows:

Definition 4.2. A continuous linear mapping T : C 8 0 pR n qbC 8 0 pR m q Ñ rC 8 0 pR n qbC 8 0 pR m qs 1 is called a paraproduct free bi-parameter Calderón-Zygmund operator if the following conditions are satisfied:

1. T is a Journé type bi-parameter δ-singular integral operator, i.e. there exists a pair pK 1 , K 2 q of δCZ-δ-standard kernels so that, for all f 1 , g 1 P C 8 0 pR n q and f 2 ,

g 2 P C 8 0 pR m q, xT pf 1 b f 2 q, g 1 b g 2 y " ż f 1 py 1 qxK 1 px 1 , y 1 qf 2 , g 2 yg 1 px 1 q dx 1 dy 1 when sptf 1 X sptg 1 " H; xT pf 1 b f 2 q, g 1 b g 2 y " ż f 2 py 2 qxK 2 px 2 , y 2 qf 1 , g 1 yg 2 px 2 q dx 2 dy 2
when sptf 2 X sptg 2 " H. 2. T satisfies the weak boundedness property |xT pχ I b χ J q, χ I b χ J y| À |I||J|, for any cubes

I Ă R n , J P R m .
3. T is paraproduct free in the sense that T p1 b ¨q " T p¨b 1q " T ˚p1 b ¨q " T ˚p¨b 1q " 0.

Recall that δCZ-δ-standard kernel is a vector valued standard kernel taking values in the Banach space consisting of all Calderón-Zygmund operators. It is easy to see that an operator defined as above satisfies all the characterizing conditions in Martikainen [START_REF] Martikainen | Representation of bi-parameter singular integrals by dyadic operators[END_REF], hence is L 2 bounded and can be represented as an average of bi-parameter dyadic shift operators together with dyadic paraproducts. Moreover, since T is paraproduct free, one can conclude from observing the proof of Martikainen's theorem, that all the dyadic shifts in the representation are cancellative.

The base case from which we pass to the general case below, is the following:

Theorem 4.3.
Let T be a paraproduct free bi-parameter Calderón-Zygmund operator, and b be a little bmo function, there holds }rb, T s} L 2 pR n ˆRm qý À }b} bmopR n ˆRm q , where the underlying constant depends only on the characterizing constants of T .

Proof. According to the discussion above, for any sufficiently nice functions f, g, one has the following representation:

(

2) xT f, gy " CE ω 1 E ω 2 8 ÿ i 1 ,j 1 "0 8 ÿ i 2 ,j 2 "0 2 ´maxpi 1 ,j 1 q 2 ´maxpi 2 ,j 2 q xS i 1 j 1 i 2 j 2 f, gy.
where expectation is with respect to a certain parameter of the dyadic grids. The dyadic shifts S i 1 j 1 i 2 j 2 are defined as

S i 1 j 1 i 2 j 2 f :" ÿ K 1 PD 1 ÿ I 1 ,J 1 ĂK 1 ,I 1 ,J 1 PD 1 ℓpI 1 q"2 ´i1 ℓpK 1 q ℓpJ 1 q"2 ´j1 ℓpK 1 q ÿ K 2 PD 2 ÿ I 2 ,J 2 ĂK 2 ,I 2 ,J 2 PD 2 ℓpI 2 q"2 ´i2 ℓpK 2 q ℓpJ 2 q"2 ´j2 ℓpK 2 q a I 1 J 1 K 1 I 2 J 2 K 2 xf, h I 1 b h I 2 yh J 1 b h J 2 ": ÿ K 1 pi 1 ,j 1 q ÿ I 1 ,J 1 ĂK 1 ÿ K 2 pi 2 ,j 2 q ÿ I 2 ,J 2 ĂK 2 a I 1 J 1 K 1 I 2 J 2 K 2 xf, h I 1 b h I 2 yh J 1 b h J 2 .
The coefficients above satisfy a

I 1 J 1 K 1 I 2 J 2 K 2 ď ? |I 1 ||J 1 ||I 2 ||J 2 | |K 1 ||K 2 |
, which also guarantees that

}S i 1 j 1 i 2 j 2 } L 2 ÑL 2 ď 1.
Moreover, since T is paraproduct free, all the Haar functions appearing above are cancellative.

It thus suffices to show that for any dyadic grids D 1 , D 2 and fixed i 1 , j 1 , i 2 , j 2 P N, one has

(3) }rb, S i 1 j 1 i 2 j 2 sf } L 2 À p1 `maxpi 1 , j 1 qqp1 `maxpi 2 , j 2 qq}b} bmo }f } L 2 ,
as the decay factor 2 ´maxpi 1 ,j 1 q , 2 ´maxpi 2 ,j 2 q in (2) will guarantee the convergence of the series.

To see (3), one decomposes b and a L 2 test function f using Haar bases:

rb, S i 1 j 1 i 2 j 2 sf " ÿ

I 1 ,I 2 ÿ J 1 ,J 2 xb, h I 1 b h I 2 yxf, h J 1 b h J 2 yrh I 1 b h I 2 , S i 1 j 1 i 2 j 2 sh J 1 b h J 2 .
A similar argument to that in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF] implies that rh

I 1 b h I 2 , S i 1 j 1 i 2 j 2 sh J 1 b h J 2 is nonzero only if I 1 Ă J pi 1 q 1 or I 2 Ă J pi 2 q
2 , where J pi 1 q 1 denotes the i 1 -th dyadic ancestor of J 1 , similarly for J pi 2 q 2 . Hence, the sum can be decomposed into three parts:

I 1 Ă J pi 1 q 1 and I 2 Ă J pi 2 q 2 (regular), I 1 Ă J pi 1 q 1 and I 2 Ľ J pi 2 q 2 , I 1 Ľ J pi 1 q 1 and I 2 Ă J pi 2 q 2 (mixed).

1) Regular case:

Following [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF] one can decompose the arising sum into sums of classical bi-parameter dyadic paraproducts B 0 pb, f q and its slightly revised version B k,l pb, f q: for any integers k, l ě 0, B k,l is the bi-parameter dyadic paraproduct defined as

B k,l pb, f q " ÿ I,J β IJ xb, h I pkq b u J plq yxf, h ǫ 1 I b u ǫ 2 J yh ǫ 1 1 I b u ǫ 1 2 J |I pkq | ´1{2 |J plq | ´1{2 ,
where β IJ is a sequence satisfying |β IJ | ď 1. When k ą 0, all Haar functions in the first variable are cancellative, while when k " 0, there is at most one of h ǫ 1 I , h

ǫ 1 1
I being noncancellative. The same assumption goes for the second variable. Observe that when k " l " 0, B k,l becomes the classical paraproduct B 0 . It is proved in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF] that

}B k,l pb, f q} L 2 À }b} BMO }f } L 2
with a constant independent of k, l and the product BMO norm on the right hand side.

Then since little bmo functions are contained in product BMO, this part can be controlled. More specifically, write rb, S i 1 j 1 i 2 j 2 sf " ÿ

I 1 ,I 2 ÿ J 1 ,J 2 xb, h I 1 b h I 2 yxf, h J 1 b h J 2 yh I 1 b h I 2 S i 1 j 1 i 2 j 2 ph J 1 b h J 2 q ´ÿ I 1 ,I 2 ÿ J 1 ,J 2 xb, h I 1 b h I 2 yxf, h J 1 b h J 2 yS i 1 j 1 i 2 j 2 ph I 1 h J 1 b h I 2 h J 2 q ": I `II,
then one can estimate term I and II separately. According to the definition of dyadic shifts, term I is equal to ÿ

J 1 ,J 2 ÿ I 1 :I 1 ĂJ pi 1 q 1 ÿ I 2 :I 2 ĂJ pi 2 q 2 xb, h I 1 b h I 2 yxf, h J 1 b h J 2 yh I 1 b h I 2 ÿ J 1 1 :J 1 1 ĂJ pi 1 q 1 ℓpJ 1 1 q"2 i 1 ´j1 ℓpJ 1 q ÿ J 1 2 :J 1 2 ĂJ pi 2 q 2 ℓpJ 1 2 q"2 i 2 ´j2 ℓpJ 2 q a J 1 J 1 1 J pi 1 q 1 J 2 J 1 2 J pi 2 q 2 h J 1 1 b h J 1 2 ‹ ‹ ‹ ' " ÿ K 1 ,K 2 pi 1 q ÿ J 1 :J 1 ĂK 1 pi 2 q ÿ J 2 :J 2 ĂK 2 ÿ I 1 :I 1 ĂK 1 ÿ I 2 :I 2 ĂK 2 xb, h I 1 b h I 2 yxf, h J 1 b h J 2 yh I 1 b h I 2 pj 1 q ÿ J 1 1 :J 1 1 ĂK 1 pj 2 q ÿ J 1 2 :J 1 2 ĂK 2 a J 1 J 1 1 K 1 J 2 J 1 2 K 2 h J 1 1 b h J 1 2 ' " ÿ I 1 ,I 2 xb, h I 1 b h I 2 yh I 1 b h I 2 ÿ K 1 ĄI 1 K 2 ĄI 2 ¨pi 1 ,j 1 q ÿ J 1 ,J 1 1 ĂK 1 pi 2 ,j 2 q ÿ J 2 ,J 1 2 ĂK 2 a J 1 J 1 1 K 1 J 2 J 1 2 K 2 xf, h J 1 b h J 2 yh J 1 1 b h J 1 2 ' " ÿ I 1 ,I 2 xb, h I 1 b h I 2 yh I 1 b h I 2 ÿ J 1 1 :J 1pj 1 q 1 ĄI 1 ÿ J 1 2 :J 1pj 2 q 2 ĄI 2 xS i 1 j 1 i 2 j 2 f, h J 1 1 b h J 1 2 yh J 1 1 b h J 1 2 .
Because of the supports of Haar functions, the inner sum above can be further decomposed into four parts, where

I " ÿ I 1 ,I 2 ÿ J 1 1 ĽI 1 ÿ J 1 2 ĽI 2 , II " ÿ I 1 ,I 2 ÿ J 1 1 ĽI 1 ÿ J 1 2 :J 1 2 ĂI 2 ĂJ 1pj 2 q 2
III " ÿ

I 1 ,I 2 ÿ J 1 1 :J 1 1 ĂI 1 ĂJ 1pj 1 q 1 ÿ J 1 2 ĽI 2 , IV " ÿ I 1 ,I 2 ÿ J 1 1 :J 1 1 ĂI 1 ĂJ 1pj 1 q 1 ÿ J 1 2 :J 1 2 ĂI 2 ĂJ 1pj 2 q 2 .
Hence, similarly as in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF], one has

I " ÿ I 1 ,I 2 xb, h I 1 b h I 2 yxS i 1 j 1 i 2 j 2 f, h 1 J 1 1 b h 1 J 1 2 yh I 1 b h I 2 |I 1 | ´1{2 |I 2 | ´1{2 ,
which is a bi-parameter paraproduct B 0 pb, f q. Moreover, one has

II " ÿ I 1 ,I 2 xb, h I 1 b h I 2 yh I 1 b h I 2 ÿ J 1 2 :J 1 2 ĂI 2 ĂJ 1pj 2 q 2 xS i 1 j 1 i 2 j 2 f, h 1 I 1 b h J 1 2 y|I 1 | ´1{2 h J 1 2 " j 2 ÿ l"0 ÿ I 1 ,J 1 2 β J 1 2 xb, h I 1 b h J 1plq 2 yxS i 1 j 1 i 2 j 2 f, h 1 I 1 b h J 1 2 yh I 1 b h J 1 2 |I 1 | ´1{2 |J 1plq 2 | ´1{2 " j 2 ÿ l"0 B 0l pb, S i 1 j 1 i 2 j 2 f q,
where constants β J 1 2 P t1, ´1u, and B 0l are the generalized bi-parameter paraproducts of type p0, lq defined in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF] whose L 2 Ñ L 2 operator norm is uniformly bounded by }b} BMO product BMO. Similarly, one can show that III "

j 1 ÿ k"0 B k0 pb, S i 1 j 1 i 2 j 2 f q, IV " j 1 ÿ k"0 j 2 ÿ l"0 B kl pb, S i 1 j 1 i 2 j 2 f q.
Since }b} BMO À }b} bmo , all the forms above are L 2 bounded. This completes the discussion of term I.

To get an estimate of term II, we need to decompose it into finite linear combinations of S i 1 j 1 i 2 j 2 pB kl pb, f qq. By linearity, one can write S i 1 j 1 i 2 j 2 on the outside from the beginning, and we will only look at the inside sum. Recall that in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF], one splits for example the sum regarding the first variable into three parts:

I 1 Ĺ J 1 , I 1 " J 1 , J 1 Ĺ I 1 Ă J pi 1 q
1 . If we split the second variable as well, there are nine mixed parts, and it's not hard to show that each of them can be represented as a finite sum of B kl pb, f q. We omit the details.

2) Mixed case. Let's call the second and the third 'mixed' parts, and as the two are symmetric, it suffices to look at the second one, i.e.

I 1 Ă J pi 1 q 1 , I 2 Ľ J pi 2 q
2 . In the first variable, we still have the old case I 1 Ă J pi 1 q 1 that appeared in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF], so morally speaking, we only need to nicely play around with the stronger little bmo norm to handle the second variable. For any fixed I 1 , J 1 , I 2 , J 2 , since I 2 Ľ J pi 2 q 2 , the definition of dyadic shifts implies that

h I 1 b h I 2 S i 1 j 1 i 2 j 2 ph J 1 b h J 2 q " h I 1 S i 1 j 1 i 2 j 2 ph J 1 b h I 2 h J 2 q and S i 1 j 1 i 2 j 2 ph i 1 h J 1 b h I 2 h J 2 q " h I 2 S i 1 j 1 i 2 j 2 ph I 1 h J 1 b h J 2 q.
Hence, we still have cancellation in the second variable, which converts the mixed case to ÿ

I 1 ĂJ pi 1 q 1 ÿ I 2 ĽJ pi 2 q 2 xb, h I 1 b h I 2 yxf, h J 1 b h J 2 yrh I 1 , S i 1 j 1 i 2 j 2 sph J 1 b h I 2 h J 2 q " ÿ I 1 ĂJ pi 1 q 1 ÿ J 2 xf, h J 1 b h J 2 yrh I 1 , S i 1 j 1 i 2 j 2 sph J 1 b ÿ I 2 ĽJ pi 2 q 2 xb, h I 1 b h I 2 yh I 2 h J 2 q " ÿ I 1 ĂJ pi 1 q 1 ÿ J 2 xf, h J 1 b h J 2 yrh I 1 , S i 1 j 1 i 2 j 2 sph J 1 b xb, h I 1 b h 1 J pi 2 q 2 yh 1 J pi 2 q 2 h J 2 q " ÿ I 1 ĂJ pi 1 q 1 ÿ J 2 xb, h I 1 b h 1 J pi 2 q 2 y|J pi 2 q 2 | ´1{2 xf, h J 1 b h J 2 yrh I 1 , S i 1 j 1 i 2 j 2 sph J 1 b h J 2 q " ÿ I 1 ĂJ pi 1 q 1 ÿ J 2 xxby J pi 2 q 2 , h I 1 y 1 xf, h J 1 b h J 2 yrh I 1 , S i 1 j 1 i 2 j 2 sph J 1 b h J 2 q,
where xby J pi 2 q 2 denotes the average value of b on J pi 2 q 2 , which is a function of only the first variable.

In the following, we will once again estimate the first term and second term of the commutator separately, and the L 2 norm of each of them will be proved to be bounded by }b} bmo }f } L 2 . a) First term. By definition of the dyadic shift, the first term is equal to ÿ

I 1 ĂJ pi 1 q 1 ÿ J 2 xxby J pi 2 q 2 , h I 1 y 1 h I 1 xf, h J 1 b h J 2 yÿ J 1 1 ĂJ pi 1 q 1 ℓpJ 1 1 q"2 i 1 ´j1 ℓpJ 1 q ÿ J 1 2 ĂJ pi 2 q 2 ℓpJ 1 2 q"2 i 2 ´j2 ℓpJ 2 q a J 1 J 1 1 J pi 1 q 1 J 2 J 1 2 J pi 2 q 2 h J 1 1 b h J 1 2 ‹ ‹ ‹ ' ,
which by reindexing K 1 :" J pi 1 q 1

is the same as

ÿ I 1 ,J 2 xxby J pi 2 q 2 , h I 1 y 1 h I 1 ÿ K 1 :K 1 ĄI 1 pi 1 q ÿ J 1 ĂK 1 pj 1 q ÿ J 1 1 ĂK 1 pj 2 q ÿ J 1 2 ĂJ pi 2 q 2 a J 1 J 1 1 K 1 J 2 J 1 2 J pi 2 q 2 xf, h J 1 b h J 2 yh J 1 1 b h J 1 2 " ÿ I 1 ,J 2 xxby J pi 2 q 2 , h I 1 y 1 h I 1 ÿ J 1 1 :J 1pj 1 q 1 ĄI 1 h J 1 1 b xS i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 q, h J 1 1 y 1 ,
where the inner sum is the orthogonal projection of the image of xf, h J 2 y 2 b h J 2 under S i 1 j 1 i 2 j 2 onto the span of th J 1 1 u such that J

1pj 1 q 1 Ą I 1 .
Taking into account of the supports of the Haar functions in the first variable, one can further split the sum into two parts where

I :" ÿ J 2 ÿ I 1 ĹJ 1 1 , II :" ÿ J 2 ÿ J 1 1 ĂI 1 ĂJ 1pj 1 q 1 . Summing over J 1 1 first implies that I " ÿ J 2 ÿ I 1 xxby J pi 2 q 2 , h I 1 y 1 h I 1 `h1 I 1 b xS i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 q, h 1 I 1 y 1 ": ÿ J 2 B 0 pxby J pi 2 q 2 , S i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 qq
where B 0 pb, f q :" ř I xb, h I yxf, h 1 I yh I |I| ´1{2 is a classical one-parameter paraproduct in the first variable. Note that the L 2 norm of it is bounded by }b} BMO }f } L 2 . Moreover, according to the definition of S i 1 j 1 i 2 j 2 , for any fixed J 2

S i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 q " ÿ J 1 2 :J 1pj 2 q 2 "J pi 2 q 2 xS i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 q, h J 1 2 y 2 b h J 1 2 .
In other words, S i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 q only lives on the span of th J 1 2 : J 1pj 2 q 2 " J pi 2 q 2 u. Hence, by linearity there holds

I " ÿ J 2 ÿ J 1 2 :J 1pj 2 q 2 "J pi 2 q 2 B 0 `xby J pi 2 q 2 , xS i 1 j 1 i 2 j 2 pxf, h J 2 y 2 b h J 2 q, h J 1 2 y 2 ˘b h J 1 2 " ÿ J 1 2 ¨B0 `xby J 1pj 2 q 2 , xS i 1 j 1 i 2 j 2 p ÿ J 2 :J pi 2 q 2 "J 1pj 2 q 2 xf, h J 2 y 2 b h J 2 q, h J 1 2 y 2 ˘' b h J 1 2 .
Thus, orthogonality in the second variable implies that

}I} 2 L 2 pR n ˆRm q " ÿ J 1 2 }B 0 `xby J 1pj 2 q 2 , xS i 1 j 1 i 2 j 2 p ÿ J 2 :J pi 2 q 2 "J 1pj 2 q 2 xf, h J 2 y 2 b h J 2 q, h J 1 2 y 2 ˘}2 L 2 pR n q À ÿ J 1 2 }xby J 1pj 2 q 2 } 2 BMOpR n q }xS i 1 j 1 i 2 j 2 p ÿ J 2 :J pi 2 q 2 "J 1pj 2 q 2 xf, h J 2 y 2 b h J 2 q, h J 1 2 y 2 } 2 L 2 pR n q .
As the second term by linearity is the same as

S i 1 j 1 i 2 j 2 ¨ÿ J 2 ÿ I 1 ĂJ pi 1 q 1 xxby J pi 2 q 2 , h I 1 y 1 xf, h J 1 b h J 2 yh I 1 h J 1 b h J 2 ',
the L 2 Ñ L 2 boundedness of the shift implies that it suffices to estimate the L 2 norm of the term inside the parentheses. Since I 1 X J 1 ‰ H, one can further split the sum into two parts:

I :" ÿ J 2 ÿ I 1 ĹJ 1 , II :" ÿ J 2 ÿ J 1 ĂI 1 ĂJ pi 1 q 1 .
Summing over J 1 first implies that

I " ÿ J 2 ÿ I 1 xxby J pi 2 q 2 , h I 1 y 1 xf, h 1 I 1 b h J 2 yh I 1 h 1 I 1 b h J 2 ": ÿ J 2 B 0 pxby J pi 2 q 2 , xf, h J 2 y 2 q b h J 2 ,
where B 0 pb, f q :" ř I xb, h I yxf, h 1 I yh I |I| ´1{2 is a classical one-parameter paraproduct in the first variable. Hence,

}I} 2 L 2 pR n ˆRm q " ÿ J 2 }B 0 pxby J pi 2 q 2 , xf, h J 2 y 2 q} 2 L 2 pR n q À ÿ J 2 }xby J pi 2 q 2 } 2 BMOpR n q }xf, h J 2 y 2 } 2 L 2 pR n q ď }b} 2 bmo ÿ J 2 }xf, h J 2 y 2 } 2 L 2 pR n q " }b} 2 bmo }f } 2 L 2 pR n ˆRm q .
For part II, note that it can be decomposed as

II " i 1 ÿ k"0 ÿ J 2 ÿ J 1 xxby J pi 2 q 2 , h J pkq 1 y 1 xf, h J 1 b h J 2 yh J pkq 1 h J 1 b h J 2 " i 1 ÿ k"0 ÿ J 2 ÿ J 1 β J 1 ,k |J pkq 1 | ´1{2 xxby J pi 2 q 2 , h J pkq 1 y 1 xxf, h J 2 y 2 , h J 1 y 1 h J 1 b h J 2 ": i 1 ÿ k"0 ÿ J 2 B k pxby J pi 2 q 2 , xf, h J 2 y 2 q b h J 2 ,
where coefficients β J 1 ,k P t1, ´1u and the L 2 norm of the generalized paraproduct B k is uniformly bounded as mentioned before. Therefore, a same argument as for part I shows that }II} L 2 pR n ˆRm q À p1 `i1 q}b} bmo }f } L 2 pR n ˆRm q , which completes the discussion of the second term, and thus proves that the mixed case is bounded.

The upper bound result we just proved can be extended to R d , to arbitrarily many parameters and an arbitrary number of iterates in the commutator. To do this, consider multiparameter singular integral operators studied in [START_REF]Multi-parameter singular integral operators and representation theorem[END_REF], which satisfy a weak boundedness property and are paraproduct free, meaning that any partial adjoint of T is zero if acting on some tensor product of functions with one of the components being 1. And consider a little product BMO function b P BMO I pR d q. One can then prove Theorem 4.4. Let us consider R d , d " pd 1 , . . . , d t q with a partition I " pI s q 1ďsďl of t1, . . . , tu as discussed before. Let b P BMO I pR d q and let T s denote a multi-parameter paraproduct free Journé operator acting on function defined on  kPIs R d k . Then we have the estimate below }rT 1 , . . . rT l , bs . . .s} L 2 pR d qý À }b} BMO I pR d q .

The part of the proof that targets the Journé operators proceeds exactly the same as the bi-parameter case with the multi-parameter version of the representation theorem proven in [START_REF]Multi-parameter singular integral operators and representation theorem[END_REF]. Certainly, as the number of parameters increases, more mixed cases will appear. However, if one follows the corresponding argument above for each variable in each case, it is not hard to check that eventually, the boundedness of the arising paraproducts is implied exactly by the little product BMO norm of the symbol. The difficulty of higher iterates is overcome in observing that the commutator splits into commutators with no iterates, as was done in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF]. We omit the details.

The assumption that the operators be paraproduct free is sufficient for us. The general case is currently under investigation by one of the authors. Important to our arguments for lower bounds with Riesz transforms is the corollary below, which follows from the control on the norm of the estimate in Theorem 4.4 by an application of triangle inequality. It is a stability result for characterising families of Journé operators. Corollary 4.5. Let for every 1 ď s ď l be given a collection T s " tT s,js u of paraproduct free Journé operators on  kPIs R d k that characterise BM O I pR d q via a two-sided commutator estimate }b} BMO I pR d q À sup j }rT 1,j 1 , . . . rT l,j l , bs . . .s} L 2 pR d qý À }b} BMO I pR d q .

Then there exists ε ą 0 such that for any family of paraproduct free Journé operators T 1 s " tT 1 s,js u with characterising constants }T 1 s,js } CZ ď ε, the family tT s,js `T 1 s,js u still characterises BM O I pR d q.

Real variables: lower bounds

In this section, we are again in R d . It is our aim to prove the following characterisation theorem of the space BMO I pR d q.

Theorem 5.1. The following are equivalent with linear dependence in the respective norms.

(1) b P BMO I pR d q (2) All commutators of the form rR k 1 ,j k 1 , . . . , rR k l ,j k l , bs . . .s are bounded in L 2 pR d q where k s P I s and R ks,j ks is the one-parameter Riesz transform in direction j ks . (3) All commutators of the form r R 1, j p1q , . . . , r R l, j plq , bs . . .s are bounded in L 2 pR d q where j psq " pj k q kPIs , 1 ď j k ď d k and the operators R s, j psq are a tensor product of Riesz transforms R s, j psq " Â kPIs R k,j k .

Corollary 5.2. Let j " pj 1 , . . . , j t q with 1 ď j k ď d k and let for each 1 ď s ď l, j psq " pj k q kPIs be associated a tensor product of Riesz transforms R s, j psq " Â kPIs R k,j k ; here R k,j k are j th k Riesz transforms acting on functions defined on the k th variable. We have the two-sided estimate

}b} BM O I pR d q À sup j }r R 1, j p1q , . . . , r R t, j ptq , bs . . .s} L 2 pR d qý À }b} BM O I pR d q .
Such two-sided estimates also hold in L p for 1 ă p ă 8. Remarks will be made in section 7. From the inductive nature of our arguments, it will also be apparent that the characterisation holds when we consider intermediate cases, meaning commutators with any fixed number of Riesz transforms in each iterate. Below we state our most general two-sided estimate through Riesz transforms.

Theorem 5.3. Let 1 ă p ă 8. Under the same assumptions as Corollary 5.2 and for any fixed n " pn s q where 1 ď n s ď |I s |, we have the two-sided estimate }b} BMO I pR d q À sup j }r R 1, j p1q , . . . , r R l, j plq , bs . . .s} L p pR d qý À }b} BMO I pR d q where j psq " pj k q kPIs , 0 ď j k ď d k and for each s, there are n s non-zero choices. A Riesz transform in direction 0 is understood as the identity.

For p " 2 and n " 1 this is the equivalence (1)ô ( 2) and for n " p|I 1 |, . . . , |I l |q it is the equivalence (1)ô (3) from Theorem 5.1.

The statements above also serve as the statement of the general case for products of Hilbert transforms. Infact, when any d k " 1 just replace the Riesz transforms by the Hilbert transform in that variable. In this section, we consider the case d k ě 2 for 1 ď k ď t and thus iterated commutators with tensor products of Riesz transforms only. The special case when d k " 1 for some k is easier but requires extra care for notation, which is why we omit it here.

The proof in the Hilbert transform case relied heavily on analytic projections and orthogonal spaces, a feature that we do not have when working with Riesz transforms. We are going to simulate the one-dimensional case by a two-step passage via intermediary Calderón-Zygmund operators whose multiplier symbols are adapted to cones.

In dimension d ě 2, a cone C Ă R d with cubic base is given by the data pξ, Qq where ξ P S d´1 is the direction of the cone and the cube Q Ă ξ K centered at the origin is its aperture. The cone consists of all vectors θ that take the form pθ ξ ξ, θ K q where θ ξ " xθ, ξy and θ K P θ ξ Q. By λC we mean the dilated cone with data pξ, λQq.

A cone D with ball base has data pξ, rq for 0 ă r ă π{2 and ξ P S d´1 and consists of the vectors tη P R d : dpξ, η{}η}q ď ru where d is the geodesic distance (with distance of antipodal points being π.)

Given any cone C or D, we consider its Fourier projection operator defined via x P C f " 1 C f . When the apertures are cubes, such operators are combinations of Fourier projections onto half spaces and as such admit uniform L p bounds. Among others, this fact made cubic cones necessary in the considerations in [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] and [START_REF] Dalenc | A lower bound criterion for iterated commutators[END_REF] that we are going to need. For further technical reasons in the proof of an error term in an underlying lower estimate in [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] and [START_REF] Dalenc | A lower bound criterion for iterated commutators[END_REF], these operators are not quite good enough, mainly because they are not of Calderón-Zygmund type. For a given cone C, consider a Calderón-Zygmund operator T C with a kernel K C whose Fourier symbol x K C P C 8 and satisfies the estimate

1 C ď x K C ď 1 p1`τ qC
. This is accomplished by mollifying the symbol 1 C of the cone projection associated to cone C on S d´1 and then extending radially. We use the same definition for T D .

Given a collection of cones C " pC k q we denote by T C , P C the corresponding tensor product operators.

In [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] it has been proven that Calderón-Zygmund operators adapted to certain cones of cubic aperture classify product BMO via commutators. As part of the argument, it was observed that test functions with opposing Fourier supports made the commutator large. This observation reduced the terms arising in the commutator to those resembling Hankel operators. In [START_REF] Dalenc | A lower bound criterion for iterated commutators[END_REF] a refinement was proven, that will be helpful to us. We prefer to work with cones with round base. Lower bounds for such commutators can be deduced from the assertion of the main theorem in [START_REF] Dalenc | A lower bound criterion for iterated commutators[END_REF], but we need to preserve the information on the Fourier support of the test function that makes the commutators large in order to succeed with our argument. Information on this test function replaces the property that the Hilbert transform is a combination of orthogonal projections of two orthogonal half spaces. It is instrumental to our argument. We have the following lemma, very similar to that in [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] and [START_REF] Dalenc | A lower bound criterion for iterated commutators[END_REF], the only difference being that the cones are based on balls instead of cubes. Lemma 5.4. For every parameter 1 ď k ď t there exist a finite set of directions Υ k P S d k

´1

and an aperture 0 ă r k ă π{2 so that for every b, there exist cones D k " Dpξ k , r k q with ξ k P Υ k as well as a normalized test function f " Â t k"1 f k whose components have Fourier support in the opposing cones Dp´ξ k , r k q so that }rT 1,D 1 ..., rT t,Dt , bs...sf } 2 Á }b} BMO p1q...ptq pR d q .

The stress is on the fact that the collection is finite, somewhat specific and serves all admissible product BMO functions.

Proof. The lemma in [START_REF] Dalenc | A lower bound criterion for iterated commutators[END_REF] supplies us with the sets of directions Υ k as well as cones of cubic aperture Q k and a test function f supported in the opposing cones. Now choose the aperture r k large enough so that p1 `τ qCpξ k , Q k q Ă Dpξ k , r k q. Then we have the commutator estimate }rT 1,D 1 ..., rT t,Dt , bs...sf } 2 Á }b} BMO p1q...ptq pR d q .

In fact, both commutators with cones C and D are L 2 bounded and reduce to }T D pbf q} 2 or }T C pbf q} 2 respectively thanks to the opposing Fourier support of f . Observe that

T C pbf q " T D pT C pbf qq " T C pT D pbf qq. With }T C } 2Ñ2 ď 1, we see that }T D pbf q} 2 ě }T C pbf q} 2 .
Using this a priori lower estimate, we are going to prove the lemma below.

Lemma 5.5. Let us suppose we are in R d with a partition I " pI s q. For every 1 ď k ď t there exists a finite set of directions Υ k Ă S d k ´1 and an aperture r k so that the following hold for all b P BMO I pR d q :

(1) For every 1 ď s ď l there exists a coordinate v s P I s and a direction ξ vs P Υ vs and so that with the choice of cone D vs " Dpξ vs , r vs q and arbitrary D k for coordinates k P I s ztv s u and their tensor product D s we have }rT 1, D 1 , . . . , rT l, D l , bs . . .s} 2Ñ2 Á }b} BMO I ,

(2) The test function f " Â t k"1 which gives us a large L 2 norm in (1) has Fourier supports of the f k contained in Dp´ξ k , r k q when k " v s and in D k else.

Before we can begin with the proof of Lemma 5.5, we will need a real variable version of the facts on Toeplitz operators used earlier.

Lemma 5.6. Let D k for 1 ď k ď t denote any cones with respect to the k th variable. Let T D k denote the adapted Calderón-Zygmund operators. Let K be any proper subset of tk : 1 ď k ď tu, let D K " Â kPK D k and T D K the associated tensor product of Calderón-Zygmund operators. Let P σ D K be a tensor product of projection operators on cones Dpξ k , r k q or opposing cones Dp´ξ k , r k q. Let j R K. Then

}T D K T D j bP σ D K P D j } L 2 pR d qý " }T D K bP σ D K } L 2 pR d qý .
Proof. As in the Hilbert transform case, we will establish this by composing some unilateral shift operators and studying their Fourier transform in the j variable. Let ξ j denote the direction of the cone D j , for any l define the shift operator S l gpx j q " ż R d j ĝpη j qe 2πiplξ j `ηj qx j dη j . S l is a translation operator on the Fourier side along the direction ξ j of the cone D j . It is not hard to observe that S l " S ´l. Now define

A l " S ´lT D K T D j bP σ D K P D j S l , and B " T D K bP σ D K .
We will prove that as l Ñ `8, A l Ñ B in the strong operator topology. As in the argument in Lemma 3.6, this together with the fact that S l is an isometry will complete the proof. To see the convergence, let's first remember that S l only acts on the j variable, and one always has the identities S l S ´l " Id and S ´lbS l " b.

This implies

A l " T D K pS ´lT D j S l qpS ´lbS l qP σ D K pS ´lP D j S l q " T D K pS ´lT D j S l qbP σ D K pS ´lP D j S l q.
We claim that both S ´lT D j S l and S ´lP D j S l converge to the identity operator in the strong operator topology, which then imply that A l Ñ B as l Ñ 8. Next we will only prove S ´lT D j S l Ñ Id as the second limit is almost identical. Observe that }S ´lT D j S l f ´f } " }pT D j ´IqS l f }. Given any L 2 function f and any fixed large l ě 0. Consider the f with frequencies supported in R d 1 ˆ. . . ˆpD j ´lξ j q ˆ. . . ˆRdt . In this case, S l f has Fourier support in R d 1 ˆ. . . ˆDj ˆ. . . ˆRdt where the symbol of T D j equals 1. Thus, for such f , we have S ´lT D j S l f " f . The sets R d 1 ˆ. . . ˆpD j ´lξ j q ˆ. . . ˆRdt exhaust the frequency space. With }T D j ´I} 2Ñ2 ď 1 the operators S ´lT D j S l converge to the Identity in the strong operator topology, and the lemma is proved. Observe that the aperture of the cone D j is not relevant to the proof.

We proceed with the proof of the lower estimate for cone transforms.

Proof. (of Lemma 5.5) For a given symbol b P BMO I , there exist for all 1 ď s ď l coordinates v " pv s q, v s P I s and a choice of variables not indexed by v s , x 0 v so that up to an arbitrarily small error }b} BMO I " }bp x 0 v q} BMO pv 1 q...pv l q . By Lemma 5.4, there exist cones D vs " Dpξ vs , r vs q with directions ξ vs P Υ vs and a normalized test function f H in variables v s with opposing Fourier support such that we have the lower estimate }rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bp x 0 v qs . . .spf H q} L 2 pR dv q Á }bp x 0 v q} BMO pv 1 q...pv l q where R dv " R dv 1 ˆ. . . ˆRdv l .

We now consider the commutator with the same cones but with full symbol b " bp¨, . . . , ¨q. Due to the lack of action on the variables not indexed by v s , in the commutator, we again have rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .spf H gq " g ¨rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .spf H q for g that only depends upon variables not indexed by v s . Again using that multiplication operators in L 2 have norms equal to the L 8 norm of their symbol, for the 'worst' L 2 -normalized g we have

}rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .spf H gq} L 2 pR d q " sup x v }rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bp x 0 v qs . . .spf H q} L 2 pR dv q
ě }rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bp x 0 v qs . . .spf H q} L 2 pR dv q Á }bp x 0 v q} BMO pv 1 q...pv l q pR dv q " }b} BMO I pR d q .

Note that the test function g can be chosen with well distributed Fourier transform. Take any cones in the variables not indexed by v s and let D denote the tensor product of their projections. f T " P D g. Notice that }rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .spf H f T q} L 2 pR d q Á }rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .spf H gq} L 2 pR d q with constants depending upon smallness of the aperture of the chosen cones. Notice that the test function f :" f H f T has the Fourier support as required in part (2) of the statement of Lemma 5.5. Now build cones D s from the D vs and the other chosen cones D k as well as operators T s, Ds . Notice that the commutators rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .s and rT 1, D 1 , . . . , rT l, D l , bs . . .s reduce significantly when applied to a test function f with Fourier support as ours. When the operators T vs,Dv s or any tensor product T s, Ds fall directly on f , the contribution is zero due to opposing Fourier supports of the test function and the symbols of the operators. The only terms left in the commutators rT 1, D 1 , . . . , rT l, Dv l , bs . . .spf q and rT v 1 ,Dv 1 , . . . , rT v l ,Dv l , bs . . .spf q have the form  s T s, Ds pbf q and  s T vs,Dv s pbf q respectively. By repeated use of Lemma 5.6 we have the operator norm estimates for any symbol b, valid on the subspace of functions with Fourier support as described for f : }  s T s, Ds b} 2Ñ2 " }  s T vs,Dv s b} 2Ñ2 . We conclude the existence of a normalized test function f with Fourier support as described in the statement (2) of Lemma 5.5 so that }  s T s, Ds pbf q} 2 Á }b} BMO I pR d q . We get the desired estimate

}rT 1, D 1 , . . . , rT l, D l , bs . . .s} L 2 pR d qý Á }b} BMO I pR d q ,
and test function f as desired.

It seems not possible to pass directly to a lower commutator estimate in tensor products of Riesz transforms from that in tensor products of cone operators. Just using tensor products of operators adapted to cones merely gives us some lower bound where we are unable to control that a Riesz transform does appear in every variable such as required in (3) of Theorem 5.1. The reason for this will become clear as we advance in the argument. Instead of using operators T s, Ds directly, we will build upon them more general multi-parameter Journé type cone operators not of tensor product type that we now describe.

Let us explain the multiplier we need for i copies of S d´1 when all dimensions are the same. We will explain how to pass to the case of i copies of varying dimension d k below. A picture illustrating a base case can be found at the end of this section.

For 0 ă b ă a ă 1, let ϕ : r´1, 1s Ñ r´1, 1s be a smooth function with ϕpxq " 1 when a ď x ď 1 and ϕpxq " 0 when b ě x ě 0. And let ϕ be odd, meaning antisymmetric about t " 0. The function ϕ gives rise to a zonal function with pole ξ 1 on the first copy of S d´1 , denoted by C 1 pξ 1 ; η 1 q. This is the multiplier of a one-parameter Calderón-Zygmund operator adapted to a cone Dpξ 1 , rq for r " π{2p1 ´aq. For i ą 1 we define C k pξ 1 , . . . , ξ k ; η 1 , . . . , η k q for 1 ă k ď i inductively. In what follows, expectation is taken with respect to traces of surface measure. When η i " ˘ξi , then conditional expectation is over a one-point set.

C k pξ 1 , . . . , ξ k ; η 1 , . . . , η k q " E a k´1 pC k´1 pξ 1 , . . . , a k´1 ; η 1 , . . . , η k´1 q | dpa k´1 , ξ k´1 q " dpη k , ξ k qq.

If the dimensions are not equal take d " max d j and imbed S d j ´1 into S d´1 by the map ξ " pξ 1 , . . . , ξ d j q Þ Ñ pξ 1 , . . . , ξ d j , 0, . . . , 0q. Obtain in this manner the function C i and then restrict to the original number of variables when the dimension is smaller than d.

The multiplier C i p ξ; ¨q gives rise to a multi-parameter Calderón-Zygmund operator of convolution type (but not of tensor product type), T ξ " T C i p ξ;¨q . Infact, it is defined through principal value convolution against a kernel K ξ px 1 , . . . , x i q such that @l :

ż αă|x l |ăβ K ξ px 1 , . . . , x i qdx l " 0, @0 ă α ă β, x j P R d j fixed @j ‰ l, ˇˇˇB | n| Bx n 1 1 . . . Bx n i i K ξ px 1 , . . . , x i q ˇˇˇď A n |x 1 | ´d1 ´n1 . . . |x i | ´di ´ni , n j ě 0.
This kind of operator is a special case of the more general, non-convolution type discussed in Section 4. It has many other nice features that will facilitate our passage to Riesz transforms. One of them is its very special representation in terms of homogeneous polynomials, the other one a lower estimate in terms of the BMO I norm.

Lemma 5.7. Let C i be a multiplier in  i k"1 R d k be as described above, with any fixed direction and aperture. Let m be an integer of order d " max d k . For any δ ą 0, the function C i has an approximation by a polynomial C N i in the

ś i k"1 d k variables t ś k η k,j k | 1 ď k ď i, 1 ď j k ď d k u so that }C i ´CN i } C m pS d k ´1q ă δ in each variable separately.
C m indexes the norm of uniform convergence in functions that are m times continuously differentiable. On the space side, C N i corresponds to an operator that is a polynomial in Riesz transforms of the variables

 k R k,j k .
Lemma 5.8. We are in R d with partition I " pI s q 1ďsďl . Let Υ consist of vectors ξ " pξ k q t k"1 with ξ k P Υ k . Let Υ psq consist of ξ psq " pξ k q kPIs . Let us consider the class of Journé type cone multipliers D s " C is p ξ psq ; ¨q of aperture r s with associated multi-parameter Calderón-Zygmund operators T s,Ds . Then we have the two-sided estimate }b} BMO I pR d q À sup ξP Υ }rT 1,D 1 , . . . , rT l,D l , bs . . .s} L 2 pR d qý À }b} BMO I pR d q .

In order to proceed with the proof of these lemmas, we will need some well known facts about spherical harmonics and zonal harmonics in particular, that we now review briefly. Fix a pole ξ P S d´1 . The zonal harmonic with pole ξ of degree n is written as Z pnq ξ pηq. With t " xξ, ηy P r´1, 1s, one writes Z pnq ξ pηq " P n ptq where P n is the Legendre polynomial of degree n. It is common to suppress the dependence on d in the notation for Z pnq ξ and P n . The Rodrigues formula with a specific weight useful for a change of variables on the sphere S d´1 is P n ptq " p´1q n 2 n pn `pd ´3q{2q n p1 ´t2 q p3´dq{2 ˆB Bt ˙n p1 ´t2 q n`pd´3q{2 .

Here pkq n denotes the descending factorial kpk ´1q . . . pk ´n `1q. Writing Ω d´1 for the measure of S d´1 , we have orthogonality by the formulas

0 " ż S d´1 Z pnq ξ pηqZ pmq ξ pηqdσ d´1 " Ω d´1 ż 1
´1 P n ptqP m ptqp1 ´t2 q pd´3q{2 dt and normalization ż 1 ´1 P 2 n ptqp1 ´t2 q pd´3q{2 dt " Ω d´1 N pd, nqΩ d´2 where N pd, nq is the dimension of the space of homogeneous harmonic polynomials of degree n in d variables. We have P n p1q " 1, P n p´1q " p´1q n P n p1q and |P n ptq| ď 1 for t P r´1, 1s. Z ϕpxξ, ηyqY pnq pηqdσ p´1 " Ω d´2 Y pnq pξq ż 1 ´1 ϕptqP n ptqp1 ´t2 q pd´3q{2 dt.

The addition theorem for Legendre polynomials says that for any complete system tY pnq k u of spherical harmonics of degree n:

(6) Z pnq ξ pηq " Ω d´1 N pd, nq N pd,nq ÿ k"1 Y pnq k pξqY pnq k pηq.
When Y pnq is harmonic and homogeneous of degree n with Y pnq pξq " 1 and Y pnq pRηq " Y pnq pηq for any rotation R P Opdq with Rξ " ξ, then Y pnq " Z pnq ξ , the zonal harmonic of degree n with pole ξ.

The lemma below will aid us in understanding the special form of the functions C i .

Lemma 5.9. Let ξ 1 , ξ 2 P S d´1 . We have

E a 1 pZ pnq η 1 pa 1 q | dpξ 1 , a 1 q " dpξ 2 , η 2 qq " E a 2 pZ pnq η 2 pa 2 q | dpξ 2 , a 2 q " dpξ 1 , η 1 qq " Z pnq ξ 1 pη 1 qZ pnq ξ 2 pη 2 q.
Proof. The first equality is a change of variable, thanks to symmetry of the zonal harmonic in its variables and invariance with respect to action of the measure preserving elements of the orthogonal group fixing poles ξ 1 or ξ 2 , that we now detail. By a rotation in one of the spheres, assume ξ 1 " ξ 2 " ξ. Take a small ball B ξ,η 1 pa 0 2 ; ε 2 q " ta 2 : dpa 2 , a 0 2 q ă ε 2 u X ta 2 : dpa 2 , ξq " dpη 1 , ξqu.

Note ta 2 : dpa 2 , ξq " dpη 1 , ξqu " S d´2 . Every a 2 P B ξ,η 1 pa 0 2 ; ε 2 q gives rise to a canonical orthogonal map σ a 2 along geodesics in a scaled copy of S d´2 . Lifted to S d´1 , these are orthogonal maps fixing ξ. Let σ 0 fix ξ and map a 0 2 to η 1 . Let a 0 1 " σ 0 pη 2 q. We observe that tσ 0 σ a 2 pη 2 q : a 2 P B ξ,η 1 pa 0 2 ; ε 2 qu " B ξ,η 2 pa 0 1 ; ε 1 q with ε 1 so that Ppdpa 2 , a 0 2 q ă ε 2 | dpξ, a 2 q " dpξ, η 1 qq " Ppdpa 1 , a 0 1 q ă ε 1 | dpξ, a 1 q " dpξ, η 2 qq.

Together with the symmetry and the rotation property Z pnq η paq " Z pnq a pηq " Z pnq σpaq pσpηqq, we obtain the first equality.

For fixed a 1 , the function Z pnq η 1 pa 1 q " Z pnq a 1 pη 1 q is a function harmonic in R d , n-homogeneous. These properties are preserved when taking expectation in a 1 . So the expression EpZ pnq η 1 pa 1 q | dpξ 1 , a 1 q " dpξ 2 , η 2 qq remains harmonic (regarded as a function in R d ), n-homogeneous. From the form EpZ pnq η 2 pa 2 q | dpξ 2 , a 2 q " dpξ 1 , η 1 qq we learn that its restriction to S d´1 depends only upon dpξ 1 , η 1 q. This implies that it is a constant multiple of the zonal harmonic with pole ξ 1 . Exchanging the roles of η 1 and η 2 gives

EpZ pnq η 1 pa 1 q | dpξ 1 , a 1 q " dpξ 2 , η 2 qq " c n Z pnq ξ 1 pη 1 qZ pnq ξ 2 pη 2 q.
When assuming the normalization Z pnq ξ pξq " 1 then c n " 1. This is a gernalisation of the classical symmetrising of the cosinus sum formula 1{2pcospx`yq´cospx´yqq " cospxq cospyq.

Let us now proceed with the proof of one of our lemmas Proof. (of Lemma 5.7) It is well known that zonal harmonic series have convergence properties when representing smooth zonal functions similar to that of the Fourier transform. For any given m and sufficiently smooth ϕ of the type described above, then

C 1 pξ 1 ; η 1 q " ÿ n ϕ n Z pnq ξ 1 pη 1 q
where the convergence is C m -uniform. The degree of smoothness required for ϕ to obtain convergence in the C m in the above expression depends upon m and the dimension d. For our purpose, we choose m ě d. The coefficients ϕ n are explicit by formula 5.

Let us denote this function's representation of degree N by a series of zonal harmonics by C pN q

1 pξ 1 ; η 1 q. C pN q 1 pξ 1 ; η 1 q " ÿ nďN ϕ n Z pnq ξ 1 pη 1 q.
For every δ ą 0 there exists N so that we have the estimate

}C pN q 1 pξ 1 ; η 1 q ´C1 pξ 1 ; η 1 q} C m pS d 1 ´1q ă δ.
In the case of i copies of spheres, we define inductively in the same manner as for C i . Let us for the moment make all dimensions equal by the imbedding discussed above.

C pN q

k pξ 1 , . . . , ξ k ; η 1 , . . . , η k q " E a k´1 pC pN q k´1 pξ 1 , . . . , a k´1 ; η 1 , . . . , η k´1 q | dpa k´1 , ξ k´1 q " dpη k , ξ k qq

We claim the identity ( 7)

C pN q i p ξ; η 1 , η 2 , . . . , η i q " ÿ nďN ϕ n i ź k"1 Z pnq ξ k pη k q.
This is trivially true for i " 1. For i ą 1 induct on the number of parameters using Lemma 5.9:

C pN q i p ξ; η 1 , . . . , η i q " E a i´1 pC i´1 pξ 1 , ξ 2 , . . . , a i´1 ; η 1 , . . . , η i´1 q | dpa i´1 , ξ i´1 q " dpη i , ξ i qq " E a i´1 ˜ÿ nďN ϕ n i´1 ź k"1 Z pnq ξ k pη k q | dpa i´1 , ξ i´1 q " dpη i , ξ i q " ÿ nďN ϕ n i´2 ź k"1 Z pnq ξ k pη k qE a i´1 pZ pnq ξ i´1 | dpa i´1 , ξ i´1 q " dpη i , ξ i qq " ÿ nďN ϕ n i ź k"1 Z pnq ξ k pη k q.
The first equality is the definition of C pN q i

, the second one is the induction hypothesis and the last an application of Lemma 5.9.

It follows that neither C i nor C pN q i depend on the order chosen in their definition and

C i p ξ; η 1 , . . . , η i q " ÿ n ϕ n i ź k"1 Z pnq ξ k pη k q
where the convergence is in C m in each variable.

Next, we study the terms arising in multipliers of the form C pN q i

. When all dimensions are equal, indeed, ś i k"1 Z pnq ξ k pη k q has the important property that, as a product of n-homogeneous polynomials, has only terms of the form (8

) i ź k"1 η α k k " i ź k"1 ˜d ź j k "1 η α k,j k k,j k
where η k P S d´1 and α k " pα k,j k q are multi-indices with |α k | " ř j k α k,j k " n for all k. This form is inhertited by C pN q i with varying n. It shows that C pN q i is indeed a polynomial in the variables ś i k"1 η k,j k . When the dimensions are not equal, observe that by restricting to actually arising number of variables we certainly lose harmonicity but not n-homogeneity or the required form of our polynomials.

Proof. (of Lemma 5.8) By Lemma 5.5 we know that for each parameter 1 ď s ď l there exists a tensor product of cones D s " Â kPIs Dpξ k , r k q with r s :" ř kPIs r k ă π{2 and ξ k P Υ k and test functions f s supported as described in Lemma 5.5 part (2) so that }rT 1, D 1 , . . . , rT l, D l , bs . . .spf q} 2 Á }b} BMO I pR d q where f " Â l s"1 f s . We make a remark about the apertures r s . Let dp¨, ¨q denote geodesic distance on S d´1 , where antipodal points have distance π. Let ξ psq be the set of directions of D s . Remember that according to Lemma 5.5, one component had a specific direction ξ psq v P Υ v and possibly large aperture with p1`τ qr psq v ă π{2. Let us choose the other directions arbitrarily but with apertures r psq v small enough so that p1 `τ qpr psq v `pi ´1qr psq v q ă π{2. Now choose an aperture r s ă π{2 so that p1 `τ qpr psq v `pi ´1qr psq v q ă r s ă π{2. Writing i s " |I s |, we find Journé type cone multipliers D s " C is p ξ psq ; ¨q according to the construction above with center ξ psq and aperture r s .

We are going to observe now that D s " 1 on sptp D s q and D s " ´1 on the Fourier support of f s . Let us drop the dependence on s for the moment. We see in an inductive manner that C i p ξ; ¨q takes the value 1 in a certain ℓ 1 ball of radius r ă π{2 centered at ξ. We show that ÿ k dpξ k , η k q ă r ñ C i p ξ, η 1 , . . . , η i q " 1.

When i " 1, the assertion is obviously true: dpξ 1 , η 1 q ă r ñ C 1 pξ 1 ; η 1 q " 1 by the choice of ϕ, r and definition of C 1 . For i ą 1, we proceed by induction. Assume that ř i´1 k"1 dpξ k , n k q ă r ñ C i´1 pξ 1 , . . . , ξ i´1 ; η 1 , . . . , η i´1 q " 1. Let us assume that ř i k"1 dpξ k , η k q ă r. Remembering the definition of C i p ξ; ¨q we assume dpa i´1 , ξ i´1 q " dpη i , ξ i q. By the triangle inequality ř i´2 k"1 dpξ k , η k q `dpa i´1 , η i´1 q ď ř i´2 k"1 dpξ k , η k q `dpa i´1 , ξ i´1 q `dpξ i´1 , η i´1 q " ř i k"1 dpξ k , η k q ă r . So C i´1 pξ 1 , ξ 2 , . . . , a i´1 ; η 1 , . . . , η i´1 q " 1 for all a i´1 we condition by. The statement for i follows.

Since C i p ξ; ¨q does not depend upon the order of the variables in its construction, we are also able to see exactly as done above that when σ k " ´1 for exactly one choice of k, then ř k dpσ k ξ k , η k q ă r ñ C i p ξ; η 1 , . . . , η i q " ´1. Let us illustrate the base case pS 1 q 2 by a picture. The picture is simplified in the sense that the odd function ϕ above is replaced by an indicator function of an interval.

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ r S 1 S 1 ξ Figure 1: R 2 ˆR2
Cone functions based on the oblique strips containing ξ are averaged. In the two-dimensional case, S 1 , expectation is over a one or two point set only. The rectangle around ξ with sides parallel to the axes illustrates the support of the tensor product of cone operators with direction ξ. The longer side is the aperture that arises from the Hankel part [START_REF] Lacey | Multiparameter Riesz commutators[END_REF]. The short sides can be chosen freely as they arise from the Toeplitz part and is chosen small so that the rectangle fits into the oblique square. The other small rectangle corresponds to the Fourier support of the test function f . Now, we are ready to prove our main theorem.

Proof. (of Theorem 5.1)

Different from the Hilbert transform case, both lower bounds require separate proofs. This is a notable difference that stems from the loss of orthogonal subspaces in conjunction with the special form of the Hilbert transform only seen in one variable. It does not seem possible to get a lower estimate (3)ñ(2) directly.

(1)ô(2) The upper bound (1)ñ( 2) is trivial and follows like the estimate 4.1. The lower bound (2)ñ(1) follows from Wiener's lemma in combination with the main result in [START_REF] Lacey | Multiparameter Riesz commutators[END_REF], the two-sided estimate for iterated commutators with Riesz transforms, similar to the first arguments used in 5.5.

(1)ô(3) The upper bound (1)ñ(3) follows from 4.1. The lower bound (3)ñ(1) uses the considerations leading up to this proof: Suppressing again the dependence on s, we see that the multiplier C i is an odd, smooth, bounded function in each η k when the other variables are fixed. Furthermore, since ϕ, written as a function of t " xξ, ηy is odd with respect to t " 0, then the above series has ϕ n ‰ 0 at most when n is odd and thus Z pnq ξ is odd. So C pN q i is as a sum of odd functions odd.

We are now also ready to see that T ξ , the Journé operator associated to the cone C i p ξ; ¨q as well as the operator associated to C pN q i p ξ; ¨q are paraproduct free. Infact, applied to a test function f " Â k f k with f k acting on the k th variable and where f l " 1 for some l gives T ξ pf q " 0. To see this, apply the multiplier C pN q i p ξ; ¨q in the l variable (acting on 1) first, leaving the other Fourier variables fixed. The multiplier function

η l Þ Ñ C pN q i p ξ; η 1 , . . . , η i q " ÿ nďN ϕ n Z pnq ξ l pη l q i ź k‰l,k"1 Z pnq ξ k pη k q
is, as a sum of odd functions, odd on S d l ´1, bounded by 1 and uniformly smooth for all choices of η k with k ‰ l. As such it gives rise to a paraproduct-free convolution type Calderón-Zygmund operator in the l variable whose values are multi-parameter multiplier operators.

Due to the convergence properties proved above, the difference C i p ξ; ¨q ´CpNq i p ξ; ¨q gives rise to a paraproduct free Journé operator with Calderón-Zygmund norm depending on N . This is seen by an application of an appropriate version of the Marcinkievicz multiplier theorem. By our stability result, Corollary 4.5, there exist integers N s so that C pNsq s p ξ s ; ¨q with ξ k P Υ k is a characterising set of operators via commutators for BMO I pR d q. This is a finite set of possibilities because of the universal choice of the r s and finiteness of the set Υ. Using the multi-parameter analog of the observation rAB, bs " ArB, bs `rA, bsB and the special form of the C pNsq s p ξ; ¨q, leaves us with the desired lower bound: Observe that when rAB, bs has large L 2 norm then either rA, bs or rB, bs has a fair share of the norm. We use this argument finitely many times in a row for operators that are polynomials in tensor products in Riesz transforms  kPIs R k,j k . This finishes ((3)ñ(1)).

Weak Factorization

It is well known, that theorems of this form have an equivalent formulation in the language of weak factorization of Hardy spaces. We treat the model case R d " R pd 1 ,d 2 ,d 3 q and BMO p13qp2q pR d q only for sake of simplicity. The other statements are an obvious generalization. For the corresponding collections of Riesz transforms R k,j k and b P BMO p13qp2q pR d q, 1 ď s ď 3, by unwinding the commutator one can define the operator Π j such that xrR 2,j 2 , rR 1,j 1 R 3,j 3 , bssf, gy L 2 " xb, Π j pf, gqy L 2 .

Consider the Banach space L 2 ˚L2 of all functions in L 1 pR d q of the form f " ř j ř i Π j pφ j i , ψ j i q normed by }f } L 2 ˚L2 " inft ÿ

j ÿ i }φ j i } 2 }ψ j i } 2 u
with the infimum running over all possible decompositions of f . Applying a duality argument and the two-sided estimate in Corollary 5.2 we are going to prove the following weak factorization theorem.

Theorem 6.1. H 1 Re pR pd 1 ,d 2 q q b L 1 pR d 3 q `L1 pR d 1 q b H 1 Re pR pd 2 ,d 3 q q coincides with the space L 2 ˚L2 . In other words, for any f P H 1 Re pR pd 1 ,d 2 q q b L 1 pR d 3 q `L1 pR d 1 q b H 1 Re pR pd 2 ,d 3 q q there exist sequences φ j i , ψ j i P L 2 such that f " ř j ř i Π j pφ j i , ψ j i q and }f } "

ř j ř i }φ j i } 2 }ψ j i } 2 .
Proof. Let's first show that L 2 ˚L2 is a subspace of H 1 Re pR pd 1 ,d 2 q q b L 1 pR d 3 q `L1 pR d 1 q b H 1

Re pR pd 2 ,d 3 q q. Recalling the remark after Theorem 2.3, this is the same as to show @f P L 2 ˚L2 , f is a bounded linear functional on BMO p13qp2q pR d q. This follows from the upper bound on the commutators since xb, ÿ j ÿ i Π j pφ j i , ψ j i qy " ÿ j ÿ i xrR 2,j 2 , rR 1,j 1 R 3,j 3 , bssφ j i , ψ j i y.

Now we are going to show

sup f PL 2 ˚L2 ! | ż f b| : }f } L 2 ˚L2 ď 1 ) « }b} BMO p13qp2q
which gives the equivalence of H 1 Re pR pd 1 ,d 2 q q b L 1 pR d 3 q `L1 pR d 1 q b H 1 Re pR pd 2 ,d 3 q q norm and the L 2 ˚L2 norm, thus showing that the two spaces are the same.

To see this, note that the direction À is trivial, and the direction Á is implied by the lower bound of commutators. For any b P BMO p13qp2q , there exists j such that }b} BMO p13qp2q À }rR 2,j 2 , rR 1,j 1 R 3,j 3 , bss}. Hence, there exist φ, ψ P L 2 with norm 1 such that }b} BMO p13qp2q À |xrR 2,j 2 , rR 1,j 1 R 3,j 3 , bssφ, ψy| " |xb, Π j pφ, ψqy| ď LHS, which completes the proof.

Remarks about our results in L p

As mentioned before, the two-sided estimates stated in section 5 and in particular Theorem 5.3 hold for all 1 ă p ă 8. The fact that upper estimates hold in L p in the case where no tensor products are present is proved in [START_REF] Lacey | Multiparameter Riesz commutators[END_REF] as well as [START_REF]Iterated Riesz commutators: a simple proof of boundedness[END_REF]. It stems from the fact that endpoint estimates for multi-parameter paraproducts hold for all 1 ă p ă 8 [START_REF] Muscalu | Bi-parameter paraproducts[END_REF], [START_REF]Multi-parameter paraproducts[END_REF]. The tensor product case can be derived following the argument for Theorem 4.1. The lower estimate or the necessity of the BMO condition can be derived from interpolation. Infact, suppose we have uniform boundedness of our commutators with operators running through all choices of Riesz transforms and some symbol b in L p . Then by duality, we have boundedness in L q where 1{p `1{q " 1. In fact, rT, bs ˚f " ´rT ˚, bsf " ´rT ˚, bs f shows that the boundedness of adjoints is inherited. The same reasoning holds for iterated commutators of tensor products. Thus by interpolation, the boundedness holds in L 2 and the symbol function b necessarily belongs to the required BMO class.

  pnq ξ are reproducing and for any spherical harmonic of degree n, Y pnq , we have (4) Y pnq pξq " N pd, nq Ω d´1 ż S d´1 Y pnq pηqZ pnq ξ pηqdσ d´1 as well as for any function ϕ zonal with respect to the pole ξ

  P K i P 2 bP i P K 2 , P i P K 2 bP K i P 2 , P K i P K 2 bP i P 2 with i P t1, 3u are bounded on L 2 pT 3 q. (2) The commutator rH 2 , rH 3 H 1 , bss is bounded on L 2 pT 3 q if and only if all four iterated Hankels P 2 Q 13 bQ K 13 P K 2 , P K 2 Q K 13 bQ 13 P 2 , P 2 Q K 13 bQ 13 P K 2 , P K 2 Q 13 bQ K 13 P 2 are bounded on L 2 pT 3 q.

  Consider associated multi-parameter Calderón-Zygmund operators T s,Ds and Ý Ñ Id s " Â kPIs Id k and Id k the identity on the k th variable. Now rT 1,D 1 , . . . , rT l,D l , bs . . .spf q " rT 1,D 1 T s, Ds pbf q} 2 " }rT 1, D 1 , . . . , rT l, D l , bs . . .spf q} 2 Á }b} BMO I pR d q we get the desired lower bound }rT 1,D 1 , . . . , rT l,D l , bs . . .s} L 2 pR d qý Á }b} BMO I pR d q .

				"	l â s"1	`Ý Ñ Id 1 , . . . , rT l,D l pT s,Ds `Ý Ñ Id s qpbf q	`Ý Ñ Id l , bs . . .spf q
	With }	Â l s"1 pT s,Ds	`Ý Ñ Id s qpbf q} 2 ě }	Â l s"1

1. Research partly supported by NSF-DMS 0901139. 2. Research partly supported by ANR-12-BS01-0013-01. The author is a junior member of IUF.

Observing that }xby J 1pj 2 q 2 } BMOpR n q ď x}b} BMOpR n q y J 1pj 2 q 2 ď }b} bmo , one has

Note that the sum in the L 2 norm is in fact very simple: ÿ

Hence, the uniform boundedness of the L 2 Ñ L 2 operator norm of dyadic shifts implies that

In order to handle II, we split it into a finite sum depending on the levels of I 1 upon J 1 1 , which leads to

where B k pb, f q :" ř I β I,k xb, h I pkq yxf, h I yh I |I pkq | ´1{2 is a generalized one-parameter paraproduct studied in [START_REF] Dalenc | Upper bound for multi-parameter iterated commutators[END_REF], whose L 2 norm is uniformly bounded by }b} BMO }f } L 2 , independent of k and the coefficients β I,k P t1, ´1u. Then one can proceed as how we dealt with part I to conclude that }II} L 2 pR n ˆRm q À p1 `j1 q}b} bmo }f } L 2 pR n ˆRm q , which together with the estimate for part I implies that }First term} L 2 pR n ˆRm q À p1 `j1 q}b} bmo }f } L 2 pR n ˆRm q .

b) Second term.