Paraproducts and Hankel Operators of Schatten Class Via P-John–Nirenberg Theorem

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Journal of Functional Analysis 217 (2004) 38–78

Paraproducts and Hankel operators of Schatten class via p-John–Nirenberg Theorem

Sandra Pott and Martin P. Smith Department of Mathematics, Analysis Research Group, University of York, Heslington, York, England YO10 5DD, UK

Received 24 October 2003; revised 5 March 2004; accepted 12 March 2004

Communicated by G. Pisier

Abstract

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of d Schatten–von Neumann class Sp; if and only if its symbol is in the dyadic Besov space Bp : Our main tools are a product formula for paraproducts and a ‘‘p-John–Nirenberg-Theorem’’ due to Rochberg and Semmes. We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols. Using an averaging technique by Petermichl, we retrieve Peller’s characterizations of scalar and vector Hankel operators of Schatten–von Neumann class Sp for 1opoN: We then employ vector techniques to characterise little Hankel operators of Schatten–von Neumann class, answering a question by Bonami and Peloso. Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts. r 2004 Elsevier Inc. All rights reserved.

MSC: 47B35; 47B10

Keywords: Hankel operators; Paraproducts; Schatten–von Neumann classes

Corresponding author. Fax: +44-1904-43-3071. E-mail addresses: [email protected] (S. Pott), [email protected] (M.P. Smith).

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 39

1. Introduction and notation

In recent years, dyadic paraproducts have been successfully employed in the study of Hankel operators in various settings, see e.g. [Pet,PS]. In this paper, we want to look at Schatten class membership of scalar, vector and multivariable dyadic paraproducts and use these to study Schatten class membership of Hankel operators. Boundedness, compactness and membership of Schatten classes of dyadic paraproducts have been characterised in terms of oscillatory properties of their symbols, see e.g. [ChPen,JPee,Pen,RS2] and the references within. Dyadic paraproducts on vector valued spaces (with matrix or more generally operator valued symbols) have also been studied, see e.g. [K,NPiTV]. It has not thus far been possible to characterise the boundedness of paraproducts with operator valued symbols in terms of oscillation properties of the symbol. These difﬁculties are closely connected with a breakdown of a form of the John–Nirenberg Theorem in the operator-valued setting (see [K,BlPo]). Here, we want to consider a ‘‘p-John– Nirenberg Theorem’’, which generalises easily to the operator setting. The purpose of the paper is threefold. First, we show that a ‘‘p-John–Nirenberg Theorem’’ which can be found in [RS2] can be used to give a comparatively simple, interpolation-free proof of the characterisation of Schatten class paraproducts in terms of oscillatory properties of their symbols. Our approach is related to Rochberg and Semmes’ method of nearly weakly orthonormal sequences—indeed, scalar dyadic paraproducts are in some sense the model case for nearly weakly orthonormal sequences, but technically simpler. Using an averaging technique from [Pet],itis possible to retrieve the known characterisation of Schatten class Hankel operators at least for 1opoN (see [Pel1], for a second proof with different methods, see [CoR,R]). Our approach is again interpolation-free and has the advantage that one does not need any nontrivial properties of Besov spaces, for example the atomic 1 decomposition of B1: Secondly, in contrast to the classical John–Nirenberg Theorem, the version of ‘‘p- John–Nirenberg Theorem’’ we require extends both to the operator-valued and the multivariable setting, and we thus obtain characterisations for the membership of Schatten classes for vector paraproducts and paraproducts in several variables. Again using averaging, we also obtain known results for vector Hankel operators (see [Pel2]) and new results on ‘‘little Hankel’’ operators, answering a question left open in [BoPelo1]. Finally, using a sesquilinear version of our method, we obtain necessary and sufﬁcient conditions for boundedness, compactness and Schatten class membership of products of dyadic paraproducts. This part is motivated by the literature about products of Hankel operators, where characterisations of compactness of products of Hankel operators are known [Zh,ACS,V], but the corresponding questions about boundedness and Schatten class membership of products of Hankel operators are still open (see e.g. [TVZh,Pel3]). The paper is organised as follows. In Section 2, we give a ‘‘p-John–Nirenberg’’ proof for the characterisation of dyadic paraproducts of Schatten class for 1ppoN: In Section 3, we characterise dyadic paraproducts of Schatten class with ARTICLE IN PRESS

40 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 operator-valued symbols for 1ppoN: In Section 4, we give an interpolation-free proof of the characterisation of Hankel operators of Schatten class with operator symbols for 1opoN: In Section 5, we use the vector results to characterise little n Hankel operators of Schatten class on H2ðCþ Þ and multivariable dyadic paraproducts of Schatten class. In Section 6, we characterise boundedness, compactness and Schatten class membership of products of paraproducts.

1.1. A dyadic decomposition

Let D denote the collection of all dyadic intervals on the real line, R; so

Àn Àn D :¼fI ¼ In;k :¼½2 k; 2 ðk þ 1ÞÞ: n; kAZg:

Àn Let Dn denote the collection of intervals in D of length 2 : For IAD; let I˜ denote the parent interval of I; let Iþ and IÀ denote the left and right halves of I; respectively, DðIÞ the collection of dyadic intervals contained in I; DðIÞ0 the collection of dyadic intervals contained properly in I; and DnðIÞ the intersection of Dn and DðIÞ: For 0 JAD ðIÞ; we write signðJ; IÞ¼1 for JADðIþÞ; signðJ; IÞ¼À1 for JADðIÀÞ: We let hI denote the Haar function corresponding to I; that is 1 hI ¼ ðwI À wI Þ; jIj1=2 þ À where wJ denotes the characteristic function of an interval J: It is well known that 2 fhI : IADg forms an orthonormal basis of the Hilbert space L ðRÞ: For further details on dyadic harmonic analysis, we refer to the survey article [Per]. Throughout the article, let Cp and Kp denote various constants, depending only on p:

1.2. Schatten classes

For a Hilbert space H; let LðHÞ and KðHÞ denote the collections of bounded operators and compact operators on H; respectively. Any operator TAKðHÞ has a Schmidt decomposition, so there exist orthonormal bases feng and fsng of H and a sequence flng with lnX0andln-0 such that

XN Tf ¼ ln/f ; enSsn ð1Þ n¼0 for all f AH: For 0opoN; a compact operator T with such a decomposition belongs to the Schatten–von Neumann p-class, SpðHÞ; if and only if ! XN 1=p T lp N: 2 jj jjSp ¼ n o ð Þ n¼0 ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 41

We shall frequently use the following elementary facts: For 0opp2; () X jjTjjp ¼ inf jjTe jjp: fe g orthonormal basis of ð3Þ Sp n n nAN H nAN for 2ppoN; () X jjTjjp ¼ sup jjTe jjp: fe g orthonormal basis of ; ð4Þ Sp n n nAN H nAN see e.g. [G, p. 95].

2. Scalar dyadic paraproducts and Besov spaces

For a locally integrable function f on R and IAD; let mI f denote the mean value of f on I; i.e. Z 1 mI f ¼ f ðtÞ dt; jIj I and fI denote the Haar coefﬁcient of f ; i. e. Z

fI ¼ /f ; hI S ¼ f ðtÞhI ðtÞ dt: I P For f locally integrable and IA ; we write P f ¼ w ð f À m f Þ¼ h f ; P D I I I JADðIÞ J J 0 2 0 and PI f ¼ JAD0ðIÞ hJ fJ : On L ðRÞ; PI and PI are the orthogonal projections on 0 spanfhJ : JADðIÞg and spanfhJ : JAD ðIÞg; respectively. We shall repeatedly use the following fact: for I; JAD; 1 mJ ðhI Þ¼7 ; when JADðI7Þð5Þ jIj1=2 and zero otherwise. For a locally integrable function b; the densely deﬁned dyadic paraproduct with symbol b; pb is given by X pbf ¼ mI fbI hI : IAD

2 It is easy to see that the adjoint of pb on L ðRÞ is given by X w pf ¼ I f b% : ð6Þ b I I I IAD j j

We want to denote this adjoint operator by Lb%: ARTICLE IN PRESS

42 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

Necessary and sufﬁcient conditions on the symbol b for pb to be bounded on L2ðRÞ or belong to a Schatten class have been obtained. We shall say a locally integrable function b belongs to the dyadic BMO space BMOd ðRÞ if 1 b d P b N jj jjBMO ¼ sup 1=2 jj I jjo : IAD jIj

Note that 0 1 Z 1=2 1 1 1=2 1 X jjP bjj ¼ jbðtÞÀm bj2 dt ¼ @ jb j2A : 1=2 I jIj I jIj J jIj I JADðIÞ

We say that bABMOd belongs to the dyadic VMO space VMOd ðRÞ if 1 lim jjPI bjj ¼ 0; ð7Þ jIj-0 jIj1=2

1 lim jjPI bjj ¼ 0 and ð8Þ jIj-N jIj1=2

1 A lim jjPIn;k bjj ¼ 0 for each n Z: ð9Þ k -N 1=2 j j jIn;kj

Here, the limits in (7) and (8) are meant to be uniform limits as jIj-0orjIj-N; respectively. Somewhat loosely, we will write 1 lim jjPI bjj ¼ 0 ð10Þ I-N jIj1=2 if conditions (7)–(9) above hold. We understand I-N as I converging to the point N in the locally compact space D with the discrete topology. For 0opoN; a locally integrable function b belongs to the dyadic Besov space d Bp ðRÞ if !! X p 1=p jbI j jjbjj d ¼ N: Bp 1=2 o IAD jIj

We then have

d 2 Theorem 2.1. (1) bABMO if and only if pbALðL ðRÞÞ; d 2 (2) For 0opoN; bABp if and only if pbASpðL ðRÞÞ: ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 43

d 2 (3) bAVMO if and only if pbAKðL ðRÞÞ:

These results are well known, see e.g. [ChPen,Per] for the boundedness result and [ChPen,RS2] for the result concerning Schatten classes Sp; 1ppoN: Before we give a ‘‘John–Nirenberg type’’ proof of Theorem 2.1(2), we need some more notation. For a locally integrable function j; let Qj be the so-called ‘‘dyadic sweep’’ or the square of the dyadic square function of j; that is, X w ðtÞ Q ðtÞ¼ I jj j2; tAR: ð11Þ j I I IAD j j

We need the following elementary property of Qj:

Lemma 2.2. PI Qj ¼ PI QPI j

2 Let Dj be the operator on L ðRÞ which is diagonal in the Haar basis and deﬁned by 1 X D h ¼ h jj j2 ðIADÞ: j I I jIj J JADðIÞ0

The following identity relates the paraproducts pj and pQj (see [BlPo]).

Proposition 2.3.

p p p p D : j j ¼ Qj þ Qj þ j

Proof. It suffices to show that /p p h ; h S / p p D h ; h S for j j I J ¼ ð Qj þ Qj þ jÞ I J I; JAD: Note that pj is superdiagonal in the Haar basis in the sense that pjhI has nontrivial Haar coefficient only for JD! I; and pj is subdiagonal in the Haar basis in the sense that pjhI has nontrivial Haar coefficient only for J+! I: Furthermore, supp p h DI and supp p p D h DI for all IA ; so we j I ð Qj þ Qj þ jÞ I D only have to consider the cases (1) I ¼ J: X 1 2 /ðpQ þ p þ DjÞhI ; hI S ¼ /DjhI ; hI S ¼ jj j ¼ /pjhI ; pjhI S j Qj jIj J JADðIÞ0

(2) I+! J: X / S 2 pjhI ; pjhJ ¼ jjK j mK hI mK hJ KAD ARTICLE IN PRESS

44 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

0 1 X X signðJ; IÞ @ j 2 j 2A ¼ 1=2 1=2 j K j À j K j jIj jJj KAD ðJþÞ KADðJÀÞ

sign J; I ð Þ / S / S ¼ ðQjÞJ ¼ pQj hI ; hJ ¼ ðpQj þ pQ þ DjÞhI ; hJ jIj1=2 j

(3) ID! J:

/p h ; h S /h ; p h S /h ; p p h S /p p h ; h S Qj I J ¼ I Qj J ¼ I j j J ¼ j j I J by (2). &

We now need to temporarily introduce a further scale of function spaces. For 0oppN and 1pqoN; we say that bAL2ðRÞ belongs to the d space Bp;q if !! p 1=p X 1 jjbjj d ¼ jjPI bjj N; ð12Þ Bp;q 1=q q o IAD jIj

q where jj jjq denotes the norm in L ðRÞ: A continuous version of the following ‘‘p-John–Nirenberg Theorem’’ can be found in [RS2, Proposition 4.1].

d Theorem 2.4. Let 0opoN: Then the spaces Bp;q; 1pqoN; all coincide with the d dyadic Besov space Bp : The corresponding norms are equivalent.

d d In the case p ¼ N; all the spaces BN;q coincide with BMO ; as known from the classical John–Nirenberg Theorem. We shall present a proof of Theorem 2.4 here, which only uses the dyadic square function, a bootstrap argument and the following proposition from [RS1, Lemma 4.3], which covers the case 1pqp2:

Proposition 2.5. Let 0opoN: There exists a constant ap such that for each nonnegative sequence ðaI ÞIAD indexed by the dyadic intervals, 0 1 p X X X p @ 1 A aI aJ pap : jIj jIj IAD JADðIÞ IAD

Note that the reverse inequality trivially holds, with constant equal to 1. Note also that the p ¼ N version of the above statement fails, i.e. there exists no ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 45 constant C such that X 1 aI sup aJ pC sup ; IA jIj IA jIj D JADðIÞ D simply take aI ¼jIj; for all IAD: We include the proof of Proposition 2.5 for the convenience of the reader.

Proof of Proposition 2.5. We shall ﬁrst suppose that 0opp1: Then, for all IAD; we have 0 1 p X X @ 1 A 1 p aJ p a and hence jIj jIjp J JADðIÞ JADðIÞ 0 1 p ! X X X X X XN @ 1 A 1 p 1 p aJ p a ¼ a ; jIj jIjp J ð2kjJjÞp J IAD JADðIÞ IAD JADðIÞ JAD k¼0 as each dyadic interval J is contained in exactly one dyadic interval of size 2kjJj; for k ¼ 0; 1; y : Summing the inﬁnite geometric series, we see that 0 1 p X X p X p @ 1 A 2 aJ aJ p jIj 2p À 1 jJj IAD JADðIÞ JAD as required. We now consider the case 1opoN: We see that, for I ¼ In;kAD; 0 1 0 1 p p X XN X @ 1 A @ n A aJ ¼ 2 aJ jIj JADðIÞ m¼n JADmðIÞ 0 1 p XN X @ À2 2 nÀm m A ¼ ðm À n þ 1Þ ðm À n þ 1Þ 2 2 aJ

m¼n JADmðIÞ 0 1 p XN X 2pÀ2 pðnÀmÞ@ m A p Cp ðm À n þ 1Þ 2 2 aJ

m¼n JADmðIÞ

P N À2 2 for some constant Cp by Jensen’s inequality, since m¼n ðm À n þ 1Þ ¼ p =6 for all m and t/tp is convex. Applying Ho¨ lder’s inequality, where 1=p þ 1=q ¼ 1; ARTICLE IN PRESS

46 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 we see that

0 1 0 1 p p=q X X X @ m A m p@ qA 2 aJ p ðÞ2 aJ 1 A A A J DmðIÞ J DmðIÞ X J DmðIÞ ðmÀnÞðpÀ1Þ m p ¼ 2 ð2 aJ Þ

JADmðIÞ

mÀn as jDmðIn;kÞj ¼ 2 ; when mXn: Consequently, we get

0 1 p X X X XN X @ 1 A 2pÀ2 ðnÀmÞ m p aJ p Cp ðm À n þ 1Þ 2 ð2 aJ Þ jIj IAD JADðIÞ n;kAZ m¼n JADmðIn;kÞ X Xm 2pÀ2 ðnÀmÞ m p ¼ Cp ðm À n þ 1Þ 2 ð2 aIm;j Þ ; m;jAZ n¼ÀN changing the order of summation. But, for all mAZ;

Xm XN 2pÀ2 ðnÀmÞ 2pÀ2 Àðlþ1Þ ðm À n þ 1Þ 2 ¼ l 2 ¼ Kp; n¼ÀN l¼1 say. Therefore,

0 1 p X X X p @ 1 A aJ aJ pCpKp ; jIj jJj IAD JADðIÞ JAD as required. &

d Corollary 2.6. Let b be a locally integrable function, and let 0opoN: Then bABp if A d and only if b Bp;q for 1pqp2:

Moreover, jjbjj d is equivalent to the expressions in (12). Bp

d D d D d Proof. Applying Ho¨ lder’s inequality, it is easy to see that Bp;2 Bp;q Bp;1 for d D d 1pqp2; and it is also easy to see that Bp;1 Bp : All of these embeddings are d D d bounded. So it only remains to prove that Bp Bp;2; and that the embedding is bounded. ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 47

By Proposition 2.5,

0 1 p=2 X p=2 X X 1 2 @ 1 2A jjbjjBd ¼ jjPI bjj ¼ jbJ j p;2 jIj 1=2 IAD IAD jIj JA ðIÞ !D X p jbI j p p ap ¼ apjjbjj d : & 1=2 Bp IAD jIj

Before we can deal with the case q42; we need

Proposition 2.7. Let 0opoN and 1pqoN: Then

2 2 d jjQbjj d p2C jjbjj d ðbAB Þ; Bp;q 2q B2p;2q 2p;2q

2q where C2q is the norm of the dyadic square function on L ðRÞ: Conversely,

2 2 2 jjbjj d pc lpðjjQbjj d þjjbjj d Þ; B2p;2q 2q Bp;q B2p;2

2q where c2q is the lower bound of the dyadic square function on L ðRÞ; and lp is a constant depending only on p; lp ¼ 1 for pX1:

q Proof. Note that the projections ðPI ÞIAD are uniformly bounded on each L ðRÞ; A d 1pqoN; with norms bounded by 2, independent of q: Let b B2p;2q: Then

! ! X 1=p X 1=p 1 p 1 p jjQbjj d ¼ jjPI Qbjj ¼ jjPI QP bjj Bp;q p=q q p=q I q IA jIj IA jIj D D ! 1=p X 1 2 Q p p p=q jj PI bjjq IA jIj D ! 1=p X 1 2C2 P b 2p p 2q 2p=2q jj I jj2q IAD jIj 2 2 ¼ 2 C jjbjj d ; 2q B2p;2q where the ﬁrst equality follows from Lemma 2.2. ARTICLE IN PRESS

48 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

Conversely, ! 1=p X 1 P b 2p 2p=2q jj I jj2q IA jIj D ! 1=p X 1 c2 Q p p 2q 2p=2q jj PI bjjq IA jIj D ! 1=p X 1 c2 P Q w m Q p ¼ 2q 2p=2q jj I PI b þ I I ð PI bÞjjq IAD jIj !! p 1=p X 1 1 c2 P Q Q p 2q 2p=2q jj I bjjq þ 1À1=q jj PI bjj1 IAD jIj jIj !! p 1=p X 1 1 c2 P Q P b 2 ¼ 2q 2p=2q jj I bjjq þ 1À1=q jj I jj2 IAD jIj jIj

2 2 pc lpðjjQbjj d þjjbjj d Þ: & 2q Bp;q B2p;2

N D Proof of Theorem 2.4. Let 2oqo : Because of the trivial inclusion Bp;q2 Bp;q1 for n 1pq1pq2oN; we can assume that q ¼ 2 ; n41: We prove by induction over n that for all nAN and all 0opoN there exists a constant Kp;n such that

d jjbjj Kp;njjbjj d ðbAB Þ: Bp;2n p Bp p

By Corollary 2.6, this is true for n ¼ 1: Suppose that the statement holds for some d nAN: Then by Proposition 2.7, for each bABp ;

2 2 2 jjbjj d l c nþ1 ðjjQ jj d þjjbjj d Þ B p p 2 b B n B p;2nþ1 p=2;2 p;2 2 2 p c nþ1 lpðKp=2;njjQbjjBd þjjbjjBd Þ 2 p=2;1 p;2

2 2 2 p 2c nþ1 lpðKp=2;njjbjj d þjjbjj d Þ 2 Bp;2 Bp;2

2 2 2 jjbjj d 2a c n lpðKp=2;n þ 1Þ: p Bp p 2

The theorem follows now with an appropriate choice of Kp;nþ1: &

N A d A d Corollary 2.8. Let 0opo and let b Bp : Then Qb Bp=2; and there exists a constant 2 Cp40 depending only on p such that jjQbjjBd pCpjjbjjBd : p=2 p

Proof. Corollary 2.6 (or Theorem 2.4) and Proposition 2.7. & ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 49

Now we can give our ‘‘p-John–Nirenberg’’ proof of Theorem 2.1(2). We will give the full proof only for pX1:

Proof of Theorem 2.1(2). Notice that

X X p X p % wI 1 p p jjpbhI jj ¼ bI ¼ jbI j ¼jjbjjBd jIj p=2 p IAD IAD IAD jIj for 0opoN by Proposition 2.5. Thus ‘‘)’’ follows immediately for 0opp2 from Eq. (3). To prove ‘‘)’’ for 2ppoN; note ﬁrst that for 0opoN X X jjD jjp=2 ¼ j/pp h ; h Sjp=2 ¼ jjp h jjp b Sp=2 b b I I b I IA IA D D 0 1 p=2 X X 1 @ 2A p ¼ jbJ j pKpjjbjj d p=2 Bp IAD jIj JAD0ðIÞ by Proposition 2.5, and that therefore for 2ppp4

jjp jj2 ¼jjpp jj 2jjp jj þjjD jj b Sp b b Sp=2 p Qb Sp=2 b Sp=2

2 2jjpQ jj þ Kpjjbjj d p b Sp=2 Bp

0 2 2 p Kp ðjjQbjjBd þjjbjjBd ÞpCpjjbjjBd p=2 p p by Corollary 2.8 and the ﬁrst part of the proof. Inductively, we obtain the result for all p with 2ppoN: To obtain the reverse direction, we deﬁne a bounded operator 2 - 2 A ˜ R: L ðRÞ L ðRÞ of norm 1 by RhI ¼ hI˜ for I D; where I denotes the parent interval of I: Recalling that

XN /Te ; s S p T p 13 j n n j pjj jjSp ð Þ n¼1 for any orthonormal bases feng; fsng; pX1andTASp (this follows immediately from e.g. [G, p. 94]), we ﬁnd that X X p p p p wI % jjpbjj XjjpbRjj X j/RhI ; p hI Sj ¼ h ˜; bI Sp Sp b I jIj IAD IAD ! X p 1 jbI j 1 p ¼ ¼ jjbjjBd 2p=2 1=2 2p=2 p IAD jIj for 1ppoN: & ARTICLE IN PRESS

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The implication ‘‘(’’ in Theorem 2.1(2) for 0opo1 is more difﬁcult to deal with and was ﬁrst shown by Peng [Pen]. We will return to this when looking at the operator case in the next chapter.

3. Operator valued Besov spaces and vector paraproducts of Schatten class

Dyadic paraproducts with matrix or operator symbols have been considered in e.g. [K,NPiTV,Pet]. We ﬁrst introduce some notation for dyadic paraproducts acting on a vector valued Hilbert space, with operator valued symbols. Let H denote separable Hilbert space and L2ðR; HÞ the corresponding vector valued Hilbert space, so Z 2 - 2 2 N L ðR; HÞ :¼fg: R H: jjgjjL2ðR;HÞ ¼ jjgðtÞjjH dto g: R

We may consider L2ðR; HÞ as the Hilbert space tensor product L2ðRÞ#H and, for f AL2ðRÞ and xAH; we let f #x denote the element of L2ðR; HÞ deﬁned for almost all tAR by f #xðtÞ¼f ðtÞx: Let B be a locally SOT-integrable operator valued function on R; so BðtÞALðHÞ for almost all tAR; and for IAD; we may formally deﬁne the operator BI ALðHÞ given by Z

/BI x; yS ¼ hI ðtÞ/BðtÞx; yS dt; x; yAH: I

(For the definition of SOT integrability, see e.g. [Ni, Section 3.11].) We then define 2 the (dyadic) paraproduct PB; acting on elementary tensors in L ðR; HÞ by X 2 PBð f #xÞ¼ mI fhI #BI x; f AL ðRÞ; xAH ð14Þ IAD and extending by linearity. One would anticipate that the boundedness of such an operator would be characterised by an operator bounded mean oscillation criterion. However, it was shown in [NTV] that the naive generalisation of the scalar BMO condition to the operator case does not imply boundedness of the operator paraproduct. We shall show, however, that Schatten class membership may be characterised by an operator Besov condition analogous to the scalar condition. These results can also be obtained using Rochberg and Semmes’ method of nearly weakly orthonormal sequences from [RS1], although the vector case does not seem to appear in the literature. We shall first derive an expression for PB: ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 51

Lemma 3.1. If X w L ð f #xÞ¼ f I #Bx; f AL2ðRÞ; xAH B I jIj I IAD

extending by linearity, then LB ¼ PB:

Proof. This can easily be veriﬁed by means of elementary tensors. &

We shall follow the same approach as in the scalar case, using dyadic square functions. For an operator valued function B; let QB be the ‘‘square of the dyadic square function’’ of B; that is X w ðtÞ Q ðtÞ¼ BB I ; tAR: B I I I IAD j j

2 Let DB be the operator on L ðR; HÞ deﬁned on elementary tensors by

X 1 X D ð f #xÞ¼ f h #B B x: B jIj I I J J IAD JADðIÞ0

The following identity relates the paraproducts PB and PQB (see [BlPo]).

Proposition 3.2.

P P P P D : B B ¼ QB þ QB þ B

Proof. It is sufﬁcient to show that

/P P h #x ; h #yS / P P D h #x ; h #yS 15 B Bð I Þ J ¼ ð QB þ QB þ BÞð I Þ J ð Þ for all I; JAD and x; yAH: This is shown exactly as in Proposition 2.3. &

For 0opoN; we shall say that an operator valued function B lies in the operator d valued dyadic Besov space Bp if !! p 1=p X B jj I jjSp jjBjj d ¼ oN: Bp 1=2 IAD jIj

We shall show that Schatten class operator valued dyadic paraproducts have symbols which belong to corresponding Besov spaces, thus generalising the scalar result. ARTICLE IN PRESS

52 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

We prove an operator analogue to Corollary 2.8. For pX2; the Besov norms of B and QB are related in the same way as in the scalar case.

d d 2 Lemma 3.3. If 2ppoN and BAB then QBAB ; with jjQBjj d papjjBjj d ; for some p p=2 Bp=2 Bp universal constant ap:

Proof. Note that jj Á jj is a norm. By the deﬁnition of Q and (5), we see that Sp=2 B 0 1 X X 1 @ A ðQBÞI ¼ BJ BJ À BJ BJ ; I 1=2 j j JADðIþÞ JADðIÀÞ 1 X 1 X and so jjðQ Þ jj jjB B jj ¼ jjB jj2 ; B I Sp=2 p 1=2 J J Sp=2 1=2 J Sp jIj JADðIÞ0 jIj JADðIÞ0 0 1 p=2 X X p=2 @ 1 2 A p which gives jjQBjj d jjBJ jj apjjBjj d ; B p Sp p B p=2 jIj p IAD JADðIÞ0 by Proposition 2.5. &

The analogous statement for p ¼ N is false for inﬁnite-dimensional H [BlPo]. For IAD; let UI and VI be the bounded operators given by

- 2 # UI : H L ðR; HÞ; UI x ¼ wI x; xAH; Z 2 2 VI : L ðR; HÞ-H; VI F ¼ FðtÞhI ðtÞ dt; FAL ðR; HÞ: I

Then BI ¼ VI PBUI : It follows that, for 0opoN; if PBASp then BI ASp:

d Proposition 3.4. If 0opp2 and BAB then PBASp and jjPBjj pjjBjj d ; p Sp Bp

Proof. Again, it will be more convenient to work with adjoints. Let 0opoN and IAD: Suppose (without loss of generality, by the discussion above) that BI has a Schmidt decomposition,

XN I / I S I BI x ¼ ln x; en sn; xAH; n¼0

I I where feng and fsng are orthonormal bases for H: Therefore, ! XN 1=p B B lI p : jj I jjSp ¼jj I jjSp ¼ ð nÞ n¼0 ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 53

# I 2 It follows that fhI en: IAD; n ¼ 0; 1; yg is an orthonormal basis for L ðR; HÞ: It # I wI # I I is clear from Lemma 3.1 that PBðhI enÞ¼jIj lnsn: Consequently, ! XN p jjBI jjS P h #eI p p jj Bð I nÞjj ¼ 1=2 n¼0 jIj for each IAD and therefore

X XN P p P h #eI p B p : jj BjjSp p jj Bð I nÞjj ¼jj jj d Bp IAD n¼0 by (3). &

The rest of this section will be concerned with showing that the statement in Proposition 3.4 extends to p42; and that also the reverse holds. We shall A d A ﬁrst use Proposition 3.4 and Lemma 3.3 to show that B Bp implies PB Sp; for 2opoN:

A d A Proposition 3.5. If 2ppoN and B Bp then PB Sp: Moreover, there exists a constant Cp40 depending only on p such that jjPBjj pCpjjBjj d : Sp Bp

Proof. The proof runs along the lines of the proof of Theorem 2.1(2). We shall suppose that 2nopp2nþ1 for n ¼ 0; 1; y and proceed by induction. The base case (n ¼ 0) is covered by Proposition 3.4 so suppose that nX1: We shall first consider the operator DB: By definition, DB has block diagonal form DB ¼ðEI ÞIAD with respect 2 " / to the Hilbert space decomposition L ðR; HÞ¼ IAD H given by f ð fI ÞIAD: Here, E is defined by /E x; yS ¼ /P P h #x; h #yS for x; yAH: That is, E ¼ P I I B B I I I 1 jIj JADðIÞ0 BJ BJ : Thus

p=2 X X X p=2 p=2 1 jjDBjj ¼ jjEI jj ¼ B BJ Sp=2 Sp=2 jIj J IAD IAD JADðIÞ0 S 0 1p=2 p=2 X X @ 1 2 A p jjBJ jj jIjp Sp IAD JADðIÞ0 X 1 p p Kp jjBI jj ¼ KpjjBjj d p=2 Sp Bp IAD jIj by Proposition 2.5, since jj Á jj is a norm. Sp=2 ARTICLE IN PRESS

54 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

nÀ1 n Also, by Lemma 3.3, QBABp=2 and 2 op=2p2 : Hence, by the inductive A 2 hypothesis, PQB Sp=2 and jjQBjj d pCpjjBjj d : Consequently, by Proposition 3.4, Bp Bp we have

jjP jj2 ¼jjP P jj 2jjP jj þjjD jj B Sp B B Sp=2 p QB Sp=2 B Sp=2

2 p 2Cp=2jjQBjj d þjjDBjj pCpjjBjj d & Bp=2 Sp=2 Bp

as required for an appropriate choice of Cp:

A A d Finally, we must show that, for 0opoN; PB Sp implies that B Bp : We will ﬁrst deal with the case 1ppoN:

d Proposition 3.6. If 1ppoN and PBASp then BAB ; moreover jjBjj d pCpjjPBjj p Bp Sp for some universal constant Cp:

2 2 Proof. Let 1ppoN; and let T: L ðR; HÞ-L ðR; HÞ be in the Schatten class Sp:

The block diagonal E ¼ðEI ÞIAD of T; taken with respect to the Hilbert space 2 " / decomposition L ðR; HÞ¼ IAD H; f ð fI ÞIAD; is then given by

EhI #e ¼ hI #EI e;

for IAD; where EI : H-H is deﬁned by /EI e; f S ¼ /Te#hI ; f #hI S for IAD; e; f AH: We will use the inequality

X E p E p T p : 16 jj jjSp ¼ jj I jjSp pjj jjSp ð Þ IAD

(see e.g. [G, p. 94]). Similarly to the proof of Theorem 2.1(2), one deﬁnes a bounded linear operator 2 - 2 # # A A R: L ðR; HÞ L ðR; HÞ of norm 1 by RhI e ¼ hI˜ e for I D; e H: Suppose now that PBASp: For each IAD; let BI have the Schmidt decomposition

XN I / I S I BI x ¼ ln x; en sn; xAH; n¼0

I I where feng and fsng are orthonormal bases for H: ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 55

Applying (16) with T ¼ PBR and using (13), we obtain X jjP jjp X jjP Rjjp X jjE jjp B sp B Sp I Sp IAD X XN X XN / I I S p / # I # I S p X j EI sn; en j ¼ j PBRhI sn; hI en j IAD n¼1 IAD n¼1 X XN w p X 1 1 XN h #sI ; I BeI lI p ¼ I˜ n I n ¼ p=2 p=2 j nj A jIj A 2 jIj I D n¼1 ! I D n¼1 X p p 1 jjBI jjS 1 p B p : & ¼ p=2 1=2 ¼ p=2 jj jj d 2 2 Bp IAD jIj

Finally, we shall consider the case 0opo1: We shall generalise an argument found in [ChPen, Lemma 5.1]. Note that, for 0 p 1; is not a norm. However, if o o jj jjSp T ¼ R þ S; then

jjTjjp jjRjjp þjjSjjp ; ð17Þ Sp p Sp Sp see e.g. [Pen, p. 78]. We can obtain a partial reverse inequality, in the case that R and S have orthogonal ranges.

Lemma 3.7. Let 0opoN: If R and S are operators with orthogonal ranges and T ¼ R þ S then

1 jjTjjp X ðjjRjjp þjjSjjp Þ: Sp 2 Sp Sp

Proof. It follows that T T ¼ RR þ SS; as RS ¼ SR ¼ 0: Therefore, RRpT T: By Douglas’ Lemma (see e.g. [Yo, p. 143]), there exists a contraction Z such that R ZT and so R T : Similarly, S T ; and the result ¼ jj jjSp pjj jjSp jj jjSp pjj jjSp follows. &

2 For mAZ; we deﬁne the orthogonal projection Dm on L ðR; HÞ by X Dmð f #xÞ¼ /f ; hI ShI #x;

IADm deﬁned here on elementary tensors for f AL2ðRÞ and xAH: We also deﬁne, for m; nAZ;

n;m PB ¼ DmPBDn: ARTICLE IN PRESS

56 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

A d n;m Lemma 3.8. Let B Bp : If mpn then PB ¼ 0: If m4nand0opo1 then ! X p jjBI jjS jjP jjp 2ðnÀmÞp=2 p : B Sp p 1=2 IADm jIj

Proof. By deﬁnition, we see that, for f AL2ðRÞ and xAH; X X n;m # / S # PB ð f xÞ¼ f ; hJ mI ðhJ ÞhI BI x: JADn IADm

n;m Therefore, by (5), if mpn then PB ¼ 0 and if m4n; X X / S X X / S n;m # f ; hJ # f ; hJ # PB ð f xÞ¼ hI BI x À hI BI x: J 1=2 J 1=2 JADn IADmðJþÞ j j JADn IADmðJÀÞ j j

Let BI have Schmidt decomposition

XN I / I S I BI x ¼ ll x; el sl ; ð18Þ l¼0 so we have

X X XN I n;m # ll / # # I S # I PB ð f xÞ¼ f x; hJ el hI sl J 1=2 JADn IADmðJþÞ l¼0 j j

X X XN I ll / # # I S # I À f x; hJ el hI sl : ð19Þ J 1=2 JADn IADmðJÀÞ l¼0 j j

n;m Thus, PB has been expressed as the difference of two operators with given Schmidt decompositions, so, by (17),

! ! p p X X XN I X X XN I n;m p ll ll jjPB jjS p þ ; p J 1=2 J 1=2 JADn IADmðJþÞ l¼0 j j JADn IADmðJÀÞ l¼0 j j ! p X X B jj I jjSp ¼ 1=2 JADn IADmðJÞ jJj ! X p jjBI jjS 2ðnÀmÞp=2 p : & ¼ 1=2 IADm jIj ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 57

A d Let B Bp and N be a positive integer (to be determined later). For k ¼ 0; yN À 1; let

XN XN XN Xm Nnþk;Nmþkþ1 Nnþk;Nmþkþ1 PB;k ¼ PB ¼ PB ; n¼ÀN m¼ÀN m¼ÀN n¼ÀN by Lemma 3.8. Let

XN XN XmÀ1 ð0Þ Nnþk;Nnþkþ1 ð1Þ Nnþk;Nmþkþ1 PB;k ¼ PB ; PB;k ¼ PB ; n¼ÀN m¼ÀN n¼ÀN

ð0Þ ð1Þ so PB;k ¼ PB;k þ PB;k:

Lemma 3.9. For 0opp1 and B; N as above,

NXÀ1 Pð0Þ XC B p : jj B;kjjSp pjj jj d Bp k¼0

Proof. By (18) and (19), we see that,

X /f ; h S PNnþk;Nnþkþ1 f #x J h #B x h #B x B ð Þ¼ 1=2 ð Jþ Jþ À JÀ JÀ Þ JADNnþk jJj X XN lJþ l /f #x; h #eJþ Sh #sJþ ¼ 1=2 J l Jþ l JADNnþk l¼0 jJj X XN lJÀ l /f #x; h #eJÀ Sh #sJÀ À 1=2 J l JÀ l JADNnþk l¼0 jJj and so

XN X XN lJþ Pð0Þ f #x l /f #x; h #eJþ Sh #sJþ B;kð Þ¼ 1=2 J l Jþ l n¼ÀN JADNnþk l¼0 jJj XN X XN lJÀ l /f #x; h #eJÀ Sh #sJÀ : À 1=2 J l JÀ l n¼ÀN JADNnþk l¼0 jJj

/ S A ð0Þ Since hJþ ; hIÀ ¼ 0 for all I; J D; PB;k has been expressed as the difference of two operators with orthogonal ranges and given Schmidt decompositions. ARTICLE IN PRESS

58 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

So, by Lemma 3.7,

! XN X XN Jþ p XN X XN J p 1 ðl Þ ðl À Þ jjPð0Þ jjp X l þ l B;k Sp 2 p=2 p=2 n¼ÀN JADNnþk l¼0 jJj n¼ÀN JADNnþk l¼0 jJj XN X 1 1 p p ¼ ðjjBJ jj þjjBJ jj Þ 2 p=2 þ Sp À Sp n¼ÀN JAD jJj Nnþk ! XN X p p=2 1 jjBI jjS ¼ p : 2p=2þ1 jIj n¼ÀN IADNnþkþ1

Consequently,

! p=2 NXÀ1 X p 1 jjBI jjS 1 jjPð0Þ jjp X p ¼ jjBjjp : & B;k Sp p=2þ1 p=2þ1 d 2 jIj 2 Bp k¼0 IAD

In seeking a converse to Proposition 3.4 for 0opo1; we shall ﬁrst suppose that A d B Bp ; a simple density argument will then give the full result.

A d Proposition 3.10. Let 0opo1: There exists a constant Cp such that if B Bp then jjBjj d pCpjjPBjj : Bp Sp

Proof. Note that

! ! XN XN PB;k ¼ DNnþk PB DNmþkþ1 n¼ÀN m¼ÀN

P P and so P P ; as N D and N D are norm 1 jj B;kjjSp pjj BjjSp n¼ÀN Nnþk m¼ÀN Nmþkþ1 projections. Consequently,

NXÀ1 NXÀ1 NjjP jjp X jjP jjp X ðjjPð0Þ jjp ÀjjPð1Þ jjp Þ B Sp B;k Sp B;k Sp B;k Sp k¼0 k¼0 NXÀ1 X C jjBjjp À jjPð1Þ jjp ð20Þ p d B;k Sp Bp k¼0 ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 59 by Lemma 3.9. However, for k = 0; y; N À 1; we have by (17) and Lemma 3.8,

XN XmÀ1 Pð1Þ p PNnþk;Nmþkþ1 p jj B;kjjSp p jj B jjSp m¼ÀN n¼ÀN ! p XN XmÀ1 X jjBI jjS 2ðNnÀNmÀ1Þp=2 p p 1=2 N N A jIj m¼À n¼À I DNmþkþ1 ! p XN X XmÀ1 jjBI jj ÀðNmþ1Þp=2 Sp Nnp=2 ¼ 2 1=2 2 N A jIj N m¼À I DNmþkþ1 !n¼À p XN X Nmp=2 jjBI jjS 2 2ÀðNmþ1Þp=2 p ¼ 1=2 Np=2 N A jIj 2 À 1 m¼À I DNmþkþ 1 ! p Àp=2 XN X 2 jjBI jjS ¼ p : 2Np=2 À 1 1=2 m¼ÀN IADNmþkþ1 jIj

Therefore, ! p NXÀ1 Àp=2 X Àp=2 2 jjBI jjS 2 jjPð1Þ jjp p ¼ jjBjjp : B;k Sp p Np=2 1=2 Np=2 d 2 1 2 1 Bp k¼0 À IAD jIj À

So, by (20), we see that, for all N; 1 2Àp=2 jjP jjp X C À jjBjjp : B Sp p Np=2 d N 2 À 1 Bp

Choose N large enough so that C 2Àp=2 40 to obtain the required result. & p À 2Np=2À1

A A d Corollary 3.11. Let 0opp1: If PB Sp then B Bp : Moreover, there exists a constant Cp such that jjBjj d pCpjjPBjj : Bp Sp

Proof. For any positive integer N; let X ðNÞ ðNÞ D :¼fIn;kAD: jnjpN; jkjpNg and B ðtÞ¼ BI hI ðtÞ: IADðNÞ

ðNÞ d ðNÞ Then, B AB and so jjB jj d pCpjjP ðNÞ jj by Proposition 3.10. But, p Bp B Sp X # # ðNÞ # PBðNÞ ð f xÞ¼ mJ fhJ BJ x ¼ P PBð f xÞ; JADðNÞ ARTICLE IN PRESS

60 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 where PðNÞ is the orthogonal projection on L2ðR; HÞ deﬁned by

ðNÞ ðNÞ P ðhJ #xÞ¼hJ #x if JAD ;

¼ 0 otherwise: ð21Þ

Therefore,

ðNÞ ðNÞ jjB jj d pCpjjP PBjj pCpjjPBjj Bp Sp Sp

ðNÞ for all N: But fjjB jj d g is an increasing sequence and so we see that Bp

ðNÞ jjBjj d ¼ lim jjB jj d pCpjjPBjjS : & Bp N-N Bp p

In summary, combining Propositions 3.4, 3.5, 3.11 and Corollary 3.6, we obtain our main result.

A A d Theorem 3.12. For 0opoN; PB Sp if and only if B Bp : Moreover,

CpjjBjj d pjjPBjj pKpjjBjj d : Bp Sp Bp

4. From dyadic paraproducts to Hankel operators

Let 1opoN; and let B: R-LðHÞ be locally integrable. We say that B is in the operator Besov space BpðRÞ; if

Z Z p jjBðxÞÀBð yÞjjS jjBjjp :¼ p dx dy N: Bp 2 o R R jx À yj

Theorem 4.1 (Peller [Pel2]). Let 1opoN; and let B: R-LðHÞ be antianalytic and locally integrable. Then the following are equivalent:

2 2 (1) The vector Hankel operator GB: L ðR; HÞ-L ðR; HÞ is in Sp: (2) BABpðRÞ:

We can use our results on vector paraproducts, along with the averaging procedure found in [Pet] to obtain an alternative proof of the sufﬁciency of this d condition. One would expect Bp to be continuously included in Bp ; and we show this here. ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 61

Lemma 4.2. Let 1opoN: Then there exists a constant Kp40 such that if BABpðRÞ d then BAB and jjBjj d pKpjjBjj : p Bp Bp

Proof. We use a method from [EJPenX]. It is easily shown that, for any AASp and JAD; Z Z jjBJ jjS 1 2 p jjBðxÞÀm Bjj dx jjBðxÞÀAjj dx: 1=2 p J Sp p Sp jJj jJj J jJj J

Letting A ¼ Bð yÞ and then averaging for yAJ; we see that Z Z jjBJ jjS 2 p jjBðxÞÀBð yÞjj dx dy: 1=2 p 2 Sp jJj jJj J J

Supposing that IADn for nAZ and using Ho¨ lder’s inequality, we see that

X p X Z Z jjBJ jjS 1 p 2p jjBðxÞÀBð yÞjjp dx dy p=2 p 2 Sp JADðIÞ jJj JADðIÞ jJj J J Z Z XN X 1 ¼ 2p jjBðxÞÀBð yÞjjp dx dy nÀm 2 Sp m n A ð2 jIjÞ J J Z¼ ZJ DmðIÞ 2p ¼ K ðx; yÞjjBðxÞÀBð yÞjjp dx dy; 2 I Sp jIj R R where

XN X 2ðmÀnÞ KI ðx; yÞ¼ 2 wJ ðxÞwJ ð yÞ: m¼n JADmðIÞ

Clearly, if either xeI or yeI then KI ðx; yÞ¼0: Suppose that x; yAI; with xay and nÀm let JADmðIÞ: Then wJ ðxÞwJ ð yÞ¼0ifjx À yj4jJj¼2 jIj: So,

nþ½log2XðjIj=jxÀyjÞ 2 2ðmÀnÞ jIj KI ðx; yÞp 2 p 2: m¼n 3jx À yj

Therefore, for all nAZ; Z Z X X p p p jjBJ jjS 2 jjBðxÞÀBð yÞjjS p p dx dy: p=2 p 2 3 R R x y IADn JADðIÞ jJj j À j ARTICLE IN PRESS

62 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78

Letting n- À N we see that

X p p jjBJ jjS 2 jjBjjp ¼ p jjBjjp : & d p=2 p Bp Bp 3 JAD jJj

For aAR and rARþ; let Da;r denote the translated, dilated dyadic grid given by

Da;r ¼f½a þ r2Ànk; a þ r2Ànðk þ 1ÞÞ: n; kAZg:

a;r a;r 2 For JAD ; let hJ denote the corresponding Haar function, normalised in L ðRÞ: We define the dyadic shift Sa;r on L2ðRÞ by X Sa;rð f #xÞ¼ /f ; ha;rSðha;r#x À ha;r#xÞ I Iþ IÀ IADa;r pffiffiffi for an elementary tensor f #xAL2ðR; HÞ: Note that Sa;r has norm 2: Let H: L2ðR; HÞ-L2ðR; HÞ denote the vector Hilbert transform on R; so Z f ðÁ À yÞ Hð f #xÞ¼ p:v: dy #x: R y

N Then, (see [Pet]), there exists a function aAL ðRÞ and a constant c040 such that the operator T: L2ðR; HÞ-L2ðR; HÞ given by

2 Tð f #xÞ¼c0Hð f #xÞþðaf Þ#x ð f AL ðRÞ; xAHÞ is contained in the WOT-closed convex hull of the set fSa;r: aAR; rARþg: We begin by showing that for 1opoN; there exists a constant C˜p40 such that

Sa;r; B C˜ B BAB : 22 jj½ jjSp p pjj jjBp ð pÞ ð Þ A A þ a;r For a R; r R ; let PB denote the vector paraproduct with respect to the dyadic grid Da;r; given by X a;r # a;r# PB f x ¼ mI fhI BI x IADa;r

a;r a;r and LB its adjoint. Let RB be the operator deﬁned on elementary tensors by X a;r # a;r# RB f x ¼ fI hI mI Bx: IADa;r

Here, mI f ; fI ; BI and mI B denote Haar coefﬁcients and averages with a;r a;r a;r a;r respect to the grid D : Then it is easily shown that MB ¼ PB þ LB þ RB ; ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 63 see [Pet], and so

a;r a;r a;r a;r a;r a;r a;r a;r a;r a;r S M M S S P P S S L L S jj B À B jjSp p jj B À B jjSp þjj B À B jjSp Sa;rRa;r Ra;rSa;r : þjj B À B jjSp

For F a function (scalar, vector or operator valued) deﬁned on R; let

U a;rFðtÞ¼rÀ1=2Fððt À aÞ=rÞ; V a;rFðtÞ¼Fððt À aÞ=rÞ;

a;r 2 a;r Then U is a unitary map on L ðR; HÞ; V is an isometry on Bp and

a;r a;r a;r À1 U PB ðU Þ ¼ PV a;rB: ð23Þ

If BABp; then pffiffiffi a;r a;r a;r a;r a;r a;r S P P S 2 S P 2 2 P a;r jj B À B jjSp p jj jjjj B jjSp ¼ jj V BjjSp pffiffiffi pffiffiffi 2 2 C V a;rB 2 2 C B p pjj jjBp ¼ pjj jjBp by (23), Theorem 3.12 and Lemma 4.2. It is similarly shown that jjSa;rLa;r À pffiffiffi B La;rSa;r 2 2 C B : B jjSp p pjj jjBp Finally, it may be seen that X a;r a;r a;r a;r # À1=2 # ðS RB À RB S Þð f xÞ¼ jIj fI ðhIþ À hIÀ Þ BI x: IADa;r P I / I S I If BI has Schmidt decomposition BI ¼ ln Á; en sn; then ffiffiffi X XN p I 2 l h À h Sa;rRa;r Ra;rSa;r f #x n/f ; h S Iþ ffiffiffi IÀ #/x; eI SsI ; ð B À B Þð Þ¼ 1=2 I p n n IADa;r n¼0 jIj 2 which is an expression in Schmidt form and so

X XN p=2 I p a;r a;r a;r a;r p 2 ðl Þ p=2 p jjS R À R S jj ¼ ¼ 2 jjBjj da;r pKpjjBjj B B Sp p=2 Bp Bp IADa;r n¼0 jIj

da;r a;r by Lemma 4.2, where Bp is the dyadic Besov space deﬁned with respect to D : This shows (22). Let B: R-LðHÞ; BABp; and suppose that B is locally bounded with respect to a;r A A þ the operator norm on LðHÞ: Let ðSgÞgAG be a net in convfS : a R; r R g which converges to the operator T introduced above in the weak operator topology. It follows immediately from (22) that S ; M C˜ B : To proceed to the WOT- jj½ g BjjSp p pjj jjSp limit, we require the following elementary lemma. ARTICLE IN PRESS

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Lemma 4.3. Let H be a Hilbert space, let 1ppoN; and suppose that ðAgÞgAG is a bounded net of operators in SpDLðHÞ: Furthermore, suppose that there exists a dense subspace A of H and a sesquilinear form A A-C; ðx; yÞ//Ax; yS; such that limgAG /Agx; yS ¼ /Ax; yS for all x; yAA: Then A extends to a bounded linear operator on H; AASp; and A sup A : jj jjSp p gAG jj gjjSp

Proof. From

/Ax; yS lim /A x; yS sup A x y x y sup A j j¼ j g jp jj gjjjj jjjj jjpjj jjjj jj jj gjjSp gAG gAG gAG for all x; yAA; it follows that A extends to a bounded sesquilinear form on H H; and therefore deﬁnes a bounded linear operator on H; which we also denote by A: The estimate for the Sp norm of A is easily obtained from the identity

jjAjjp Sp () XN / S p A N ¼ sup j Aen; sn j : N N; feng; fsngn¼1 orthonormal systems in H n¼1 see (13), and the density of A in H: &

We can now ﬁnish the proof of the direct implication in the theorem. Let A ¼ ff AL2ðR; HÞ; f has compact supportg: Since B is locally bounded, the commutator ½H; MB deﬁnes a sesquilinear form on A A; and one has

1 lim /½Sg; MBx; yS ¼ /½T; Bx; yS ¼ /½H; Bx; yS gAG c0 by the WOT convergence of ðSgÞgAG to T: Thus by the previous lemma,

C˜ H; M p B : 24 jj½ BjjSp p jj jjBp ð Þ c0

It is not difficult to see that the locally bounded functions are dense in Bp for p41: For a given BABp; one can for example choose the sequence given by BnðxÞ¼ ˜ i ˜ Bðx þ nÞ; where B denotes the harmonic extension of B to the upper half plane. Then each Bn is locally bounded, and ðBnÞnAN converges to B in Bp by the Dominated Convergence Theorem, subharmonicity and a vector version of Fatou’s Thm, see e.g. [Ni, Section 3.11]. By density, we obtain (24) for all BABp: Using that > 1 2 & PH2ðR;HÞ½H; MBPH ðR;HÞ ¼ 2i GB for antianalytic B finishes the proof. We note also that a version of Theorem 4.1 holds, for 0opp1; using appropriate definitions of the operator Besov spaces, for such p; see [Pel2]. ARTICLE IN PRESS

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5. Little Hankel operators and dyadic paraproducts in several variables

In this section, we want to use the vector valued results above to obtain a characterisation of Schatten class dyadic paraproducts in several variables and of Schatten class little Hankel operators on certain product domains. As in the case of vector paraproducts, the method of nearly weakly orthonormal sequences from [RS2] provides an alternative route to obtain the characterisation of the symbols of Schatten class paraproducts, although this does appear not explicitly in the literature. Let nAN: We write R ¼ Dn for the collection of dyadic rectangles in Rn: For ? y ? R ¼ I1 In; let hRðt1; ; tnÞ¼hI1 ðt1Þ hIn ðtnÞ: The collection ðhRÞRAR is then the product Haar basis of L2ðRnÞ: n / S For a locally integrableR function f on R ; we denote the Haar coefficient f ; hR 1 y ? by fR; and the average jRj R f ðt1; ; tnÞ dt1 tn by mRf : Let bAL2ðRnÞ: The densely defined linear mapping on L2ðRnÞ given by X f / hRbRmRf ð25Þ RAR is the multivariable dyadic paraproduct with symbol b; denoted by pb: ðnÞ If we want to make clear that we take the paraproduct in n variables, we write pb : 2 n 2 n For 1pipn; let Pi: L ðR Þ-L ðR Þ denote the Riesz projection in the ith variable > 2 n and Pi denote I À Pi: Then P ¼ P1?Pn is the orthogonal projection from L ðR Þ n onto the Hardy space H2ðRnÞ: We identify functions in the Hardy space H2ðCþ Þ on the n-fold product of the upper half planes with their boundary values in n n n % n H2ðR Þ in the usual manner, and we write H2ðR Þ¼H2ðCþ Þ: Let H2ðR Þ¼ ff AL2ðRnÞ: f%AH2ðRnÞg: n % n The densely defined linear map on H2ðR Þ-H2ðR Þ given by

/ >? > f P1 Pn bf ð26Þ

ðnÞ is the little Hankel operator with symbol b; denoted by gb: Again, we will write gb if we want to emphasize that the Hankel operator is taken with respect to n variables. The characterisations of bounded multivariable dyadic paraproducts and little Hankel operators in terms of their symbols are by no means a simple extension of the one-dimensional results. For n ¼ 2; boundedness of dyadic paraproducts was characterised in terms of an oscillation property of the symbol over all open sets in Rn by [Chang], and this gave n rise to a characterisation of the dual of the Hardy space H1ðCþ Þ in terms of oscillation properties [ChangFef]. Only recently, it was shown that also the 2 boundedness of little Hankel operators on H2ðCþ Þ can be characterised in terms of an oscillation property over open sets, in the course of the solution of the ARTICLE IN PRESS

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2 long-standing weak factorisation problem on H1ðCþ Þ [FSa,FLa]. For nX3; no such characterisation is known. Little Hankel operators on the unit ball in Cn; or more generally, on smoothly bounded strictly pseudoconvex sets, are much better understood [BoPelo2]. The main point of this section is to show that because of the good behaviour of Schatten class vector paraproducts and vector Hankel operators, multivariable paraproducts and little Hankel operators of Schatten class on certain product domains can be characterised quite easily in terms of their symbols. 2 þ2 In [BoPelo1], it is shown that for bAH ðC Þ; the little Hankel operator gb on 2 þ2 2 2 þ2 H ðC Þ is of trace class, if and only if @1 @2 b is integrable on C ; that is, b is 2 in the Besov space B1ðR Þ: This appears as a special case of a consideration of tube domains over symmetric cones. It is conjectured that these results extend at least to 1opp2 (see also [Ru] for an overview of known results and open questions of little Hankel operators on various types of domains in several dimensions). We will give here a Besov space characterisation of the symbols for 1opoN for little Hankel operators, and for 0opoN for multivariable dyadic paraproducts.

Theorem 5.1. Let nAN; 0 p N; and let bAL2ðRnÞ: Then p AS if and only if o o b p P 1=p 1 p A p=2 jbRj oN; and the Sp norm of pb is equivalent to this expression. R R jRj

Proof. We prove this statement by induction over n: For n ¼ 1; this is just Theorem 2.1(2). Suppose that the statement is true for some nAN: Given bAL2ðRnþ1Þ; we ðnþ1Þ understand the multivariable paraproduct pb as a vector-valued paraproduct in one variable, deﬁning BðtÞ¼pðnÞ for tAR: We write b for the function on Rn R bðÁ;y;Á;tÞ I y - y given by ðt1; ; tnÞ I bðt1; ; tn; tÞhI ðtÞ dt: Then pðnÞ ¼ B ; and it is easy to see that pðnþ1Þ: L2ðRnþ1Þ-L2ðRnþ1Þ is unitarily bI I b 2 2 n 2 2 n equivalent to PB: L ðR; L ðR ÞÞ-L ðR; L ðR ÞÞ via the natural unitary equivalence L2ðRnþ1Þ-L2ðR; L2ðRnÞÞ: Applying the induction hypothesis and Theorem 3.12, we obtain

X 1 X 1 jjpðnþ1Þjjp ¼jjP jjp ^ jjB jjp ¼ jjpðnÞjjp b Sp B Sp p=2 I Sp p=2 bI Sp IAD jIj IAD jIj X X ^ 1 1 p jðbI ÞR0 j p=2 0 p=2 IAD jIj R0ADn jR j X 1 b p: & ¼ p=2 j Rj RADnþ1 jRj The same method applies for the characterisation of the symbols of little Hankel 2 þn operators on H ðC Þ of Schatten class Sp; 1opoN: ARTICLE IN PRESS

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n ðiÞ We need the following notation. For iAf1; y; ng; tAR and f : R -C; let Dt be the ﬁnite difference operator in the ith coordinate given by

ðiÞ n ðDt f ÞðxÞ¼f ðx1; y; xi þ t; y; xnÞÀf ðx1; y; xnÞ x ¼ðx1; y; xnÞAR :

n n For 1opoN; we say that b: R -C is in BpðR Þ; if Z Z ð1Þ? ðnÞ p jðDt1 Dtn bÞðxÞj Q dx1? dxn dt1? dtnoN: n n n 2 R R i¼1 jtij

We denote the seminorm defined by the pth root of the expression above by b n : jj jjBpðR Þ Applying the well-known equivalence of the ‘‘harmonic analysis’’ definition and the ‘‘complex analysis’’ definition of analytic Besov class functions coordinatewise (see e.g. [Ni, Theorem 8.7.2] for a version on the unit disc), one sees easily that for b n analytic in Cþ ; the expression on the left is equivalent to Z pÀ2 pÀ2 p jIz1j yjIznj j@z ?@z bðzÞj dz1? dzn: n 1 n Cþ

Theorem 5.2. Let 1opoN; and let bAH% 2ðRnÞ: Then the following are equivalent:

2 n - % 2 n (1) gb: H ðR Þ H ðR Þ is in Sp; n (2) bABpðR Þ;

n and the BpðR Þ norm is equivalent to the Sp norm.

Proof. For n ¼ 1; this is just Peller’s characterisation of Schatten class Hankel operators in the case 1opoN: As before, we use induction over the dimension n: Suppose that the statement above holds for some nAN: Let bAH% 2ðRnþ1Þ: We deﬁne - 2 n ðnÞ A an operator valued function B: R LðL ðR ÞÞ by BðtÞ¼gbðÁ;?;Á;tÞ: For each t R; bðÁ; ?; Á; tÞ is an antianalytic function in n variables, and bðÁ; ?; Á; tÞAH% 2ðRnÞ for almost every tAR: It is easy to verify that the vector Hankel operator GB is unitarily equivalent to the little Hankel operator gb via the canonical unitary L2ðR; L2ðRnÞÞ-L2ðRnþ1Þ: Therefore by Theorem 4.1,

Z Z p jjBðtÞÀBðsÞjjS jjgðnþ1Þjjp ¼jjG jjp ^ p dt ds b Sp B Sp 2 R R jt À sj Z Z jjBðs þ tÞÀBðsÞjjp Sp ¼ 2 dt ds R R jtj ARTICLE IN PRESS

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Z Z jjgðnÞ À gðnÞ jjp bðÁ;?;Á;sþtÞ bðÁ;?;Á;sÞ Sp ¼ 2 dt ds R R jtj Z Z jjgðnÞ jjp bðÁ;?;Á;sþtÞÀbðÁ;?;Á;sÞ Sp ¼ 2 dt ds R R jtj Z Z p jjbðÁ; ?; Á; s þ tÞÀbðÁ; ?; Á; sÞjj n ^ BpðR Þ 2 dt ds R R jtj

b nþ1 : & ¼jj jjBpðR Þ

The same method applies of course for little Hankel operators on domains of the n form D ¼ Cþ ODCnþm in the case where we have a Besov space type characterisation of Schatten class little Hankels on H2ðOÞ; for example, if ODCm is a smoothly bounded convex domain of finite type (see [BoPelo2, Theorem 1.3]). For such domains, we can define the Hardy class H2ðDÞ¼ n H2ðCþ Þ#H2ðOÞDL2ðRn @OÞ and, for bAH% 2ðDÞ; define the little Hankel 2 operator gb on a dense subspace of H ðDÞ:

n Theorem 5.3. Let D ¼ Cþ OCCnþm; where O is a smoothly bounded convex domain of ﬁnite type in Cm: Let bAH% 2ðDÞ; and let 1opoN: Then the following are equivalent. (1) Z Z ð1Þ ðnÞ p jjD ?D bðx1; y; xn; Á; ?; ÁÞjj t1 tn BpðOÞ Q dx1? dxn dt1? dtnoN; n n n 2 R R i¼1 jtij

2 - % 2 (2) gb: H ðDÞ H ðDÞ is in Sp:

2 (For the deﬁnitions of H ðOÞ and BpðOÞ; see [BoPelo2].) It would be interestingS to see whether this method is also useful for domains of the form zACþ fzg nþ1 n þ OzCC ; where Oz is a ‘‘sufﬁciently nice’’ domain in C for each zAC : The case of the light cone, which was studied by Bonami and Peloso [BoPelo1], would be an interesting candidate for this approach.

6. Products of dyadic paraproducts

Operator-theoretic properties of the product Gf Gg of a Hankel operator and the adjoint of a Hankel operators have been studied for a long time, partly motivated by the identity Gf Gg ¼½Tf ; TgÞ; where ½Tf ; TgÞ denotes the semi-commutator Tf Tg À 2 Tfg of the Toeplitz operators Tf and Tg on H ðDÞ; the Hardy space of the unit disc. ARTICLE IN PRESS

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One example for this is the Axler–Chang–Sarason–Volberg Theorem, which characterises compact products of Hankel operators in terms of certain Douglas algebras ([V,ACS]). The study of such products of Hankel operators is in general much more difﬁcult than the study of single Hankel operators. There is still no full characterisation of boundedness and Schatten class membership in terms of oscillation properties of the symbols. In [Zh], it was shown that the natural reproducing kernel condition

lim jjGgkzjj2jjGf kzjj2 ¼ 0 ð27Þ jzj-1

is equivalent to compactness of the product Gf Gg: Here, fkzgzAD denote the normalised reproducing kernels on H2ðDÞ: However, it is an open question whether the reproducing kernel condition

sup jjGgkzjj2jjGf kzjj2oN; ð28Þ zAD which can be understood as a ‘‘combined’’ oscillation condition and was shown to be necessary in [Zh], implies the boundedness of the product GgGf : Slightly stronger sufﬁcient conditions have been found in [Zh,TVZh]. It is also open for which symbols g; f AL2ðTÞ the product of Hankel operators GgGf is in the Schatten–von Neumann class Sp [Pel3], although partial results were found in [VI], and estimates for the singular values of such products have been obtained in [RS1, Section VII]. In this section, we will again consider dyadic paraproducts as a model case for Hankel operators and study operator-theoretic properties for products pf pg of dyadic paraproducts. A sesquilinear version of the dyadic sweep from (11), given by X w Q½f ; g¼ I f g% ð f ; gAL2ðRÞÞ ð29Þ jIj I I IAD

(see also [BlPo]), allows us to address this dyadic analogue in a very simple fashion. As before, we collect some elementary properties of the sesquilinear map Q:

Lemma 6.1. (1) jjQ½f ; gjj1pjj f jj2jjgjj2; 0 0 (2) PI Q½f ; g¼PI Q½PI f ; PI g

2 We ﬁrst need an analogue of Proposition 2.3. For f ; gAL ðRÞ; let D½f ;g be deﬁned on the Haar basis by 0 1 1 X D h ¼ @ f g% Ah : ½f ;g I jIj J J I JAD0ðIÞ ARTICLE IN PRESS

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Lemma 6.2.

pgpf ¼ pQ½f ;g þ DQ½f ;g þ D½f ;g:

Proof. Exactly as in Proposition 2.3. &

Theorem 6.3. Let f ; gAL2ðRÞ: Then the following are equivalent:

2 - 2 (1) pgpf deﬁnes a bounded linear operator on L ðRÞ L ðRÞ: (2) Q½f ; gABMOd ; and

X 1 sup fJ g%J oN; ð30Þ IA jIj D JAD0ðIÞ

(3) 1 0 0 sup jjQ½PI f ; PI gjj1oN; IAD jIj

(4) sup jjpgpf hI jjoN: IAD Proof. (1) ) (4) obvious.

(4) ) (2): Remember from 6.2 that pQ½f ;g is the superdiagonal part of pgpf ; DQ½f ;g is the subdiagonal part, and D½f ;g is the diagonal part with respect to the Haar basis. The uniform boundedness of jjpgpf hI jj therefore implies uniform boundedness of jjD h jj ¼ j/pp h ; h Sj and thereby (30). Furthermore, note that P f ;g I g f I I 1 2 2 0 2 2 A jIj JAD0ðIÞ jðQ½f ; gÞJ j ¼jjpQ½f ;ghI jj ¼jjPI pgpf hI jj pjjpgpf hI jj for all I D; and therefore Q½f ; gABMOd : 0 0 0 0 (2) ) (3): Note the identity Q½PI f ; PI g¼wI mI ðQ½PI f ; PI gÞ þ PI ðQ½f ; gÞ: Thus

1 0 0 0 0 1 0 0 jjQ½PI f ; PI gjj p jmI ðQ½PI f ; PI gÞj þ jjPI ðQ½PI f ; PI gÞjj I 1 I 1 j j j j

X 1 1 ¼ fJ g%J þ jjPI Q½f ; gjj : ð31Þ jIj jIj 1 JAD0ðIÞ

1 (3) ) (2): By the uniform boundedness of the projections ðPI ÞIAD on L ðRÞ; the uniform boundedness of the left-hand side in (31) implies the uniform boundedness 1 N of the right-hand side. Therefore jjQ½f ; gjjBMOd ¼ supIAD jIj jjPI Q½f ; gjj1o ; and also (30) holds. ARTICLE IN PRESS

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(2) ) (1): By Theorem 2.1, pQ½f ;g and DQ½f ;g are bounded, and by (30), D½f ;g is bounded. Thus pgpf ¼ pQ½f ;g þ DQ½f ;g þ Df ;g is bounded. &

Condition (30) looks like a natural sesquilinear analogue to the BMOd condition. However, a simple example shows that it is not sufﬁcient for the boundedness of pgpf :

2 Remark 1. There exists functions f ; gAL ðRÞ such that (30) holds, but pgpf does not deﬁne a bounded linear operator on L2ðRÞ:

Proof. Let f ; g be defined by the following Haar coefficients. For kX0; let Ik ¼ Àk 4 k k ½0; 2 : Let a40; 1=2oa o1; and let f þ ¼ fIÀ ¼ g þ ¼ a ; gIÀ ¼Àa for each kX0: Ik k Ik k 2 LetP all remaining Haar coefficients of f and g be 0: Then g; f AL ðRÞ and % A JAD0ðIÞ fJ gJ ¼ 0 for each I D; but

2 X X X X 2 1 jðQ½f ; gÞ j ¼ fJ g%J À fJ g%J I jIj ID½0;1 ID½0;1 JADðI þÞ JADðI ÀÞ

2 XN X X XN 1 k 4k ¼ fJ g%J À fJ g%J ¼ 2 4a jI j k¼0 k A þ JA I À k¼0 J DðIk Þ Dð k Þ ¼þN:

The remark follows now by Theorem 6.3(2). &

A further natural candidate for a ‘‘combined’’ BMOd -condition is given by

1 X sup j fJ jjgJ j: ð32Þ IA jIj D JAD0ðIÞ

D It turns out that this condition leads even to a stronger property. ForPsAfÀ1; 1g ; 2 - 2 / Plet Ts denote the dyadic martingale transform L ðRÞ L ðRÞ; IAD hI fI IAD sðIÞhI fI : Then we have the following result:

Theorem 6.4. Let f ; gAL2ðRÞ: Then the following are equivalent:

(1) For each dyadic martingale transform Ts; pgTspf deﬁnes a bounded linear 2 operator on L ðRÞ; and the operators ðpgTspf ÞsAfÀ1;1gD are uniformly bounded. (2) 1 X sup j fJ jjgJ j: IA jIj D JAD0ðIÞ ARTICLE IN PRESS

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(3)

sup jjpgTspf hI jjoN: IAD;sAfÀ1;1gD

Proof. (1) ) (3) obvious. (3) ) (2): Let IAD: Then for each sAfÀ1; 1gD; *+ 1 X X /pT p h ; h S ¼ signðJ; IÞsðJÞh f ; signðJ; IÞh g g s f I I jIj J J J J JAD0ðIÞ JAD0ðIÞ 1 X ¼ sðJÞg% f : jIj J J JAD0ðIÞ

A D Choosing an appropriate sequence ðsðJÞÞJAD fÀ1; 1g ; we obtain

pffiffiffi 1 X pffiffiffi X 2 % / S jgJ jj fJ jp 2 sðJÞgJ fJ p sup j pgTspf hI ; hI j jIj jIj D JAD0ðIÞ JAD0ðIÞ sAfÀ1;1g pffiffiffi 2 p sup jjpgTspf hI jj: jIj sAfÀ1;1gD

A A D (2) ) (1): Observe that pgTspf ¼ pgpTsf : For all I D and all s fÀ1; 1g ;

X X 1 0 0 1 wJ 1 jjQ½PI ðTsf Þ; PI gjj ¼ sðJÞg%J fJ p jgJ jjfJ j: jIj 1 jIj jJj jIj JAD0ðIÞ JAD0ðIÞ 1 (1) follows now from Theorem 6.3(3). &

Unfortunately, when considering products of operators it is not as easy as in the situation in Section 4 to pass from results on paraproducts to results on Hankel operators via averaging. The following remark shows that products of paraproducts and products of Hankel operators behave quite differently.

Remark 2. A seemingly natural dyadic analogue to Zheng’s necessary condition (28) is the following:

sup jjpf hI jj2jjpghI jj2oN: ð33Þ IAD

This condition is easily seen to be sufﬁcient for the uniform boundedness of all D operator products pgTspf ; sAfÀ1; 1g ; by Theorem 6.4(2). However, whenever the sets fIAD: fI a0g and fIAD: gI a0g are disjoint, the product pgTspf is 0 for all sAfÀ1; 1g: Thus one sees that (33) is not necessary. ARTICLE IN PRESS

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Finally, we want to characterise Schatten class products of paraproducts. First, let us look at the compact case.

Theorem 6.5. Let f ; gAL2ðRÞ: Then the following are equivalent: 2 - 2 (1) pgpf deﬁnes a compact linear operator on L ðRÞ L ðRÞ: (2) Q½f ; gAVMOd ; and

1 X lim fI g%I ¼ 0: ð34Þ I-N jIj JD! I

(3) 1 0 0 lim jjQ½PI f ; PI gjj1 ¼ 0: I-N jIj

(4) lim jjpgpf hI jj ¼ 0: I-N

Here, the limits in (2)–(4) are meant in the sense of (10), and convergence to 0 is meant to be uniform,asjIj-0orjIj-N; respectively.

Proof. (1) ) (4): For NAN; let PðNÞ denote the orthogonal projection deﬁned in ðNÞ (21). ðP ÞNAN converges to the identity in the strong operator topology, so ðNÞ pgpf P À pgpf converges to 0 in norm, and we obtain (4). For the remainder of the proof, one shows (4) ) (2) ) (1) and (2) 3 (3) along the same lines as in the proof for Theorem 6.3, using Theorem 2.1(3) and Lemma 6.2. &

Now we look at the Schatten classes Sp; 1ppoN: In this case, it turns out that if D pgpf ASp then also pgTspf ASp for all sAfÀ1; 1g ; with uniformly bounded Sp- norm. We get a natural combined dyadic Besov space condition for the symbols f and g:

Theorem 6.6. Let f ; gAL2ðRÞ; 1ppoN: The following are equivalent: (i) A A D pgTspf Sp for each s fÀ1; 1g ; and ðpgTspf ÞsAfÀ1;1gD is bounded in Sp: (ii) pgpf ASp: d (iii) Q½f ; gABp ; and

p X X 1 fJ g%J oN: jIj IAD JAD0ðIÞ ARTICLE IN PRESS

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(iv) X 1 p j fI gI j oN: I p IAD j j

(v) For all 1pqoN; X 1 Q P 0f P 0g p N p=q jj ½ I ; I jjqo : IAD jIj

Proof. We will show (ii) ) (iv) ) (v) ) (iii) ) (ii) and (iv) ) (i) ) (ii). (ii) ) (iv): For IAD; let

1 cI ¼ ðwI þþþ À wI þÀ À Þ jI þþj1=2 and

0 1 cI ¼ ðwIþþÀ À wI ÀÀþÞ: jI þþj1=2

0 The sequences ðcI ÞIAD and ðcI ÞIAD are not orthonormal, but it is easy to see that they are the images of the orthonormal Haar basis under bounded linear 0 maps A; B: In the notation of [RS2], ðcI ÞIAD and ðcI ÞIAD are weakly orthonormal. Therefore, X X j/pp c ; c 0Sjp ¼ j/Bpp Ah ; h Sjp jjAjjpjjBjjpjjpp jjp : g f I I g f I I p g f Sp IAD IAD

Notice that 8 > 1 þþ > if J ¼ I ; > 2jI þþj1=2 > > 1 > þÀ > À if J ¼ I ; <> 2jI þþj1=2 m c ¼ 1 J I > if JDI þþþ; > þþ 1=2 > jI j > > 1 > À if JDI þÀ À; > þþ 1=2 :> jI j 0 otherwise ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 75 and 8 > 1 þ > if J ¼ I ; > 4jI þþj1=2 > > > 1 À > À if J ¼ I ; > 4jI þþj1=2 > > 1 > þþ > if J ¼ I ; <> 2jI þþj1=2 0 mJ c ¼ 1 I > À if J ¼ I ÀÀ; > þþ 1=2 > 2jI j > > 1 > if JDI þþÀ; > þþ 1=2 > jI j > > 1 > À if JDI ÀÀþ; > þþ 1=2 :> jI j 0 otherwise:

/ 0S 1 þþ Thus mJ cI ; mJ cI equals 4jI þþj for J ¼ I and 0 otherwise, giving

X X 0 p 1 1 p j/p p c ; c Sj ¼ j f þþ g þþ j : g f I I 4p jI þþjp I I IAD IAD

0 PAdjusting the deﬁnitionsP of cI and cI ; we obtainP corresponding expressions for 1 p 1 p 1 p þÀ þÀ Àþ Àþ ÀÀ ÀÀ IAD jI þÀjp j fI gI j ; IAD jI Àþjp j fI gI j ; and IAD jI ÀÀjpj fI gI j : Thus (iv) holds. P 1=2 (iv) ) (v): Let f ¼ IAD hI j fI gI j : Then

X 1 X 1 Q P 0f ; P 0g p Q P 0f p p=q jj ½ I I jjqp p=q jj ½ I jjq IAD jIj IAD jIj X 2p 1 0 2p 2p 2p p C2q jjPI fjj2q ¼ C2q jjfjj d p=q B2p;2q IAD jIj X 1 C2pK 2p C2pK f g p p 2q 2p;2qjjfjjBd ¼ 2q 2p;2q p j I I j : 2p jIj IAD

2q by Theorem 2.4, where C2q denotes the norm of the dyadic square function on L ; d d and K2p;2q denotes the equivalence constant between the Bp;q and the Bp norms from Theorem 2.4. (v) ) (iii): Suppose that (v) holds for some q; 1pqoN: Then by Ho¨ lder’s inequality, (v) holds in particular for q ¼ 1: Note that the projections ðPI ÞIAD are ARTICLE IN PRESS

76 S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 uniformly bounded on L1ðRÞ: We obtain X X X 1 p 1 0 0 p p 1 0 0 p jjPI Q½f ; gjj ¼ jjPI Q½PI f ; PI gjj pC jjQ½PI f ; PI gjj jIjp 1 jIjp 1 jIjp 1 IAD IAD IAD

d and it follows from Theorem 2.4 that Q½f ; gABp : Furthermore,

p X X X 1 1 0 0 p fJ g%J p jjQ½PI f ; PI gjj : jIj jIj 1 IAD JAD0ðIÞ IAD

Thus (iii) holds. (iii) ) (ii): This follows directly from Lemma 6.2. (iv) ) (i): Note again that pgTspf ¼ pgpTsf ; and that condition (iv) is invariant under exchanging f with Tsf : Condition (i) now follows by applying the implication (iv) ) (ii) proved above to the symbols Tsf and g: (i) ) (ii): This is immediate. &

Using Theorem 3.12, we also obtain a vector version of this result:

Corollary 6.7. Let H be a separable Hilbert space, let F; G: R-LðHÞ be weakly locally integrable, and let 1ppoN: Then the following are equivalent: (i) PGPF ASp; (ii) X 1 p p jjG FI jj oN: jIj I Sp IAD

Proof. The proof ðiiÞ)ðivÞ)ðiÞ)ðiiÞ in Theorem 6.6 also works in the vector case. We omit the details here. &

In [VI], it was shown that the condition X 1 p sup p j fI gI j oN ð35Þ kAN jIj IADk

is necessary for the product of Hankel operators GgGf to be in Sp: We have seen above that the stronger condition Theorem 6.6 (iv) holds whenever pgpf is in Sp: It would be interesting to know whether Theorem 6.6 (iv) holds (at least in some average sense) whenever GgGf is in Sp: Conversely, it would be of great interest to know whether a translation and dilation invariant version of condition 6.6 (iv) implies that pgpf is in Sp: We ﬁnish by stating this as a conjecture. ARTICLE IN PRESS

S. Pott, M.P. Smith / Journal of Functional Analysis 217 (2004) 38–78 77

As before, denote by Da;r; where aAR and rARþ; the dyadic grid obtained by dilating the standard dyadic grid D by r and then translating it by a:

Conjecture 6.8. Let 1ppoN; and let f ; gAH% 2ðRÞ: Suppose that X 1 p sup p j fI gI j oN: þ jIj aAR;rAR IADa;r

Then GgGf ASp:

Remark: After this paper had been refereed, V.V. Peller pointed out to us that a characterisation of little Hankel operators of Schatten class on the bidisk can be found in the article [LiPen] (in Chinese).

Acknowledgments

The authors gratefully acknowledge support by EPSRC Grant GR/R97610/01. Pott also acknowledges support by the Nufﬁeld Foundation.

References

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