Higher order Journé commutators and characterizations of multi-parameter BMO

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Higher order Journé commutators and characterizations of multi-parameter BMO

Yumeng Ou, Stefanie Petermichl, Elizabeth Strouse

To cite this version:

Yumeng Ou, Stefanie Petermichl, Elizabeth Strouse. Higher order Journé commutators and charac- terizations of multi-parameter BMO. Advances in Mathematics, Elsevier, In press. �hal-01977607�

Higher order Journ´ e commutators and characterizations of multi-parameter BMO

Yumeng Ou

Stefanie Petermichl

Elizabeth Strouse

aDepartment of mathematics, Brown University, 151 Thayer Street, Providence RI 02912, USA

bInstitut de Math´ematiques de Toulouse, Universit´e Paul Sabatier, Toulouse, France

cInstitut de Math´ematiques de Bordeaux, 351 cours de la Lib´eration, F-33405 Talence, France

Abstract

We characterize L^p boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms. We obtain a two-sided norm estimate that shows that such operators are bounded on L^p if and only if the symbol belongs to the appropriate multi-parameter BMO class. We extend our results to a much more intricate situation; commutators of multiplication by a symbol function and paraproduct-free Journ´e operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the tensor products of Riesz transforms are a representative testing class for Journ´e operators.

Previous results in this direction do not apply to tensor products and only to Journ´e operators which can be reduced to Calder´on-Zygmund operators. Upper norm estimate of Journ´e commutators are new even in the case of no iterations.

Lower norm estimates for iterated commutators only existed when no tensor prod- ucts were present. In the case of one dimension, lower estimates were known for products of two Hilbert transforms, and without iterations. New methods using Journ´e operators are developed to obtain these lower norm estimates in the multi- parameter real variable setting.

Key words: Iterated commutator, Journ´e operator, multi-parameter, BMO

1 Introduction

As dual of the Hardy space H¹, the classical space of functions of bounded mean oscillation, BMO, arises naturally in many endpoint results in analysis, partial differential equations and probability. When entering a setting with several free parameters, a large variety of spaces are encountered, some of which lose the feature of mean oscillation itself. We are interested in charac- terizations of multi-parameter BMO spaces through boundedness of commu- tators.

A classical result of Nehari [26] shows that a Hankel operator with anti-analytic symbol b mapping analytic functions into the space of anti-analytic functions byf ÞÑP_´bf is bounded with respect to theL² 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 transformrH, bs “Hb´bH.

Ferguson-Sadosky in [14] and later Ferguson-Lacey in their groundbreaking paper [13] study the symbols of bounded ‘big’ and ‘little’ Hankel operators on the bidisk through commutators of the tensor product or of the iterated form

rH₁H₂, bs, and rH₁,rH₂, bss.

Hereb “bpx₁, x₂qand theH_k are the Hilbert transforms acting in thek^thvari- able. A full characterization of different two-parameter BMO spaces, Cotlar- Sadosky’s little BMO and Chang-Fefferman’s product BMO space, is given through these commutators.

Through the use of completely different real variable methods, in [6] Coifman- Rochberg-Weiss extended Nehari’s one-parameter theory to real analysis in the sense that the Hilbert transform was replaced by Riesz transforms. These one-parameter results in [6] were treated in the multi-parameter setting in Lacey-Petermichl-Pipher-Wick [18]. Both the upper and lower estimate have proofs very different from those in one parameter. In addition, in both cases it is observed that the Riesz transforms are a representative testing class in the sense that BMO also ensures boundedness for (iterated) commutators with more general Calderon-Zygmund operators, a result now known in full generality due to Dalenc-Ou [8]. Notably the Riesz commutator has found

Email address: [email protected] (Stefanie Petermichl).

URL: http://math.univ-toulouse.fr/˜petermic(Stefanie Petermichl).

1 Research supported in part by NSF-DMS 0901139.

2 Research supported in part by ANR-12-BS01-0013-02. The author is a member of IUF.

3 Correponding author, Tel:+33 5 61 55 76 59, Fax: +33 5 61 55 83 85

striking applications to compensated compactness and div-curl lemmas, [3], [20].

Our extension to the multi-parameter setting is two-fold. On the one hand we replace the Calderon-Zygmund operators by Journ´e operators J_i and on the other hand we also iterate the commutator:

rJ₁, ...,rJ_t, bs...s.

We prove the remarkable fact that a multi-parameter BMO class still en- sures boundedness in this situation and that the collection of tensor products of Riesz transforms remains the representative testing class. The BMO class encountered is a mix of little BMO and product BMO that we call a little product BMO. Its precise form depends upon the distribution of variables in the commutator. Our result is new even when no iterations are present: in this case, lower estimates were only known in the case of the double Hilbert trans- form [14]. The sufficiency of the little BMO class for boundedness of Journ´e commutators had never been observed.

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.

Necessity of the little product BMO condition is shown through a lower es- timate on the commutator. There is a sharp contrast when tensor products of Riesz transforms are considered instead of multiple Hilbert transforms and when iterations are present.

In the Hilbert transform case, Toeplitz operators with operator symbol arise naturally. Using Riesz transforms in R^d as a replacement, there is an absence of analytic structure and tools relying on analytic projection or orthogonal spaces are not readily available. We overcome part of this difficulty through the use of Calder´on-Zygmund operators whose Fourier multiplier symbols are adapted to cones. This idea is inspired by [18]. Such operators are also men- tioned in [31]. 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 Journ´e operators that are not of tensor product type. Through geometric considerations and an aver- aging procedure of zonal harmonics on products of spheres, we construct the multiplier of a special Journ´e operator that preserves lower commutator esti- mates and resembles the multiple Hilbert transform: it has large plateaus of constant values and is a polynomial in multiple Riesz transforms. We expect

that this construction allows other applications.

There is an increase in difficulty when the dimension is greater than two, due to the simpler structure of the rotation group on S¹. In higher dimension, there is a rise in difficulty when tensor products involve more than two Riesz transforms.

The actual passage to the Riesz transforms requires 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 Journ´e operators. We make use of recent versions ofTp1qtheorems in this setting. These recent advances are different from the corresponding theorem of Journ´e [16]. The results we allude to have the additional feature of providing a convenient representation formula for bi-parameter in [22] and even multi-parameter in [28] Calder´on- Zygmund operators by dyadic shifts.

2 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 Journ´e. By this we mean the Hardy spaces as- sociated with domains like the poly-disk orR^d:“Ât

s“1R^ds ford“ pd₁, . . . , d_tq.

While doing so, we typically do not distinguish whether we are working onR^d or T^d. In higher dimensions, the Hilbert transform is usually replaced by the collection of Riesz transforms.

The (real) one-parameter Hardy space H_Re¹ pR^dqdenotes the class of functions with the norm

d

ÿ

j“0

}R_jf}1

where R_j denotes the j^th Riesz transform or the Hilbert transform if the di- mension is one. Here and below we adopt the convention thatR₀, the 0^thRiesz transform, is the identity. This space is invariant under the one-parameter fam- ily of isotropic dilations, while the product Hardy space H_Re¹ pR^dq is invariant under dilations of each coordinate separately. That is, it is invariant under at

parameter family of dilations, hence the terminology ‘multi-parameter’ theory.

One way to define a norm on H_Re¹ pR^dq is }f}H¹ „

ÿ

0ďjlďdl

}

t

â

l“1

R_l,jlf}1.

R_l,jl is the Riesz transform in the j_l^th direction of the l^th variable, and the 0^th Riesz transform is the identity operator.

The dual of the real Hardy space H_Re¹ pR^dq^˚ is BMOpR^dq, the t-fold product BMO space. It is a theorem of S.-Y. Chang and R. Fefferman [4], [5] that this space has a characterization in terms of a product Carleson measure.

Define

kbk_BMOpRdq :“ sup

UĂR^d

´

|U|^´1 ÿ

RĂU

ÿ

εPsig_d

|xb, w^ε_Ry|²

¯1{2

. (1)

Here the supremum is taken over all open subsetsU ĂR^d with finite measure, and we use a wavelet basisw_R^ε adapted to rectanglesR “Q₁ˆ ¨ ¨ ¨ ˆQ_t, where eachQlis a cube. The superscriptεreflects the fact that multiple wavelets are associated to any dyadic cube, see [18] for details. The fact that the supremum admits all open sets of finite measure cannot be omitted, as Carleson’s example shows [2]. This fact is responsible for some of the difficulties encountered when working with this space.

Theorem 1 (Chang, Fefferman) We have the equivalence of norms }b}_pH¹

RepR^dqq^˚ „ }b}_BMOpR^d_q. That is, BMOpR^dq is the dual to H_Re¹ pR^dq.

This BMO norm is invariant under at-parameter family of dilations. Here the dilations are isotropic in each parameter separately. See also [10] and [12].

2.2 Little BMO

Following [7] and [14], we recall some facts about the space little BMO, often written as ‘bmo’, and its predual. A locally integrable function b : R^d “ R^d1 ˆ. . .ˆR^ds ÑC is in bmo if and only if

}b}bmo“ sup

Q“Q1ˆ¨¨¨ˆQs

|Q|^´1 ż

Q

|bpxq ´b_Q| ă 8

Here theQ_k ared_k-dimensional cubes andb_Q denotes the average ofboverQ.

It is easy to see that this space consists of all functions that are uniformly in BMO in each variable separately. Let x_vˆ “ px₁, . . . ., x_v´1,¨, x_v`1, . . . , x_sq.

Then bpx_ˆvq is a function in x_v only with the other variables fixed. Its BMO norm in xv is

}bpx_vˆq}BMO“sup

Qv

|Q_v|^´1 ż

Qv

|bpxq ´bpx_ˆvqQv|dx_v and the little BMO norm becomes

}b}bmo “max

v tsup

xvˆ

}bpxˆvq}BMOu.

On the bi-disk, this becomes }b}_bmo “maxtsup

x1

}bpx₁,¨q}BMO,sup

x2

}bp¨, x₂q}BMOu,

the space discussed in [14]. Here, the pre-dual is the space H¹pTq bL¹pTq ` L¹pTq bH¹pTq. All other cases are an obvious generalization, at the cost of notational inconvenience.

2.3 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 characterizing BMO spaces consisting for example of those functions bpx₁, x₂, x₃q such that bpx1,¨,¨q and bp¨,¨, x3q are uniformly in product BMO in the remaining two variables.

Definition 1 Let b : R^d Ñ C with d “ pd₁,¨ ¨ ¨, d_tq. Take a partition I “ tI_s : 1 ď s ď lu of t1,2, ..., tu so that Y91ďsďlI_s “ t1,2, ..., tu. We say that b P BMOIpR^dq if for any choices v “ pv_sq, 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 “ px₁, ..., x_tq P R^d, we define x_vˆ by removing those variables indexed by v_s, the little product BMO norm becomes

}b}BMOI “max

v tsup

xˆv

}bpxvˆq}BMOu

where the BMO norm is product BMO in the variables indexed by v_s.

For example, whend“ p1,1,1q “ 1, whent “3 and l“2 withI₁ “ p13qand I2 “ p2q, writing I “ p13qp2q the space BMO_p13qp2qpT¹q arises, which consists of those functions that are uniformly in product BMO in the variables p1,2q and p3,2q respectively, as described above. Moreover, as degenerate cases, it

is easy to see that BMO_p12...tq and BMOp1qp2q...ptq are exactly little BMO and product BMO respectively, the spaces we are familiar with.

Little product BMO spaces onT^dcan 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 The pre-dual of the space BMO_p13qp2qpT¹q is equal to the space H_Re¹ pT^p1,1qq bL¹pTq `L¹pTq bH_Re¹ pT^p1,1qq

:“ tf`g :f P H_Re¹ pT^p1,1qq bL¹pTq and g PL¹pTq bH_Re¹ pT^p1,1qqu.

Proof. The space

H_Re¹ pT^p1,1qq bL¹pTq “ tf PL¹pT³q:H₁f, H₂f, H₁H₂f PL¹pT³qu

equipped with the norm}f} “ }f}1`}H<sub>1</sub>f}<sub>1</sub>`}H₂f}1`}H₁H₂f}1 is a Banach space. Let W¹ “L¹pT³q ˆL¹pT³q ˆL¹pT³q ˆL¹pT³qequipped with the norm

}pf1, f2, f3, f4q}W1 “ }f1}1` }f2}1` }f3}1` }f4}1.

Then we see that H_Re¹ pT^p1,1qq bL¹pTqis isomorphically isometric to the closed subspace

V “ tpf, H₁pfq, H₂pfq, H₁H₂pfqq:f PH¹pT^p1,1qq bL¹pTqu

of W¹. Now, the dual ofW¹ is equal to W⁸ “L⁸pT³q ˆL⁸pT³q ˆL⁸pT³q ˆ L⁸pT³q equipped with the norm }pg₁, g₂, g₃, g₄q}8 “ maxt}g_i}8 : 1 ď i ď 4u so the dual space of V is equal to the quotient of W⁸ by the annihilator U of the subspace V inW⁸. But, using the fact that the Hilbert transforms are self-adjoint up to a sign change, we see that

U “ tpg₁, g₂, g₃, g₄q:g₁`H<sub>1</sub>g<sub>2</sub>`H₂g₃`H₁H₂g₄ “0u and so:

V^˚ –W⁸{U –Impθq where

θpg1, g2, g3, g4q “ g1`H1g2`H2g3`H1H2g4

since U “kerpθq. But

Impθq “L⁸pT³q `H<sub>1</sub>pL<sup>8</sup>pT<sup>3</sup>qq `H₂pL⁸pT³qq `H₁pH₂pL⁸pT³qqq 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 ofL¹pTq bH_Re¹ pT^p1,1qq is equal to L⁸pT³q `H<sub>2</sub>pL<sup>8</sup>pT<sup>3</sup>qq `H₃pL⁸pT³qq `H₂H₃pL⁸pT³qq, which is equal to

the space of functions that are uniformly in product BMO in variables 2 and 3.

Now, we consider the ‘L¹ sum’ of the spaces H_Re¹ pT^p1,1qq bL¹pTqand L¹pTq b H_Re¹ pT^p1,1qq; that is

M_p13qp2q “ tpf, gq:f PH_Re¹ pT^p1,1qq bL¹pTq;g PL¹pTq bH_Re¹ pT^p1,1qqu equipped with the norm

}pf, gq} “ }f}_H¹

RepT^p1,1qqbL¹pTq` }g}_L¹_pTqbH¹

RepT^p1,1qq.

We see that, if φ :M_p13qp2q Ñ L¹ppT³q is defined by φpf, gq “f `g, then the image ofφ is isometrically isomorphic to the quotient ofM_p13qp2q by the space

N “ tpf, gq PM_p13qp2q:f `g “0u

“ tpf,´fq:f PH_Re¹ pT^p1,1qq bL¹pTq XL¹pTq bH_Re¹ pT^p1,1qqu.

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 pairspφ, φqwith φin 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. QED

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_p13qp2qpR^dq is equal to H_Re¹ pR^pd1,d2qq bL¹pR^d3q `L¹pR^d1q bH_Re¹ pR^pd2,d3qq.

3 The Hilbert transform case

In this section, we characterize the boundedness of commutators of the form rH₂,rH₃H₁, bss as operators on L²pT³q. In the case of the Hilbert transform, this case is representative of 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 4.

Now letb PL¹pTⁿqand let P and Q denote orthogonal projections onto sub- spaces of L²pTⁿq. 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 it- erated Hankel or Toeplitz type operators of the form Q^KbQ (Hankel), P bP (Toeplitz),P Q^KbQP (mixed), whereb means the (not a priori bounded) mul- tiplication operator Mb onL²pTⁿq.

We shall use the following simple observation concerning Hilbert transform type operators again and again:

Remark 1 If H “ P ´P^K and T : L²pTⁿq Ñ L²pTⁿq is a linear operator then

rH, Ts “2P T P^K´2P^KT P and H is bounded if and only if P T P^K and P^KT P are.

Proof.

pP ´P^KqT ´TpP ´P^Kq “ pP ´P^KqTpP `P<sup>K</sup>q ´ pP `P^KqTpP ´P^Kq

“2P T P^K´2P^KT P.

QED We state the main result of this section.

Theorem 3 Let b P L¹pT³q. Then the following are equivalent with linear dependence on the respective norms

(1) bP BMO_p13qp2q

(2) The commutators rH₂,rH₁, bss and rH₂,rH₃, bss are bounded on L²pT³q (3) The commutator rH₂,rH₃H₁, bss is bounded on L²pT³q.

Corollary 1 We have the following two-sided estimate

}b}BMO_p13qp2q À }rH₂,rH₃H₁, bss}L²pT³qÑL²pT³qÀ }b}BMO_p13qp2q.

It will be useful to denote by Q₁₃ orthogonal projection on the subspace of functions which are either analytic or anti-analytic in the first and third vari- ables; Q₁₃ “ P₁P₃ `P<sub>1</sub><sup>K</sup>P<sub>3</sub><sup>K</sup>. Then the projection Q<sup>K</sup><sub>13</sub> onto the orthogonal of this subspace is defined by Q<sup>K</sup><sub>13</sub> “ P<sub>1</sub><sup>K</sup>P<sub>3</sub> `P₁P₃^K. We reformulate properties (2) and (3) in the statement of Theorem 3 in terms of Hankel Toeplitz type operators.

Lemma 1 We have the following algebraic facts on commutators and projec- tion operators.

(1) The commutators rH2,rH1, bss and rH2,rH3, bss are bounded on L²pT³q if and only if the operatorsP_iP₂bP_i^KP₂^K, P_i^KP₂bP_iP₂^K, P_iP₂^KbP_i^KP₂, P_i^KP₂^KbP_iP₂ with iP t1,3u are bounded on L²pT³q.

(2) The commutator rH₂,rH₃H₁, bss is bounded on L²pT³q if and only if all four operators P₂Q₁₃bQ^K₁₃P₂^K, P₂^KQ^K₁₃bQ₁₃P₂, P₂Q^K₁₃bQ₁₃P₂^K, P₂^KQ₁₃bQ^K₁₃P₂ are bounded on L²pT³q.

Proof. Using Remark 1 it is easy to see that rH2,rH1, bss “4`

pP2P1bP₁^KP₂^K´P2P₁^KbP1P₂^Kq ´ pP₂^KP1bP₁^KP2´P₂^KP₁^KbP1P2q˘ and that the corresponding equation for rH₂,rH₃, bss is also true. This, along with the observation that the ranges of all arising summands are mutually orthogonal, gives assertion (1). To prove (2) we just notice that H₁H₃ “ Q₁₃´Q^K₁₃ is a Hilbert transform type operator which permits us to repeat the

above argument replacingP1 byQ13. QED

The following lemma will allow us to insert an additional Hilbert transform into the commutator without reducing the norm.

Lemma 2 }P₃P₁^KP₂^KbP₁P₂P₃}L²ÑL² “ }P₁^KP₂^KbP₁P₂}L²ÑL². Proof.

The inequality ďis trivial, since P₃ is a projection which commutes with P₁^K and P₂^K. To see ě, notice that P3P₁^KP₂^KbP1P2P3 is a Toeplitz operator with symbolP₁^KP₂^KbP₁P₂. So}P₃P₁^KP₂^KbP₁P₂P₃} “sup_x3}P₁^KP₂^Kbp¨,¨, x₃qP₁P₂}.The latter is just }P₁^KP₂^KbP₁P₂}. For convenience we include a sketch of the facts about Toeplitz operators we use. LetW3 be the operator of multiplication by z₃, W₃pfq “z₃f, acting on L²pT³q. If we define B “P₁^KP₂^KbP₁P₂ as well as

An“W₃^˚npP3P₁^KP₂^KbP1P2P3qW₃ⁿ and Cn “W₃ⁿpP₃^KP₁^KP₂^KbP1P2P₃^KqW₃^˚n as operators acting onL²pT³qthen the sequencesA_n andC_n converge toB in the strong operator topology: it is easy to see thatW3 ,W₃^˚; and P3 commute with P₁, P₂, P₁^K and P₂^K. The multiplier b satisfies the equation W₃^˚nbW₃ⁿ “b and W₃ⁿW₃^˚n“Id. So we see that

A_n“P₁^KP₂^KpW₃^˚nP₃W₃ⁿqbP₁P₂pW₃^˚nP₃W₃ⁿq. But if f P L²pT³q, then, since W₃ⁿ is a unitary operator:

}W₃^˚nP₃W₃ⁿpfq´f} “ }P₃W₃ⁿpfq´W₃ⁿpfq} “ }pP₃´IqpW₃ⁿqpfq} Ñ0 pn Ñ 8q, as tail of a convergent Fourier series. This means that W₃^˚nP₃W₃ⁿ converges to the identity in the strong operator topology. Thus, for each f PL²pT³q we

have}pA_n´Bqpfq} Ñ0. So }P₁^KP₂^KbP1P2} ďsup

nPN

}W₃^˚npP3P₁^KP₂^KbP1P2P3qW₃ⁿ} ď }P₃P₁^KP₂^KbP₁P₂P₃},

QED Now, we are ready to proceed with the proof of the main theorem of this section.

Proof. (of Theorem 3) We show p1q ô p2qand p2q ô p3q.

p1q ô p2q. Consider f “ fpx1, x2q and g “ gpx3q. Then rH2,rH1, bsspf gq “ g ¨ rH₂,rH₁, bsspfq. So }rH₂,rH₁, bsspf gq}²_L²_pT³_q “ }F g}²_L²_pTq where Fpx₂q “ }rH₂,rH₁, bsspfq}_L²_pT²_q. The mapg ÞÑF g hasL²pTqoperator norm}F}8. Now change the roles ofx₁andx₃. The Ferguson-Lacey equivalences}rH₂,rH_i, bss} „ }b}BMO give the desired result.

p2q ñ p3q. Boundedness of the commutatorsrH₂,rH₁, bssandrH₂,rH₃, bssim- plies the boundedness of the mixed commutatorrH₂,rH₁H₃, bssby the identity rH₂,rH₁H₃, bss “H₁rH₂,rH₃, bss ` rH₂,rH₁, bssH₃.

p3q ñ p2q. This part relies on Lemma 2. We wish to conclude from the bound- edness ofrH₂,rH₃H₁, bss the boundedness ofrH₂,rH₁, bss andrH₂,rH₃, bss. To see boundedness ofrH₂,rH₁, bss, let us look at one of the Hankels from Lemma 1. Lemma 2 shows that P₂^KP₁^KbP₂P₁ is bounded if and only if the operator P₃P₁^KP₂^KbP₁P₂P₃ is. And the latter is an operator found in the list from part (2) of Lemma 1. The analogous reasoning shows that all eight Hankels in 1

are bounded and so (2) is proved. QED

4 Real variables: lower bounds

In this section, we are again in R^d with d “ pd₁, . . . , d_tq and a partition I “ pI_sq1ďsďloft1, . . . , tu. It is our aim to prove the following characterization theorem of the space BMOIpR^dq.

Theorem 4 The following are equivalent with linear dependence of the re- spective norms.

(1) bP BMOIpR^dq

(2) All commutators of the form rR_k1,jk

1, . . . ,rR_kl,j

kl, bs. . .s are bounded in L²pR^dq where k_s PI_s and R_ks,jks is the one-parameter Riesz transform in direction j_ks.

(3) All commutators of the form rR_1,jp1q, . . . ,rR_l,jplq, bs. . .s are bounded in L²pR^dq where j^psq “ pj_kqkPIs, 1 ďj_k ď d_k and the operators R_s,jpsq are a tensor product of Riesz transforms R_s,jpsq “Â

kPIsR_k,jk.

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 characterization 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 Let 1ăpă 8. Under the same assumptions as Corollary 2 and for any fixed n“ pn_sq where 1ďn_sď |I_s|, we have the two-sided estimate

}b}BMOIpR^dq Àsup

j

}rR_1,jp1q, . . . ,rR_l,jplq, bs. . .s}L^ppR^dqýÀ }b}BMOIpR^dq

where j^psq “ pj_kqkPIs, 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|I₁|, . . . ,|I_l|qit is the equivalence (1) ô(3) from Theorem 4.

Our main focus is of course on a two-sided estimate when n “ p|I₁|, . . . ,|I_l|q when the tensor product is a paraproduct-free Journ´e operator:

Corollary 2 Let j “ pj1, . . . , jtq with 1ďjkďdk and let for each 1ďsďl, j^psq “ pj_kqkPIs be associated a tensor product of Riesz transforms R_s,jpsq “ Â

kPIsR_k,jk; here the R_k,jk arej_k^th Riesz transforms acting on functions defined on the k^th variable. We have the two-sided estimate

}b}_{BM OIpR}^d_q Àsup

j

}rR_1,jp1q, . . . ,rR_t,jptq, bs. . .s}_L^p_pR^d_qýÀ }b}_{BM OIpR}^d_q.

The statements above also serve as the statement of the general case for prod- ucts of Hilbert transforms. In fact, when any d_k “ 1 just replace the Riesz transforms by the Hilbert transform in that variable. In this section, we con- sider the cased_k ě2 for 1ďkďt and thus iterated commutators with tensor products of Riesz transforms only. The special case whend_k “1 for somek 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 Calder´on-Zygmund operators whose multiplier symbols are adapted to cones.

In dimensiondě2, a coneC ĂR^d with cubic base is given by the datapξ, Qq where ξ P S^d´1 is the direction of the cone and the cube QĂ ξ^K centered at the origin is its aperture. The cone consists of all vectorsθ that take the form pθξξ, θ_Kq where θξ “ xθ, ξy and θ_K P θξQ. 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 coneC orD, we consider its Fourier projection operator defined via Px_Cf “χ_Cf .ˆ When the apertures are cubes, such operators are combinations of Fourier projections onto half spaces and as such admit uniformL^p bounds.

Among others, this fact made cubic cones necessary in the considerations in [18] and [9] that we are going to need. For further technical reasons in the proof these operators are not quite good enough, mainly because they are not of Calder´on-Zygmund type. For a given coneC, consider a Calder´on-Zygmund operatorT_C with a kernelK_C whose Fourier symbolKx_C PC⁸and satisfies the estimate χ_C ďKx_C ď χ_p1`τqC. This is accomplished by mollifying the symbol χ_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_kq we denote byT_C, P_C the corresponding tensor product operators.

In [18] it has been proved that Calder´on-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. In [9] 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 [9], but we need to preserve the information on the Fourier support of the test function in order to succeed with our argument. Information on this test function is instrumental to our argument: it reduces the terms arising in the commutator to those resembling Hankel operators. We have the following lemma, very similar to that in [18] section 7 and [9] section 3, the only difference being that the cones are based on balls instead of cubes.

Lemma 3 For every parameter 1ďk ďt there exist a finite set of directions ΥkPS^dk´1 and an aperture 0ărkăπ{2 so that, for every symbol b belonging to product BMO, there exist cones D_k “ Dpξ_k, r_kq with ξ_k P Υ_k as well as a normalised test functionf “Ât

k“1f_k whose components have Fourier support

in the opposing cones Dp´ξ_k, r_kq such that

}rT_1,D1...,rT_t,Dt, bs...sf}2 Á }b}BMO_p1q...ptqpR^dq.

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

Proof. The lemma in [9] supplies us with the sets of directions Υk 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`τqCpξ_k, Q_kq Ă Dpξk, rkq. Then we have the commutator estimate

}rT1,D1...,rTt,Dt, bs...sf}2 Á }b}_{BMOp1q...ptqpR}^d_q.

In fact, both commutators with conesC and Dare L² bounded and reduce to }T_Dpbfq}2 or }T_Cpbfq}2 respectively thanks to the opposing Fourier support of f. Observe that T_Cpbfq “ T_DpT_Cpbfqq “ T_CpT_Dpbfqq. With }T_C}2Ñ2 ď1,

we see that }T_Dpbfq}2 ě }T_Cpbfq}2. QED

Using this a priori lower estimate, we are going to prove the lemma below.

Lemma 4 Let us suppose we are in R^d with d “ pd₁, . . . , tq and a partition I “ pI_sq1ďsďl. For every 1ďk ďt there exists a finite set of directions Υ_k Ă S^dk´1 and an aperture r_k so that the following hold for all bPBMOIpR^dq: (1) For every 1ďsďl there exists a coordinate v_s PI_s and a direction ξ_vs P

Υ_vs and so that with the choice of cone D_vs “Dpξ_vs, r_vsqand arbitraryD_k for coordinates k P I_sztv_su and if D_s denotes their tensor product, then we have

}rT_1,D1, . . . ,rT_l,Dl, bs. . .s}2Ñ2 Á }b}BMOIpR^dq, (2) The test functionf “Ât

k“1f_k which gives us a largeL² norm in (1) has Fourier supports of the fk contained inDp´ξk, rkq whenk“vs and in Dk

otherwise.

Before we can begin with the proof of Lemma 4, we will need a real variable version of the facts on Toeplitz operators used earlier.

Lemma 5 Let D_k for 1ďk ďt denote any cones with respect to the k^th vari- able. Let T_Dk denote the adapted Calder´on-Zygmund operators. Let K be any proper subset of tk : 1ďk ďtu, let DK “Â

kPKDk and TDK the associated tensor product of Calder´on-Zygmund operators. Let P_D^σ

K be a tensor product of projection operators on cones Dpξ_k, r_kq or opposing cones Dp´ξ_k, r_kq. Let j RK. Then

}T_DKT_DjbP_D^σ

KP_Dj}L²pR^dqý“ }T_DKbP_D^σ

K}L²pR^dqý.

Proof.

We will establish this by composing some unilateral shift operators and study- ing their Fourier transform in thej variable. Let ξ_j denote the direction of the cone D_j, for any l define the shift operator

S_lgpx_jq “ ż

R^dj

ˆ

gpη_jqe^{2πiplξj`ηjqxj}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_DKT_DjbP_D^σ

KP_DjS_l, and B “T_DKbP_D^σ

K.

We will prove that asl Ñ `8,A_l ÑB in the strong operator topology. As in the argument in Lemma 2, 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_lS_´l “Id and S_´lbS_l“b.

This implies

A_l“T_DKpS_´lT_DjS_lqpS_´lbS_lqP_D^σ

KpS_´lP_DjS_lq

“TDKpS´lTDjSlqbP_D^σ_KpS´lPDjSlq.

We claim that both S_´lT_DjS_l and S_´lP_DjS_l converge to the identity operator in the strong operator topology, which then implies thatA_lÑBasl Ñ 8. We will only proveS_´lT_DjS_l ÑIdas the second limit is almost identical. Observe that}S_´lT_DjS_lf´f} “ }pT_Dj´IqS_lf}. Given anyL² functionf and any fixed large l ě 0. Consider the f with frequencies supported in R^d1 ˆ. . .ˆ pD_j ´ lξ_jqˆ. . .ˆR^dt. In this case,S_lf has Fourier support inR^d1ˆ. . .ˆD_jˆ. . .ˆR^dt where the symbol of T_Dj equals 1. Thus, for such f, we have S_´lT_DjS_lf “f. The sets R^d1 ˆ. . .ˆ pD_j´lξ_jq ˆ. . .ˆR^dt exhaust the frequency space. With }T_Dj´I}_2Ñ2 ď1 the operatorsS_´lT_DjS_lconverge 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. QED

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

Proof. (of Lemma 4) For a given symbolbP BMOI, there exist for all 1 ďsďl coordinates v “ pv_sq, v_s P I_s and a choice of variables not indexed by v_s, x_vˆ⁰ so that up to an arbitrarily small error

}b}BMOI “ }bpx_vˆ⁰q}BMO_pv

1q...pvlq.

By Lemma 3, there exist conesD_vs “Dpξ_vs, r_vsqwith directionsξ_vs P Υ_vs and a normalised test function f_H in variables v_s with opposing Fourier support

such that we have the lower estimate

}rTv1,Dv1, . . . ,rTv_l,D_vl, bpx_ˆv⁰qs. . .spfHq}L²pR^dvq Á }bpx_ˆv⁰q}BMO_pv

1q...pvlq

where R^dv “R^dv1 ˆ. . .ˆR^dvl.

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 vs, in the commutator, we have

rT_v1,Dv

1, . . . ,rT_vl,Dvl, bs. . .spf_Hgq “g¨ rT_v1,Dv

1, . . . ,rT_vl,Dvl, bs. . .spf_Hq for g that only depends upon variables not indexed by v_s. Again using that multiplication operators in L² have norms equal to the L⁸ norm of their symbol, for the ‘worst’ L²-normalised g we have

}rT_v1,Dv

1, . . . ,rT_vl,Dvl, bs. . .spf_Hgq}_L²_pR^d_q

“sup

xvˆ

}rT_v1,Dv

1, . . . ,rT_vl,Dvl, bpx_ˆv⁰qs. . .spf_Hq}_L²_pR^dv_q ě }rT_v1,Dv

1, . . . ,rT_vl,Dvl, bpx_vˆ⁰qs. . .spf_Hq}L²pR^dvq

Á }bpx_vˆ⁰q}BMO_pv

1q...pvlqpR^dvq“ }b}_BMOIpR^d_q.

Note that the test function g can be chosen with well distributed Fourier transform. Take any cones in the variables not indexed byv_sand letDdenote the tensor product of their projections. fT “PDg. Notice that

}rTv1,Dv1, . . . ,rTv_l,D_vl, bs. . .spfHfTq} Á }rTv1,Dv1, . . . ,rTv_l,D_vl, bs. . .spfHgq}

with constants depending upon how small the aperture of the chosen cones is.

Notice that the test function f :“ f_Hf_T has the Fourier support as required in part(2) of the statement of Lemma 4.

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_v1,Dv

1, . . . ,rT_vl,Dvl, bs. . .s and rT1,D1, . . . ,rTl,D_l, bs. . .s reduce significantly when applied to a test func- tion f with Fourier support like ours. When the operators T_vs,Dvs or any tensor product T_s,Ds fall directly on f, the contribution is zero due to op- posing Fourier supports of the test function and the symbols of the opera- tors. The only terms left in the commutators rT_1,D1, . . . ,rT_l,Dvl, bs. . .spfqand rT_v1,Dv

1, . . . ,rT_vl,Dvl, bs. . .spfq have the form Â

sT_s,Dspbfq and Â

sT_vs,Dvspbfq respectively.

By repeated use of Lemma 5 we have the operator norm estimates for any symbolb, valid on the subspace of functions with Fourier support as described forf:}Â

sT_s,Dsb}2Ñ2 “ }Â

sT_vs,Dvsb}2Ñ2.We conclude that a normalised test functionf with Fourier support as described in the statement (2)of Lemma 4