CHARACTERIZATIONS OF BOUNDED MEAN OSCILLATION ON THE POLYDISK IN TERMS OF HANKEL OPERATORS AND CARLESON MEASURES

SARAH H. FERGUSON AND CORA SADOSKY*

Abstract. A scale of BMO spaces appears naturally in product spaces corre- sponding to the different, yet equivalent, characterizations of the class of functions of bounded mean oscillation in one variable. S.-Y. Chang and R. Fefferman charac- terized product BMO, the dual of the (real) H~e on product domains, in terms of Carleson measures. Here we describe two other BMO spaces, one con- tained in and the other containing product BMO, in terms of Carleson measures and Hankel operators. Both of these spaces play a significant role in on the polydisk and in multivariable operator theory.

0 Introduction

Recall that a ~b, integrable on the circle T, is said to be of bounded mean oscillation, r ~ BMO(T) if

I1r = sup T~ [r162 - r < oo, Ic_T I I Jz where Ct =~ft r162 is the mean value of the function r over the I, [JoNi]. In their seminal paper [FeSt], C. Fefferman and E. M. Stein gave alternative characterizations of the space BMO (T) in terms of and Carleson measures on the unit disk D. The famous duality result asserts that BMO(T) is the real dual of the Hardy space H~e (T) [Fe]. It is this characterization, together with Nehari's theorem, that connects the theory of BMO with the study of Hankel operators on H2( r). In passing from the disk to the polydisk, there are natural extensions of BMO in terms of bounded mean oscillation, duality and Carleson measures. An early example of Carleson [Carl] showed that these extensions are not equivalent. In *First author supported in part by a grant from the National Science Foundation. Second author supported in part by a grant from the Department of Energy.

239 JOURNALD'ANALYSE MATH[~MATIQUE, Vol. 81 (2000) 240 S. H. FERGUSON AND C. SADOSKY particular, the product analogue of the Carleson measure condition does not char- acterize the dual of H~e(T 2) [Fell. However, there is a description of the dual of H~e ('/~) in terms of Carleson measures (appropriately defined) on the bidisk, [Ch] and [Fell. Later in [ChFef], an atomic decomposition for H~e (~) (by atoms supported on open sets) was established, providing a version of bounded mean oscillation to characterize the dual of Hl(~). The space BMO(~) := H~e('Fd) *, d > 1, is called product BMO. The paper [ChFefl] is an excellent survey on this space and the product space program of Chang and Fefferman. The success in extending the central duality result to Hl(qrd), d > 1, which required deep analysis, deflected interest from other notions of bounded mean oscillation in product spaces, even at the cost of losing essential features of the one-dimensional theory. In fact, many of the defining properties of BMO, including bounded mean oscillation itself, do not carry over to product BMO. In this paper, we show how to recover the lost features of the theory in the context of the d-torus ql~, d > 1. For this, it is necessary to deal with two different spaces, whose definitions are natural analogues of BMO on the circle. The first space is the so-called little BMO space, denoted bmo(Td), studied by M. Cotlar and the second author in connection with the product and weighted inequalities on the d-torus [CoSa]. Little BMO coincides with the class of functions which are of bounded mean oscillation on rectangles. A function r E Ll('Iff2) is in bmo(q~) if and only if

[lr = sup _w Ir162215 dm(~)dm(A) < c~, LJCV II} IJ} where the constant Cr• is the average of r over the rectangle I • J. In one variable, bmo(T) = BMO('F); but, for all d > 1, we have the proper containment bmo(']~) C BMO(qI~), hence the name little BMO; cf. [CoSa2]. The second space considered here is the rectangular BMO space, denoted BMORect(qFd), which was introduced and studied in [Fet] and, more recently, in [Fefl] in connection with singular operators in product domains. A function r ~ L~(~) is in BMORect(Te) if

[{~bllRect= z,J_CTsup ff[1 ~1 fxfJ 1r A) - ~bj(ff) - Cz(A) + ~bz•162 < ~, where for each Iffl = 1, Cj(~) = ~ fj r n)dm(~) and, similarly, Cz(A) is the mean value of r A) over the interval I. In terms of Carleson measures on the bidisk, BMORect(qF2) coincides with the space of functions whose associated measure (determined by the double Poisson integral) satisfies the Carleson condition on rectangles. Using a construction due BOUNDED MEAN OSCILLATION ON THE POLYDISK 241 to Carleson [Car I ], R. Fefferman showed directly that BMO (q~) c BMO n~ct(q~) by producing a function in rectangular BMO which did not determine a bounded linear functional on H~ (T2) (via the usual integral pairing) [Fef]. It turns out that both bmo(Td) and BMOnect(T a) can be described completely in terms of Hankel operators on H2(qI~). We motivate this by recalling the one- variable relation between bounded mean oscillation and norms of Hankel operators. Let P+ : L 2(~7) --+ H 2 (qF) be the orthogonal projection onto the Hardy space H 2 ('IF) and let pl = I - P+. For ~ E L2('F), the operator FO, densely defined on analytic polynomials by the formula I'of := Pl(~bf), is called a Hankel operator with symbol ~b. Note that the symbol of the Hankel operator Fo is not unique, since it is determined only by P• Nehari's theorem [Ne] says that Fo extends to a bounded operator, defined on all of H 2 (7), if and only if there is a function/3 ~ L ~ ('~) such that F~ = F~ and that furthermore, in this case, IIr ll = infpJ-~=v• 11/311 , The Hilbert transform H acts on L 2 (I?) as H = -~P+ + iP-L; and since H~(qF) is the subspace of functions that are in L 1(T) together with their Hilbert transform, the duality BMO(qF) = H~e(qF)* yields the identification BMO(qF) = L~ + H(L~ In terms of orthogonal projections, we see that a function ~b E L1($) is in BMO(qF) if and only if there are functions ~1,/32 E L~176('1~) such that P+~b = P+/31 and P• = P• Thus, by Nehari's theorem together with the fact that H2(T) • is the space of complex conjugates of functions in H02(T), the space BMO('F) coincides with the space of functions ~b E L ~(T) for which both Hankel operators, F~ and F~, determine bounded operators on H2(qr). This, in turn, is equivalent to the L 2 boundedness of the commutator [M~, H ], where M~ is multiplication by the function ~b. Thus the characterization of BMO(qF) as dual to H~(~) yields the equivalence of the seminorms, I1~11. (defined above) and max{llr, ll, IIr~ll}. We shall show that the exact analogue of the one-variable relationship between bounded mean oscillation, bounded commutators and Hankel operators carries over to the space bmo(~) and the big Hankel operators on H 2 (T t). Furthermore, using new norm estimates for these operators, we show that the big Hankel operator F~ is bounded on H2('I~) if and only if sup~ew liFq~K~[I < c~, where K~ denotes the normalized Szeg6 kernel on the polydisk. A proof of this in the one variable case can be found in [Bon] and [Nik]. This result provides us with a uniform Carleson criterion to describe the space bmo(~). The Carleson criterion is then used to characterize the analytic functions of bounded mean oscillation on rectangles in terms of analytic multipliers between H ~(T a) and the d spaces of partial derivatives of functions in H2(qra), denoted by O~H2(II~), j = 1,..., d. Comparing with the analytic versions of results in [Ch] and [Fef], which describe the space BMOA('Fa) ( = P+(L~176 ) as multipliers from Hz(Da) into the space of mixed partials 242 S. H. FERGUSON AND C. SADOSKY

01... OaH2(~)a), yields examples of analytic functions which act boundedly on H x(T 2) but are not of bounded mean oscillation on rectangles. We also give two descriptions of the pre-dual of bmo(Ta), one in terms of an atomic decomposition and the other in terms of weak factorization. The relationship between rectangular BMO and little Hankel operators is more subtle than that for little BMO and big Hankel operators. For a function, being in rectangular BMO is equivalent to the boundedness of its associated little Hankel operator applied to normalized Szeg6 kernels; however, it is not known whether this suffices to imply the boundedness of the little Hankel on all of H2(Ta). It is sufficient for the big Hankel operators (proven here for d > 1 and in [Bon] in the case d = 1), but may be insufficient for the little Hankel operators for d > 1. If it were sufficient, then it would follow that weak factorization does not hold for HI(Ta), as it does for H 1 of the unit ball in C a, d > 1 [CRW]. Weak factorization for H 1('1~ 2) was asserted in [Lin], using an argument based on the atomic decomposition of product BMO and a reduction to rectangular atoms. Unfortunately, there is a counterexample to that reduction [Bo], and the question is still open. A different approach to this problem is through the study of the different BMO spaces, as discussed in Section 3. The paper is organized as follows. Section 1 contains the norm estimates for the big Hankel operators and commutators used to characterize bmo('~), an extension to the big Hankel operators of the estimate in [Bon] and [Nik] and a weak factorization for the pre-annihilator of the subspace of bounded functions supported on a half-plane, Corollary 1.5. This weak factorization, together with the norm estimates for Hankel operators, shows that functions in L ~('~) orthogonal to H ~176(Ta) do not necessarily have a corresponding weak factorization. Section 2 contains the new results on the space bmo(Ta), in particular, the Carleson criterion for a function to be of bounded mean oscillation on rectangles, its characterization in terms of big Hankel operators, and the characterization of the analytic sub- space bmoa(T a) := bmo (Ta) f3 H 2('~) in terms of analytic multipliers. An atomic decomposition and a weak factorization of the predual of bmo(q3z) are also given. Section 3 contains the new results on rectangular BMO and little Hankel operators, and a relation between rectangular mean oscillation and product BMO. The estimates established for the rectangular BMO norm and the little BMO norm yield, in particular, the proper inclusions for d > 1

bmoa(T~) C BMOA(']I~) C BMOARect(T~).

Finally, the characterization of the space of symbols of little Hankel operators, sitting between BMOA(Ta) and BMOAR,ct(Ta), in terms of the boundedness of BOUNDED MEAN OSCILLATIONON THE POLYDISK 243

the nested commutator, gives an appropriate function-space setting for the weak factorization problem for H x(T 2). In what follows, we limit the statements, as well as their proofs, to the case d = 2, in order to simplify the exposition. Since here we do not deal with the atomic decomposition of product BMO, -dependent covering lemmas necessary to handle atoms supported on open sets are not required, and all results remain valid, with obvious modifications, for d > 2.

0.1 Preliminary definitions and notation Throughout the paper, we denote by H 2 | L 2 the right half-plane of L2(~2), defined to be the subspace of functions whose biharmonic extension to the bidisk is analytic in the first variable. As a Hilbert space, the right half-plane is isomorphic to the Hilbert space tensor product of H 2 (~) and L 2 (T), so we are slightly abusing notation here. The upper half-plane ofL 2 ('1~) will be denoted by L ~ | H 2. Let ] denote the Fourier transform of function f; then a function f is in L 2 | if and only if ](m, n) = 0 for all indices (m, n) with n < 0. Similarly, f is in H 2 | L 2 if and only if ](m, n) -- 0 for all indices (m, n) with m < 0. Let P1 (respectively, P2) denote the orthogonal projection of L ~(qI~) onto H 2 | L 2 (respectively, L 2 | H2). Then the product P1P2 is the orthogonal projection onto H 2 (]l~). We recall the definition of the real Hardy space on the toms and the BMO duality. Let H~ = -iP~ + iP~ be the Hilbert transform in the jth variable, j = 1, 2. A function f E Ll(ql~) is in H~(T 2) if Hlf, H2f and H1H2f are in Ll(ql~). Equivalently, f E H~e(T2) if and only if the functions P1P2f, P~Pg.f, P1P~f and P~P~f are all integrable. The space BMO(T2) is by definition the dual ofH~ ('1~) and thus coincides with the space of functions ~ e L2(T2) for which there exist functions ~i E L~176 i = 0,... ,3, such that r =/3o + Hlf31 + H2~2 + H1H2~3; cf. [ChFefl]. The BMO norm is defined by IIr := infmaxo

1 Norm estimates of big Hankel operators on the torus

We begin by first estimating the norm of the big Hankel operator on H 2 (~I~2). We should point out that there is some overlap with the earlier work of M. Cotlar and the second author in [CoSal]. In that paper, the authors prove that the symbols of bounded big Hankel operators on H2('~) are in BMOr(~), the so- called restricted BMO class; the estimates provided there are in terms of the restricted BMO norm. It was known that the little BMO space bmo(~) was 244 S. H. FERGUSON AND C. SADOSKY properly contained in restricted BMO [CoSa2], and a problem posed by the second author was to determine the precise geometric relationship between bmo(Te) and restricted BMO. What we know now is that the relationship between these two BMO spaces, (1 - P1P2)BMOr = (1 - P1P2)bmo, is best understood in terms of big Hankel operators; for this reason we (re)prove, using different techniques, an analogue of the lifting theorem in [CoSal] which fits more naturally into our setting. It will follow from the results in this section that a function r is in bmo(T2) if and only if both r and r are in the restricted BMO class. Recall that the big Hankel operator with symbol r E L2(qI~) is densely defined on analytic polynomials (in 2 variables) by the formula Fcf = P•162 where P• : L2(]/'2) ~ H2(~) • is the orthogonal projection onto the orthogonal complement of the Hardy space H2(~). The Hankel operator PC is said to be bounded if it is bounded on analytic polynomials with respect to the L 2 norm. In this case, Fr extends to a bounded operator defined on all ofH 2 (~). It is clear that L ~ functions determine bounded big Hankel operators, and we always have the estimate

IIr ll < distr162 H~(qr2)) := sup I1r ZlI flcL~ (T2 ) for all r E L ~ (~). However, unlike in the case d = 1, there are bounded big Hankel operators on H2(T 2) which do not have bounded symbols [CoSa2]. It follows, by a standard duality argument, that the Nehari distance formula fails in two-variables. In fact, there is no constant C > 0 satisfying dist~(r H~(qI~)) < cllr~l/for every r E L ~ (~). A constructive proof of this appears in IF] and in [BaTi]. There is a generalization of Nehari's theorem to the multivariable setting which allows us to estimate the norm of a big Hankel operator in terms of the two quotient norms corresponding to the two half-planes ofL 2(q~). Associated to the big Hankel operator Fr there are two half-plane Hankel operators, P3~Fr j = 1, 2, defined on analytic polynomials by P]-Fcf = P~-(q)f). Clearly, IIP~r~ll < tlr~ll, J = 1, 2. On the other hand, [[P•162 < [[p/a_(r + [[p~(r for any analytic polynomial f; and so we have the estimate

(1) m {llP rr IIP r ll} < IIrr < m x{llP rr Ile r ll}.

The two key ingredients in the proof of the lifting result below are the operator- valued Nehari theorem [Pal and the existence of an expectation of B(H 2 (T)) onto the subspace of Toeplitz operators, T(L ~176(T)) = {Tr J~b E L ~ (T)}, [Arv].

Theorem 1.1. Let r E L2(T2). Then the half-plane Hankel operator P~FV is bounded on H2(T 2) if and only if there exists a function fle L~(T 2 ) such that BOUNDED MEAN OSCILLATION ON THE POLYDISK 245

P# r = P# /3. In this case,

lIP~r+ll = inf 11/3llo~ and the infimum is attained. Consequently, the big Hankel operator Fr is bounded if and only if there exut functions /3j ~ L~(~) with PiLe = P~/35, J = 1,2. In this case, we can find /3j E L~(T 2) such that max{H/31Ho~, [[/32[!~} < I[Fr _< v~ma~{ll/3111~, 11/321t~}. Proof. Note that if P~r = P~/3 for some bounded function/3, then P~ (el) = P~(/3f) for any polynomial f E H2(T2). In this case, P~F~ is bounded with IIp~r~ll _< 11/311o~. Hence lle~r+ll _< inf 11/311~.

Now suppose that the Hankel operator P~F+ is bounded. For A = (A,, A2) E q~, we can write ~b(A) = )~,~=_~ A~r where (r c_ L2(qI'). Let f E H2(qF2) be a polynomial and write f(A) = ~=0 A~fm(A2), where each fm is an analytic polynomial. Then for A E q~, P~(r = Ek=IA1oo - k Era=0oo ~b_(,,~+k)(A2)fm(A2); thus the norm of the Hankel operator P~Fr is equal to the norm of the matrix of multiplication operators, (M~_(,,+,+,)), as an operator from ~)~,H~(T) into t~)~IL2(qF). But the norm of this operator matrix is equal to the norm of the matrix of Toeplitz operators (Tr acting on t~)~1 H2 01"); and so, by Page's theorem [Pal, we can find a sequence of operators X,~ E B(H 2 (qF)) such that

oo oo = E cx~ + E I(I =1 n=0 n=l Let 7r be the projection onto the Toeplitz subspace 7"(L~176 On the circle, we have that as an operator-valued function

OO OO OO OO 7r (z ~ oX.+ z (-o T+ _. )z= ("Ta,, + z (-nT 0_., n=0 n=l n=0 n:l where the sequence (/3.) c_ L~c(T). Define a function/3 on the ~ by/3(A1,A2) = E.=o ~/3.(~2) + E.=I ~1 r Then

11/311~ sup ('~/3.(.) + ('~r = sup ff #. + ( T+ lr =1 n=0 n=l oo Iffl=1 n=0 n=l It follows, since ~- is contractive, that 11/311~ = IIe~r+ll. Since/:'11/3 = P~r the proof of the statement and estimate on the norm of the half-plane Hankel operator P~rr is now complete. The statement and estimate on the big Hankel operator Fr now follows from the estimate in (1) above. [] 246 S. H. FERGUSON AND C. SADOSKY

Remark 1.2. The proof above shows that [IP~P2r~ll = II/'~r~[I; the same is valid for the upper half-plane Hankel operators, i.e., Ilele~r~ll = IIP~r~ll. To see this, let f((, A) = ~,~=0~ (fn()~)n be a function in H2('~). Then P~P2(dPf) = ~o~=~ ~k ~,~=op+(r Hence, IIP~P2r~II -- II (T~,+~+,)IIB(eT=o~). As shown in the proof above, the norm of the matrix of Toeplitz operators above is equal to IIPx~r, ll.

For r E L2(Td), d = 1,2, we let IlOll~ank(T~) = IIr~ll provided that Fr is bounded. This defines a seminorm on the space of symbols of bounded big Hankel operators defined on H2(Ta). The previous theorem provides an estimate of this seminorm in terms of two quotients of L~(~) which, as seen in the proof, is essentially a one-variable, operator-valued result. In terms of Hankel operators in one-variable, the following alternative estimate for the norm of the half-plane Hankel operator P~Fr will also be used in the sequel.

Corollary 1.3. Let (b E L~(qFe). Then IIPl~rr = sup~r I1r A)llT-ta-k(T) and IlP~r~[I = supcer limb(C,. )llnank(T). Hence, the quantities

IIr176 and max{supll~b(.,A)Jl~tank(w), supll~b(ff, .)lluan/~(r)) AET CET determine equivalent seminorms on the space of symbols of bounded big Hankel operators on H 2(~). Proof. Write P#$ (r A) = ~n~__l (n4~-,,(A). Since the function r - P#$ is analytic in the first variable, (2) I1~(', ~)lln.,~k(T) = II (~-(m+~+x)(~))IIm(e~). Note that for any doubly indexed (and possibly infinite) sequence (~bi3) c_ L ~ (T), the norm of the operator matrix (T,~,~) can be estimated via the formula

II (Tr = sup II (r )~ET One way to see this is by viewing (~)~IH~(T) as the Hardy space of e2-valued functions on T and then using the integral representation of the inner product to get the evaluation at A estimate. Applying this to the Toeplitz operator matrix (T~_(,,+,~+~)), we see from (2) that

sup I1r A)}l~a,~k(T) = ][ (Tr

As shown in the previous proof, the norm of the operator matrix with Toeplitz entries on the right hand side is equal to the norm of the half-plane Hankel operator P~Fr and so the result now follows from Theorem 1.1. [] BOUNDED MEAN OSCILLATION ON THE POLYDISK 247

Let H ~ | L ~ denote the weakly closed subspace of L ~ (q~) of functions whose biharmonic extension to the bidisk is analytic in the first variable, i.e., H r162| L ~176= L ~ (T 2) fq H 2 | L 2. Similarly, let L ~176| H ~176= L ~ (T2) fq L 2 | H 2. Note that for any q~ E L~ distoo(r H ~176| L ~) = infp~r ll]3lloo. Thus, the corollary below follows immediately from Theorem 1.1. In one variable, this is precisely Nehari's theorem.

Corollary l.4. For r E L~(q~), Ile r, II = distoo(r176 | L~176 Consequently, max{dist~(q~,H~176| L~162176 | H~176 < IIr~ll _< v~max{distcr162 H ~r | L~162dist~(r L ~ | H~)}.

It was pointed out to us by K. Davidson (personal communication) that the maximum L ~ distance a bounded function is to the space of bounded functions which are analytic in the jth variable, j = 1, 2, is not, in general, comparable to the distance to H r162(T 2). In other words, the two quantities

(3) max{distoo(r ~ |162 r162|176162 and dist(r do not determine equivalent seminorms on L ~176(~). Thus, by Corollary 1.4, the norm of the Hankel operator with bounded symbol is not equivalent to the distance the symbol is to H~(q~). This was proven in [F] and also in [BaTi]. We should point out that the non-equivalence of the seminorms above is both necessary and sufficient for the existence of a bounded big Hankel operator with no bounded symbol; this was first proven by M. Cotlar and the second author in [CoSa2]. Now since H ~162(71`2 ) = H ~176| L ~176M L ~ | H ~, we have that

(4) (H oo |177 + (L ~ | H~)• c H~(~Ia)• where for any set W c_ L ~176('I~), ~Y• c_ L 1(q~) denotes the functions orthogonal to 14/via the bilinear pairing (f, g) = fr2 fg din. A standard duality argument shows that the non-equivalence of the two seminorms in (3) above is both necessary and sufficient for the containment in (4) to be proper. As seen from the next corollary, the sum on the left hand side in (4) coincides with the functions in the pre-annhilator of H ~ (qr2) which have a certain weak factorization. Thus, the fact that the containment in (4) is proper implies that there is a function in H~(T2)• which does not have a certain weak factorization. This is discussed in more detail in the remarks below.

Corollary 1.5. If f E (H ~176| L~176177then there exist functions gi E H2('I ~z) and hi E H i | L 2 such that f = Y~i gihi and IISII1 -- Ilgilhllh ll2. 248 S. H. FERGUSON AND C. SADOSKY

Proof. We give the standard duality argument which is just the dual formulation of the lifting problem for the right half-plane Hankel operators. Let W denote the space of functions in LI(~) of the form f = ~igihi with gi E H2(~), hi E H~ | L 2 and ~i [Igill211hill2 < 00. Define a complete norm on W by Ilfllw = inf ~/IIgill2llhill2, where the infimum is over all such representations ] = ~ gihi. Note that the space W is contained in (H ~ | L~176 and, by Holder's inequality, the inclusion i : W ~-+ (H ~ | L~)~_ is contractive with dense range. We want to show that the inclusion is onto and isometric. It suffices to show that the adjoint i* is isometric. First observe that, for any function r E LI(T2), the norm of the half-plane Hankel operator P~Fr can be computed by the formula IIP~rr [I = sup I fr~ egh dm}, where the supremum is over all functions g E H 2 ('i~) and h E H02 | L 2 with [Igl[2 < 1 and Ilhl[2 < 1. It follows that the dual space of W can be identified with the space of bounded half-plane Hankel operators corresponding to the right half-plane H 2 | L z. In other words, the functional Lr := fr~ yearn, as a linear functional on W, has norm equal to IIP~r~ll. Identifying the dual of (H ~ | L~ with the quotient L~1762)/H ~ | L ~ we see that the map i* : L~ ~ | L ~ ~ W* just sends the equivalence class of a bounded symbol r to the half-plane Hankel operator P(1-'r By Theorem 1.1, this map is isometric (and onto.) []

Remark 1,6. The analogous faetorization problem for the pre-annihilator of H~(ql~) is to determine if every function f E H~(qI'2)_L has a decomposition f = ~"~.igihi with gi E H2(T2), hi E L2('J~2) CI H~('/I"2)• and ~i I]9ill2llh/ll2 < 00. Since H ~ (qr2) = H ~ | L ~ N L ~ @ H ~, we have the containment

| L%• + (L | _c

On the other hand, L2(~r~) M H~(q~)j_ = Ho2 @ L 2 + L ~ | and so by Corollary 1.5, the sum on the left of (4) coincides with the space of functions in H~('~)• which possess such a faetorization. As noted above, the seminorms, max{distoo(r ~162| L~),dist~(r ~ | H~)} and dist~(r176162 are not equivalent on L~('/I~). It follows by standard duality that the containment in (4) is proper; hence, not every function in H~('/1~)• has the desired factorization. An assertion made in [KrLi] is that the sum (H ~ | L~)• + (L ~ @ H~)• coincides with the space H~(qI ~) M H~176177 It turns out that this is not the ease. More precisely, we always have the containment H~,(T 2) M H~(qr2)• _c (H ~176| L~)• + (L ~ | H~)• but this containment is proper. To see this, suppose that f E Hle(T 2) is orthogonal to H~('~'2). Then f = Plf + P2f - P1Pzf, where each of the functions, Plf, P2f and P1P2f, is integrable. Since f E H~(T2)• Plfe (H~|177 andP2f E (L~174176176 Thus, f E (H~|162177176 BOUNDED MEAN OSCILLATION ON THE POLYDISK 249

(and so f has the desired weak factorization). Now take f E L t (T) such that P+f is not integrable and set F((, A) = (f(A). Then F E (H ~ | L~)• but the function F is not in H~ (T2), since P2 F is not integrable and thus the containment above is proper. It will follow from Corollary 2.8 that a function f orthogonal to H~(T 2) can be weakly factored if and only iff is in the pre-dual of bmo(T2); see Remark 2.9 below.

We end this section by extending to the bidisk a well-known result concerning the norm of a bounded Hankel operator in one variable. Let k~ E H 2 (~), z E D, be the normalized Szeg6 kernel on the disk defined by

i 12 k+(r - --- 1 - ~,(

In one variable, the Hankel operator F~ is bounded on H2(~?) if and only if sup~ lIP• (~bk~)][2 < ~, [Bon], [Nik]. By Theorem 1.1, a necessary and sufficient condition for this result to extend to the bidisk is that the same statement holds for the half-plane Hankel operators PjzF~, j = 1,2. For each z = (Zl,Z2) E ~, let K~ ((i, (2) = k~ ((1)k+~ ((2) denote the normalized Szeg6 kernel on the bidisk. We shall extend this result via Theorem 1.1 by showing that the quantities [[P~F~ 1[ and sup~e~ [IP~ (~bK~)l]2 determine equivalent seminorms on L ~ ('~) for j = 1, 2.

Theorem 1.7. There exists a constant C > 0 such that

Ilr+ll < c sup IIP•162 forall r E L~ zE~

Proof. Fix z = (Zl, 7,2) E ~. A straightforward computation yields

IIP~(r = ~ IIP•

Note that the function A ~ I/% (A)[2 is the on T, and so the integral above is the harmonic extension of the function u+~(A) = [IP•162 , A)kz~)l[2 to ~. It follows that supze ~ IIP~(r < ~ if and only ifsup:~ev Ilu~ II~ < ~, which holds if and only if

supsup IIPI(r , < oo. AET zED By Bonsall's estimate [Bon], the above holds if and only if

AET 250 S. H. FERGUSON AND C. SADOSKY

By Corollary 1.3, the supremum above is equal to the norm of the fight half-plane Hankel operator P(PO; and so we have shown that

sup IIP#(r < co ~ liP#roll < oo. zED2 Since SUpzE~ IIP#(Ogz)ll _< IIP#r [I and both of these seminorms become complete norms when restricted to symbols r E (H 2 | L2) -c, it follows, by the bounded inverse theorem, that there is a constant C > 0 such that lIP#r ll _< Csupz~ IIP#(OK~)ll2 for all r E L~(T2). A similar argument for the upper half plane Hankel operator P#Fr together with the estimate in (1) completes the proof. []

2 Bounded mean oscillation on rectangles

A natural analogue of bounded mean oscillation on product spaces is to replace intervals with products of intervals or rectangles. The space of functions which are of bounded mean oscillation on rectangles coincides with the little BMO space, bmo(T 2), introduced and studied by M. Cotlar and the second author in connection with weighted norm inequalities for the product Hilbert transform [CoSa]. For this reason, the space bmo(T 2) was originally defined in terms of the Hilbert transforms, one for each variable. The characterization of little BMO in terms of mean oscillation on rectangles was given later in [CoSa2]; for our purposes here, we take this characterization of bmo(T2) as our starting point. More precisely, a function O E Lt(T 2) is in bmo(T 2) if

IIr := isyPT ~ j(/jfj 'r ~)- eRI dm(()dm(A)< co,

where en is the mean value ofr over the rectangle R = I x J. It is easy to see that bmo(372) coincides with the space of integrable func- tions which are uniformly of bounded mean oscillation in each variable separately [CoSa2]. In other words, if we fix )~ E 37 and let

I[r ' A)H* =supI_CT ~1 ~/Ir A) - er(A)l dm(~),

and similarily let Hr162")11. denote the one variable oscillation in the the second variable ofr then the two conditions, sup~eT IIr A)]1. < co and sUPeeT [Ir ")]1. < co are both necessary and sufficient for the function r to be in bmo(T2). It follows that the seminorm ]lr is comparable to

(5) max{sup ][r A)I[. , sup Hr162 )]1. }" ~ET (ET BOUNDED MEAN OSCILLATION ON THE POLYDISK 251

In one variable, the seminorms [1r max{[lr IIr and the norm of the commutator II[M~, H]IIB(L=> are comparable on BMO(T); el. [CRW]. The next result is an extension of this fact to bmo(~) which links the norms of big Hankel operators and commutators to bounded mean oscillation on rectangles.

Theorem 2.1. Let r E LI(T2). The following conditions are equivalent.

(i) The function r is in bmo(T2).

(ii) The big Hankel operators Fr and F~ are both bounded on H2(T2).

(iii) The commutators [Mr , H1 ] and [ M~ , 1-12] are both bounded on L 2(T 2).

(iv) The commutator [Mr Hx 1t2 ] is bounded on L 2(T 2).

Proof. By Theorem 1.1 and Corollary 1.3, the quantity max{llrr IIr,~ll} is comparable to

max{ sup I1r :X)lluo.kcr), sup lie(if, ")ll~an~(r), sup I1r ~)ll~a.~(~), I,X[=I Ir 1)~I=1 sup I1,~(r )[In,,..kCT)}, Ir which, in turn by the one variable theory, is comparable to the quantity in (5). As mentioned above, the quantity in (5) is equivalent to the bmo norm. There- fore, conditions (i) and (ii) are equivalent. A routine computation shows that [M~,Hj] = 2z((Pj. ~_ P~) - P~F~). Since the operators P~P, and (Pjl P~) * have orthogonal ranges, [[[M~, Hj Ill is comparable to max{[lP~Fr IIP~P~II}, J = 1, 2. It follows by Theorem 1.1 that conditions (ii) and (iii) are equivalent. To see that conditions (ii) and (iv) are equivalent, let Q = P1P2 + P~P~. Note that Q is the orthogonal projection onto the first and third quadrants of L2(T2) - t2(Z x Z). A straightforward computation yields the identity, [M~, H1H2] = Q• Q - (Q2-M~Q)*. If IMp, HIH2] is bounded on L2(T2), then since the ranges of Q• Mr and (QJ-M~,Q)* are orthogonal, both Q• Mc,Q and Q• M$Q are bounded on L 2 (T2). We now show that the boundedness of Q• on L2(11'2) implies that I'~ is bounded on H2(T2). Let f E H2('~). Then IIQ•162 2 = IIP~P2(r + IIP~P~(r thus IIP~P2(r <_ cIIfll2 and IIPxP~(r <-- Cllfll2 for all f e H2(q~), where C = IIQ• Hence the operators P~P2Fr and P1P~Fr are both bounded on H2(~). But by the remarks in (1.2), we have that IIP~PzFr = IIP~Fr and [IP~P~Pr = IIP~Pr Therefore, the boundedness of the operator Q• on L 2 ('11"2) implies that both P~Fr and P~Fv are bounded on H ~(T 2 ); and 252 s. H. FERGUSONAND C. SADOSKY so, by Theorem 1.1, Fr is bounded on HZ(TZ). By replacing r with r we see that IIQ• < c~ implies that F$ is bounded on H2(T2). The proof that (iv) implies (ii) is now complete. If (ii) holds, then the operators P~P2Fr P~P2F(b, P1P~Fr and P1P2• are all bounded on H2(T2). If f E H2(q~), then by conjugation, IIP~Pz(r = [[e~P~(~bf)[[2 and [IPxP~(r = [[P~P2(4)f)[12. Since P~P~- is the projection onto ffAH2(~), it follows that both Q• and Q• are bounded on Lz(Tz); thus, the commutator in (iv) is bounded on L 2 ('It=). []

We now turn our attention to Carleson measures and give a Carleson criterion for a function to be in bmo('~). Let us first recall the connection between Hankel operators and Carleson measures in one-variable. A positive measure # on the unit disk is called a Carleson measure if there is a constant C > 0 such that #(S(1)) < C{I I for all intervals I c T, where S(I) is the Carleson sector determined by the interval I, [Car]. Carleson's embedding theorem states that a measure # is Carleson if and only if sup IIfIIL2(D,~) < e~3, where the supremum is over all analytic polynomials f with Ilfl[2 -- 1, [car]. It is well-known that p is a Carleson measure if and only if supze D }Ik~I}L~(D,~,) < OO, where k~ denotes the normalized Szeg6 kernel on the unit disk, [-Nik]. If/* is a Carleson measure, we denote the Carleson embedding constant by I}/*l]c, i.e., ]l/z]}c = suplf/ll2= I IiflIL=(D,~) • supped IIk~ IIL2(D,~). For r E L2(T), let u(z) = (r [ks[2) be the harmonic extension ore to D and define

Define the measure d#r z) = ]Vu( z)121og ( ~z~) dA( z), where dA denotes area measure on the unit disk. For z E D, the Littlewood--Paley identity yields

}lPJ-(r + IlP~-(r = ~ Ik~(w)}2d#r(w)"

Thus, by the estimate on the norm of a Hankel operator [Bon], [Nik] the operators re and F$ are both bounded on H2(T) if and only if sup,eD }lP•162 < cr and sup,eD IIP•162 < or Hence, the Hankel operators re and F$ are both bounded if and only if the measure #r is a Carleson measure on D. A proof that the equivalence r e BMO(T) r162 is Carleson was given in [TeSt], from which it follows that the quantities max{]lr ]t42]}nank(T)}, ]]/.tr and BOUNDED MEAN OSCILLATION ON THE POLYDISK 253

JIr determine equivalent seminorms on BMO(7). We extend these results to the bidisk by replacing BMO with bounded mean oscillation on rectangles, Hankel operators with big blankel operators and Carleson measures with the uniform Carleson criterion in each variable separately. For r E LI(T 2) and j = 1, 2, define the measures d#~ on D ~ by

(6) d#:(zl,z2) = [V~u(zl,z2)1210g (~) dA(z~), where u(zl, z2) = (r IK, 12} is the biharmonic extension of r to the bidisk and

]Vjul2 = (t~zj + j= 1,2.

Applying the Littlewood--Paley identity in each variable yields the identities

and

IIP#(r + IIP (r = 2 2 2 -7r Ik,2(w2)l d/zr

By Theorem 1.7, it follows that the seminorms max{ItP#r, lt, {IP#r,~l[} and sup,2eD ]]#~(-,z2)]lc are equivalent; similarly, max{JrP#r, lJ, IJP#rgl} and supz, eD [l#~(zl," )lie are comparable seminorms. The next result now follows immediately from the identities above and Theorem 1.7. Corollary 2.2. A function r e Ll(ql~) is of bounded mean oscillation on rectangles if and only if the measures

are uniformly Carleson.

2.1 The space bmoa(T2) In the one variable case, the analytic version of the Carleson embedding theorem together with the equivalent descriptions of BMO (T) in [FeSt] yields a characterization of the space BMOA(T) := H2(T) n BMO(T) in terms of analytic multipliers. To be precise, let OH2(D) := {g' [ g 6 H2(D)}. By the Littlewood--Paley identity, f 6 0H2(D) if and only if fD If(z)l 2 log(1/Izl)dA(z) < oo. Iff E H2(D), then

dlz$(z) = If'(z)l 2 log(1/Izl) dA(z); 254 S. H. FERGUSON AND C. SADOSKY thus,/zf is a Carleson measure if and only if f'H2(]D) C OH2(D). Therefore, the space of derivatives of functions in BMOA(T) coincides with the space ofpointwise multipliers of H a (D) into cgH2 (D). There is a two-variable analogue of this fact for the analytic subspace of product BMO, BMOA('~) := Hi(D2) * which follows from a Carleson criterion on the measure determined by the double Poisson integral for a function to act boundedly on H~(ql"2), [Ch], [Fef]. We state the analytic version of this result below in Theorem 2.3. In this section, we use the Carleson criteria in Corollary 2.2 to describe the space bmoa(:l[e) := bmo(~) ~ HZ(ql~) in terms of pointwise analytic multipliers. We know by results in [CoSa2] that bmoa(T2) C BMOA(qr2), and the analytic versions of the two different Carleson criteria for the two spaces clarifies the relationship between them. It is a simple matter to produce analytic functions in the dual of nle (,'][,2)which are not of bounded mean oscillation on rectangles; we do this below. Before stating the analytic results, we need some notation. For r E Ll(q~), define (7) dtzr z2) = IVu(zl ,z2)l 2 log(~~l~)log(}~2l)dA(zl)dA(z2),1 where [Vu[ 2 = [0102u1~ + I&-x~u[2 + 101-~2ul2 + I~1 ~'2u[2. Let Oloq2g2(D2) denote the Hilbert space of analytic functions on the bidisk which, as a class of functions, coincides with the set {O2g/OzlOz2 [ g E H2(]I~)}. The norm on the space 0102H2(D ~) is given by

(8) [[f[,2=(2)2fDfD[f(zl,z2)121og([-~-~)log([~l)da(zl)da(z2).

For j = 1,2, let OjH2(I~) := {Og/Ozj : g E H2(iI~)} equipped with the norm which makes the partial derivative operator 0j a partial isometry from H e (Dz ) onto OiH2( ). The main result in [FefJ is a Carleson criterion on the measure #r which is both necessary and sufficient for r to act boundedly on H~e (ql~). In [Ch], this condition was shown to be equivalent to the Carleson embedding condition: sup [[f[[ t~2(w ,,,) < ~, where the supremum is over trigonometric polynomials with [[f[[2 = 1. It follows that iff E zlz2H2(~), then f E BMOA('II~) if and only if O102f. H2(~) c_ 01c92H~(1[~). Note that the dual ofHl(T 2) coincides with the space P+(L"~(Tz)) -- {P+r I r E L~176 and so the theorem below is just the analytic version of the combined results in [Fef] and [Ch].

Theorem 2.3. Let f E ZlZ2H2(D2). Then 010zf . Hz(D2) C_ c9162H2(Dz) if and only if there exists a function r E L~ 2) such that f =/9+r BOUNDED MEAN OSCILLATION ON THE POLYDISK 255

A rather straightforward kemel argument together with Corollary 2.2 will be used to prove the following characterization of the space bmoa(~) in terms of analytic multipliers between H 2 (D2 ) and 0j H 2 (I~), j = 1, 2.

Theorem 2.4. A function f E H2(II~) is of bounded mean oscillation on rectangles if and only if Oil. H2(~) c_ 01H2(~) and 02f . H2(~) C_ 02H2(~).

The proof of this theorem relies on a fact of independent interest concerning multipliers between certain tensors of analytic reproducing kernel Hilbert spaces. Let Ki : ll) x D ~ C be an analytic kernel generating the Hilbert space H(Ki) of holomorphic functions on II), i = 1, 2. The Hilbert space tensor product, H(KI) | H(K2), can be identified as a Hilbert space of holomorphic functions on ~ with reproducing kemel (K1 | K2)(Zl, z2, Wl, w2) := Kl(zl, wl)K2(z2, w2), [Sal]. More specifically, the map defined on the algebraic tensor product which sends an elementary tensor f | g to the function (Zl, z2) ~ f(zx)9(z2) is, because of analyticity, one-to-one. If 7~ denotes the range of this map, equipped with the corresponding Hilbert tensor norm, then the completion of 7-t is easily seen to be a space of holomorphic functions on the bidisk and by construction, isomorphic to the Hilbert space tensor product H(K1) | H(K2). N. Salinas observed that this completion is isometrically equal to the space generated by the kernel K1 @ K2. In what follows, we identify the Hilbert space H(K1) | H(K2) as a subset of ho10morphic functions on the bidisk. In the statement of the proposition below, the norm ]lgll~ denotes the norm of the operator of multiplication by the function g between the intrinsic reproducing kernel Hilbert spaces.

Proposition 2.5. Let Ki be an analytic kernel on the unit disk, i = 1, 2, and let f be a holomorphic function on the bidisk. Then f H2(D) @ H(K1) C_ H 2(D) @H(K2) if and only if the function f satisfies the following two conditions:

(i) for each z E D, f(z, .).H(K1) C_ H(K2) and

(ii) supzeD [If(z,-)ll~ < ~" In this case, the multiplier norm off is equal to the supremum in condition (ii).

Proof. Iff multiplies H2(D) | H(K1) into H2(D) @ H(K2), then there exists a constant C > 0 such that the function

c2K2(z2,w2) f(zl,z2)f(wl,w2) Kl(z2,w2) 1 -- ZlZ/~I I -- Zl~/~1 defines an analytic reproducing kernel on the bidisk. Furthermore, the least con- stant C > 0 for which this is the case is equal to [[f[[~; cf. [Aron]. If we restrict to 256 S. H. FERGUSON AND C. SADOSKY the diagonal zt -- wl = A, then for each A E D, the function defined on D • D by

C2K2(z, w) - f(A, z)f(A, w)K1 (z, w) is positive in the sense of kernels. It follows that for each A E D, f(A, ) multiplies H(KI) into H(K2) and the multiplier norm satisfies [If(A, -< C for all A E D. Conversely, suppose that f satisfies conditions (i) and (ii) above and let C = sup, ev []f( z, ")[]~. If g is a function in H2(D) | H(KI), then for each A E ]1), the function z ~ g(A, z) is in H(K1) and, by assumption, IJf(A,. )g(A, .)ILK2 -< C[Jg(A, )ilK1. Thus, for any 0 < r < 1,

2--~1 fo~ Ill(re'O, )g(re'e, ")ll~= dO <_ C 21~ jfo2~r IIg(re~~ " )ll~c= dO < C 2 IIgIIH~|2

It follows that the function .fg is contained in HZ(D) | H(K2) and tlyglln2| < CIIglIn~H(K,). [] Proof of Theorem 2.4 Let f E H2(~) and consider the measure d#ll(z, w) = [Olf(z,w)l~d#(w), as in Corollary 2.2. Applying Carleson's embedding theorem in the first variable, we see that for each w E D, the measure #3(" ,w) is a Carleson measure if and only if the function aif(., w) multiplies H~(D) into 0H2(D). Furthermore, the Carleson embedding constant [[#}(. ,w)[Ic is equal to the multiplier norm [[01f(., w)[l~. Hence, by Proposition 2.5, the measure/z} is uniformly Carleson in the first variable, i.e., sup~,e v 1[#3(', w)[[c < c~, if and only if Otf multiplies HZ(D) @ H~(D) = H2(D 2) into 0H2(D) | H2(D) = 01H2(D2). The same argument works in the second variable, so the proof is now complete by Corollary 2.2. []

Theorems 2.3 and 2.4 above allow one to write down examples of func- tions in BMOA('~) = P+(L~176 which are not of bounded mean oscillation on rectangles. Take g,h E H2(D) and let ](z,w) = g(z)h(w). A necessary and sufficient condition for 0102](z,w) = g'(z)h'(w) to multiply H2(]I~) into 0102HZ(II~) = 0H2(D) | OH2(D) is that both g and h be contained in BMOA(~). However, a necessary and sufficient condition for 01 ](z, w) = g'(z)h(w) to multiply H2(I~) into oan~(~) = OHm(D) | H2(D) is that 9 E BMOA(T) and h E n~(TI). So, choosing g, h E BMOA($) with h unbounded yields an example of a function f = g | h in BMOA(~), but not contained in bmoa(qI~).

2.2 The predual of brao(T~) Since brao(T2) C BMO(T2) [CoSa2], the Hardy space H~e(T2) is contained in the predual of bmo(T2). In [ChFet], an BOUNDED MEAN OSCILLATION ON THE POLYDISK 257 atomic decomposition for the space H~, (~) was given. The atoms are supported on open sets (rectangles alone do not suffice [Fef]) and are rather cumbersome to work with. In particular, it is not clear how to use these atoms to prove a weak factorization for the space H~, ('~), as is the case for H 1 of the unit ball in C d, d > 1 [CRW]. In fact, the geometric subtlety in the atomic decomposition in [ChFef] has lent itself to an incorrect (and published) proof of weak factorization for H 1(l~), cf. [Lin], [Bo]. In this section, we define the appropriate atoms for an atomic decomposition of the predual of bmo(~) and also prove a weak factorization for this space. By an atom on '/~ we mean a function a E L~(q[e) supported on a rectangle I x J c T x T with Ilall~ < VIIIIJI and such that fr~ adm = 0. Let Atom(q~) denote the collection of all such functions. A basic example of this type of atom can be constructed as follows. Take f E L~r with Ilfll~ < and intervals I, J c_ T. Let i ifl fdm. :' • " = ffi ffi • ~ Then the function

(9) a(;, :9 - x~(r xJ(~) III IJI (:(('a)-:z• is easily seen to be in Atom(qI~). The proof of the observation below follows standard methodology.

Proposition 2.6. A function r e LI(~) is of bounded mean oscillation on rectangles if and only if SUPaeAtom(T2) ] f~ Ca dml < oo. Consequently, the predual of bmo(~ ) can be identified with the space, denoted by H1Atom(ql~ ), offunctions of the form f = ~iaiai with (ai) C Atom(T2), (ai) C_ C and ~i la, I < ~. The space H1Ator,(~) is equipped with the usual norm II/ll --- inf ~i lad, where the infimum is over all decompositions f = ~i aiai.

Proof. Let a E Atom(q~) with support in I x J and let r E LX('l~). Then fT2 Cadre = fz fj(~b - (bz• and thus r E bmo(qi~) implies that supaeato,~(rb I fv~ Ca din{ < c~. Conversely, suppose that sup,,e.4to,,,(r2) I fr~ Ca din[ < ~. By considering atoms of the form in (9) above, it follows that

sup (r r < oo foreveryI, J C T. fr:fl~=~ ~i11/,f - Hence, the function

i1.I ij f (r162 - ex• 258 S. H. FERGUSON AND C. SADOSKY is in LI(]/~), with L 1 norm independent of I and J. But this is exactly what it means for r E bmo(T2). The statement on duality follows from standard considerations. []

We shall not use atoms to prove a weak factorization for the predual of bmo(~); instead, we make use of a result from [CoSa2] and Corollary 1.5. Let H~e(T) | LI(T) denote the space of functions f E L1 ('/ff2) such that Hlf E LI(~). In other words, f E H~e(T)| and onlyifPlf ~ LX(T2) and P(f E LI(T2). Similarly, let L 1(~) | n~e(~ ) be the space of functions f E L t (T2) such that P2f and P~f are in LI(T2). It was shown in [CoSa2] that the pre-dual of bmo(T2) coincides with the sum of the spaces H~(T) | LI(T) and LI(T) | H~(T). This result, which is stated below, will be used to deduce the weak factorization.

Theorem 2.7 (Cotla~-Sadosky). A function f E Ll(q~2) satisfies

sup [fT2fCdm i < c~ ~be b'rao (T2), IIr if and only if there exist functions fl E H~e(T ) | LI(T) and f2 e LI(T) | H~(T) such that f = fl + f2. One way to see this goes as follows. Equipped with the norm Ilf[I = Ilflll + IIH1f llx, the space H~(T).| L I(T) is a Banach space whose dual can be identified (isometrically) with the sum of the spaces L~c(qI~) and HI(L~C(T2)). The term sum here means the vector space sum equipped with the natural quotient norm induced by the map (fl, ~/) ~ fl + H17 defined on the e ~ sum, L~(T 2) @~r L~r ('1~). Similarly, the dual of LI(T) | n~e(T ) with the norm I}fll = Ilfl}l + I}n2f}]l is the space L~('~) + H2(L~(~)) with the norm I1r = infmax{llflHoo, II~/ll~}, where the infimum is over all pairs 3,7 E LCC(q~) with r = 3 + H27. It is straightforward to verify that the space L~(qff2) + HI(L~C(T2)) coincides with the functions which are uniformly of bounded mean oscillation in the first variable. Similarly, a function r is in L~(q~) + Hz(L~(TZ)) if and only if r is uniformly of bounded mean oscillation in the second variable. The intersection of these spaces first appeared in [CoSa] in the characterization of the weights for which the product Hilbert transform HIH2 is bounded in weighted L2(~). It was shown in [CoSa2] that a function is of bounded mean oscillation on rectangles if and only if it is uniformly in BMO(T) in each variable separately. Therefore, the space bmo(qff2) coincides with the intersection of the spaces L ~ (~) + H1 (L ~ (T2)) and L~(~) + H2(Lcr It follows by standard duality that the pre-dual of brno(q[e) can be identified with the range of the "'sum" map defined on the e 1 sum H~e(T) | LI(T) *1 LI(T) @ H~e(T). BOUNDED MEAN OSCILLATION ON THE POLYDISK 259

Let f E LI(T2). The Fourier series of the function Plf is supported on the right half-plane; and so by Corollary 1.5, Plf E L~(T2) if and only if Plf = ~j gjhi, where (gi) c_ H2(T2), (hi) c_ H 2 | 2 and )"~j Ilgjll2llhjll2 < ~. Similarly, P~f E LI(T2) if and only if P~f = f~, where the function f2 decomposes as Ejgjhj with (~j) C_ H2(T2), (ib) c_ H~ | L 2 and Ej IIDJll2llhjlla < ~. Since f = Plf + P~f, the space H~(T) | LI(T) coincides with the space of functions f = fl + f2, where each of the functions fi decomposes as a sum of products of functions in H2(T 2 ) times functions in H 2 | L 2. Now the theorem above says that bmo($2). = n~e(qr) | L~(T) + L~(T) | H~($); and so, by applying the same argument in the second variable, we have the following weak factorization for functions in the pre-dual of bmo(T2).

Corollary 2.8. The space H~e (T) | L 1(T) + L1 (T) | H~ (T) coincides with the space of functions f = fl + f2, where each of the functions fl and f2 has a factorization fi = ~~jgjh~ with (gj) C H2(T2), (h~.) C H 2 | L 2 + L 2 | H 2 and Ej tlajll211h~llz < ~, i = 1,2.

Remark 2.9. We saw in Remark 1.6 that the space H~,(T 2) fq H~177 is properly contained in (H ~ | L~177 + (L ~ | H~177 and this latter space coincides with the subspace of H~ of functions which have a weak factorization of the form f = ~],jgjhj, where gj E H2(qF2), hj E H ~ @ L 2 -4r L 2 | H 2 and ~-:~j Ilgjll21[hj[12 < ~x~. It follows, by the corollary above, that (H ~ | L~r177+ (L ~176| H ~)• = bmo (7i~). N H ~ (7~2)_L. Therefore, by Proposition 2.6, a function f e L t ('IF) which is orthogonal to H ~ (~) can be weakly factored as above if and only if f = ~]j aia i, where ~]i Jail < c~ and each of the atoms aj is bounded, supported on a rectangle and satisfies fr~ a dm = O.

3 Rectangular BMO and little Hankel operators

The rectangular BMO space, introduced in [Fef], is by definition the space of functions r E L2(T2) for which

I/r I,jCT_~.~.~,ff~sup 1 1 ~ fzfJ /r -- Cj(ff) -(fii(A)-{-r < 00, where Cj(~) = FJI fJ r A)dm(A), [~l = 1, and r is the mean value of r A) over the interval I, [AI = 1. Rectangular BMO is the dual of the space spanned by the so-called rectangulm" atoms in L2('/F). A rectangular atom is a function a E L~(T2) supported on a rectangle I x J c_ T x T with Ilal[2 < 1/[I[IJ [, satisfying fr a(i, A) dm(r = 0 for almost all IAI = 1 and fr a(~, A) drn(A) = 0 for almost all 260 S. H. FERGUSON AND C. SADOSKY

[(1 = 1. A basic example of this type of atom has the form

(10) a(r A) = Xx(r Xj(A) III IJI (f(C'A) - fl(A) - fd(() + flxd), where I, J _C T and f E L~ 2) with Ilfll~ -< An open question is whether the corresponding L ~ atoms (supported on rect- angles) can be used to describe the pre-dual of rectangular BMO. In other words, can we remove the power of 2 in the formula above and compute the rectangular oscillation above with respect to the L x norm? In [Fef], a construction due to L. Carleson [Carl] was modified to exhibit a function in rectangular BMO which is not in product BMO, i.e., does not act boundedly on HI,(T 2). Thus, the rectangular atoms are not enough for an atomic decomposition of the space H~, (ql'2). However, it was pointed out more recently in [Fell ] that in order for an L 2-bounded operator to be bounded from L~(T ~) to BMO(T2) it is sufficient to check its action on the rectangular atoms. The next proposition illustrates the subtlety of the relation between BMO and rectangular atoms.

Proposition 3.1. lf$ E LI(T2) satisfies

X~T'~I JZ JC_T I I JJ then ~ is in BMO(q~2).

Proof. By definition,

II4,11,,. sup /" I1r162 ) - ,r (.)llB~o ,a~(r I_CT I..tl j./-

If we identify ~ with the vector-valued function : T --+ LI(T) defined by ~(~) = ~(~, ), then the BMO-valued BMO norm of is exactly II~ll,,. More precisely, recall that the BMO(BMO) norm of@ is defined by

II~IIBMO(BMO) = sup~_cr --III IlO(() - CzllBMo dm((), where @i = ~ ft r162 ef. [GR]. Since Oz = Cz(" ), IIr = IIOlISMOCBMO~. Now the scalar-valued function ( ~ II~ (r IIBMO has BMO norm no greater than 211OlIBMO~BMO)= 211~11,*" Thus, ifh is a trigonometric polynomial, then BOUNDED MEAN OSCILLATION ON THE POLYDISK 261

-< fT IIh(r )11111r162 dm(r < 211h11111r [::]

In terms of Carleson measures, rectangular BMO coincides with the functions whose measure determined by the double Poisson integral, as defined in (7), satisfies the product form of the Carleson criterion [ChFefl]. In other words, IIr < oo if and only if

#r x S(J)) sup < oo, z,JC_W IIIIYl where for an interval I C_ 7, S(I) denotes the corresponding Carleson region. The standard one-variable estimates with normalized Szeg6 kernels k, can be used iteratively in each variable to establish the following fact. We omit the proof and refer the reader to the literature [ChFefl], [Gar], [Sar].

Proposition 3.2. Let r E L 2(T 2) and let #r be the measure defined in (7). The following are equivalent.

(1) IIr < o0"

(2) supz,jc r #r x S(J)) < oo. - IIIIJ[ (3) sup,,,z2e D Ilkz, | kz, IIL'(W,~+) < 0r Let BMO nect ( Te ) denote the rectangular BMO space and let BMOARect (qIr denote the analytic subspace. We shall describe both of these spaces in terms of the so-called little Hankel operators. The little Hankel operator with (non-unique) symbol r E L 2 (,~e), denoted by 7r is densely defined on analytic polynomials by the formula 7r = P~P~(r If ] = P~P~r then f E ZlZ2H2('~2), 7r = ~/1 and the function f uniquely determines the little Hankel operator q'r A straightforward computation yields the identity

II"L/(k,,, | k.~)ll~ =

(1)2f~ f~ I/(fil, r -/(z~,fi2)- I(r l(zx,z2)121k,,(fx)k,,(fi2)l 2 dr162 which, via the Littlewood-Paley formula, becomes

II~i(k,,, | k,,2)ll~ = 262 S. H. FERGUSON AND C. SAD|

In terms of the measure

#f(ZI,Z2)= ]Ol02f(Zl, z2)l 9. log (11[) log (_~T2I)da(zl)dA(z2),

the identity above can be written as

(11)

The next result follows immediately from the equivalence of conditions (1) and (3) in Proposition 3.2 and the identity above.

Proposition 3.3. A function f E H 2 (D2) is in BMOARect (Te ) if and only if supze~ [l"/](kzl | kz:)][2 < c~.

Using Proposition 3.3, we can now estimate the rectangular oscillation Hr R~ct in terms of four quantities associated to the little Hankel operators corresponding to the four corners ofr Ifr E L2('~) and f = P~P~r then "~4, = ~/]; thus, by (11), [l%(kz, | k~2)ll~ = (-~ Ikzl(Wl)kz,(W2)]]Ol02~Z(Wl,W2)1210g(]-~ll)log([~2])dd(wl)dA(w2),

where u(zl, z2) = (r [kz~ @ k~2 [2) is the biharmonic extension ofr to the bidisk. Define the conjugation operators tl, t2 and t by t1r162 (2) = r162 (2), t2r (2) = r (2) and t = tit2. Note that conjugation in the first variable, tl, followed by the Poisson integral intertwines the operators 0~ and 0~. In other words, 01 (t1r [kzl | kz~[2) = t101(r [k~1 | k~212). It follows by the identity above that

II'r,,,(k~, | k,~)ll~ =

(2)2L fD 'k~'(Wl)k~'(w2)12'O~'O2u(w~'w2)l~l~ (~)log (]w-~)da(w~,dA(w~).

The analogous identities involving the operators Oxo~ and 0102 follow by considering ")'t2r and 7tr respectively. Now let d#r denote the measure defined in (7). Then for Zl, z2 E D

2 ~=)[12 + [l'yt,r | k~=)ll~

By Proposition 3.2, the supremum over the bidisk of the quantity I1kzl | IlL2(m .us) is comparable to IIC[l~ect. The next result is now immediate. BOUNDED MEAN OSCILLATION ON THE POLYDISK 263

Theorem 3.4. The quantifies I[~[]Rect and sup max{llTr | kz~)ll2,117,,r | k~)l12,117t2r | kz2)ll2, z~ II'Y,r | kz2)l12} determine equivalent seminorms on BMO nect (qr2). Theorem 3.4 says that we can estimate the rectangular oscillation of a function r using the four little Hankel operators corresponding to the four corners of r In particular, were the quantities sup~,~eD 1171(k~ | k~)ll~_ and infp+~=/I1~11~ to determine equivalent seminorms on zlz2H~ it would follow from Theorem 3.4 that the two seminorms I1"II Re~t and I1"II BMO are equivalent on BMO (~). But the containment BMO(qr2) C BMOn~t(~) is proper [Fef], so the quantities IIr and IIr are not comparable semi-norms on BMO(q~). This observation is recorded below. Corollary 3.5. The quantities

sup II'y/(k~,| and inf I1~11~ zl ,z2ED P+~3=I are not comparable on z xz2 H ~ (q~). Thus, BMOA (q~) = P+ (L ~r (q~)) is properly contained in B M OA a~a (T2). Let us quickly point out what this means in terms of analytic multipliers. By the identity (11), we see that for a function f E ZlZ2H2(']~2), 117y(kz~ @ k~)ll2 = 110102/ k~ | k~ IIo~o~H~<~). By Theorem 2.3, the quotient norm infp+z=/I1~11o~ is comparable to II0x0~fllJ-, which is the multiplier norm of 0102f viewed as a multiplier of H 2 (I~) into the space of mixed partials 0102H 2 (lI~). We always have the estimate supzl,~2e D 110102f. kz~ | k,~llo~o=H2(~) <- 110~02/11,~, however, the proper containment BMOA(~) c BMOAn,a(q~) implies that the two quantities supz~,~2eD I[OtO2f.kz, | O2H2(W) and [10x02fll~ do not determine comparable seminorms on BMOA(T e). Returning to little Hankel operators, note that iff E z~ z2 H 2 (Te ) and ~ E L ~ (qr2 ) with P+/~ = f, then "/] = ~/~, and so I1~/11 < I1~11oo. Hence, II'y/ll -< infp+z=/II~ll~- It is an open question if a reverse estimate holds.

Problem 3.6. Does there exists a constant C > 0 such that infp+~=f 1[~[[~ < cI1%,11 for all f ~ z~z2H~r

Equivalently, does every bounded little Hankel operator have a bounded symbol? The dual formulation of this problem is the weak factorization prob- lem for H~(qr2). Note that II'~fll = sup l fr~ fghdml, where the supremum is 264 S. H. FERGUSON AND C. SADOSKY over all functions g 6 H2(T2), h 6 zlz2H2(]~ "2) with IIg[12 = [Ihl[2 = 1. Since infp+~=y {I/311~ = sup l fw2 fhdml, where the supremum is over the unit ball of zlz2Hl(~), an equivalent formulation of the question above asks if every function f 6 Hx(~) decomposes as f --- Y]4gihi with gi,hi 6 H2(~) and Ei Ilodf2tlhdb < ~. Another open problem is whether the analogue of the main result in [Bon] holds for the little Hankel operators. In other words, Problem 3.7. Does there exists a constant C > 0 such that 117111 < C sup~l,~ 117r | k~=)ll for all f 6 zlz2H~('~a) ?

By Theorem 3.4, this is equivalent to asking if BMOAae~t (~) coincides with the space of symbols of bounded little Hankel operators. We see that sitting between BMOA(qI~) and BMOARer 2) is another space, namely, the space of symbols of bounded little Hankel operators on H~(~). This space can be identified with the minimal tensor product BMOA('IF) | BMOA(T) and can be described in terms of the boundedness of the nested commutator [ [ Me, H1 ], H2 ] on L 2 (,][,2). Recall that in one variable, IIr is comparable to/[[Mr H]IIB(L~), [CRW]. Because of this, we may regard BMO(7) as a concrete operator space sitting inside B(L~(T)) via the map r ~ [My, H]. We use this fact to characterize the minimal tensor product BMO(T) | BMO(T) C_ B(L2(T) | LZ(T)) ~ B(L2('~2)). To do this, define on the algebraic tensor product BMO(T) | BMO(qF) the seminorm n I%

j =0 j=0 Lemma 3.8. lfu(ff, A) = E~=0 r then

n

j=0 Proof. Since Hj = -i(Pj - P~-) = -2iPj + iI, j = 1, 2, [[ M~,, H1 ], He ] = --4[[M,L,/'11, Psi; and so [[Mu, H~ ], H2] = -4( PI P~M,,P~ P~ - P~ P2M,,P~P~ - P~P~ M,,P~ P~ + PI~P~ M,,P~P2). Let P : L2(T) ~ H:(T) be the orthogonal projection and r e L2(T). Then for Y | g e L~(T) | L~(T), [Me ,H] | [Mz, H](f | =[M, ,H]f | [Mz, H]g = - 4(P:McPf | PxM~Pg - PMcP• | pxM~Pg - piMcpf | PM~PIg + PMcPIf | PMzP-~g). BOUNDED MEAN OSCILLATION ON THE POLYDISK 265

Making the identification L2('Jr) | L2('/r) = L2(qr2), we see that the sum of the tensors above can be identified with the function [IMp, H1], H2]h ~ L2(T2), where h((, A) = f(()g(A) and u((, A) = r The result now follows. []

It follows by the lemma above that the completion of the algebraic tensor product BMO(T) | BMO('~) with respect to the norm

j=0 j=0

can be identified with the space of functions u E L2('I~) such that [[Mu,//1 ], H2 ] is bounded on L 2 (T2), with the norm

(12, Ilull~oo, =-I1[[M~, H~ ], H2 ]IIB('(T~,, + [[ f u(, ~)dm(~)[I=

We let BMOF~ct (qr2) denote this space. We show that BMOFact (qr2 ) contains BMO(T 2) by showing that the pre-dual of BMOFact(T 2) coincides with the functions in H~(T 2) which have a certain weak factorization. We first introduce some notation. If M, N are closed subspaces of L2(T2), define the space M | N C_ LX(ql"2 ) by

M| = {f = Zhjgj: (hi) c M, (gj) c N, and )-'~ Ilhyll2llgjll2 < ~}- j J Theorem 3.9. A function r E L2(T2) is in BMOF~t(~"2) if and only if the little Hankel operators 7r162162 and 7tr are bounded on H2(T2). Hence, the predual of BMOFact(qF2) can be identified with the space of functions f E H~e(T2) satisfying P1P2f E H2(T2) | H2(T2), ta(P~P2f) E zlH2(qIr 63 H2(T2), t2(P1P~ f) E z2H2(ql"2) 63 H2(T 2) and t(P~P~f) E zlz~H2(T 2) 63 H2(qr2).

Proof. As shown in the proof of Lemma 3.8,

- Me, H11,//21 =P1P2MoP~ P~ - P~ P~McP, P~ - PI P~ McP~ P2 + P~ P~ M~P~P~.

Since these four operators have orthogonal ranges, H[[Mr is comparable to max{ llP,P~M~Pr Pr ll, IIP~P~ M~P~ Pr II, IIP~Pr M~Pr P~II, IIPr Pr M~P~P~ ) II}. 266 S. H. FERGUSON AND C. SADOSKY

By definition, )]P~P~Mr P2 II = H?oI]. To compute the norm ofIIPI~P2MoP~ P~ II, notethat for any function f E L2(~), IlP~(f)llz = [[Pl(Uxtlf)llz where Ulg(~,A) = ~g(r A); and thus,

IIP~P=MoP,P~II= sup IIP~P2(r = sup IIP~P~(tl~)tf)l)2 fEz2H 2 frzlz~H 2 [Ifl[2=l 11I[12=1

Similarly, IIP1P~McP~P2[I = [17,1r and IIPxPzMr = II~r,~{[. The first statement of the theorem now follows. Note that

,,3,r = sup [/r sup fEH2 grzlz2H 2 2 fEz~z2H~QH 2 llfllz=llgl[~=l II.r112=1 Similarly, the norm of Tt,r can be computed by pairing r with functions in ziH2QH 2 of norm 1, i = 1, 2; and the norm of Ttr can be computed by pairing r with functions in H 2 | H z of norm 1. So we may identify the pre-dual of BMOFact(~ffz) with functions f E LI(T2) whose comers P1P2f, Pr P1P~f and P~P~f have the prescribed factorization above. By definition, such a function will necessarily be in H]~ ('/I~), since each of the four comers are in La (I"~). []

It follows by Theorems 3.9 and 3.4 that BMO(Tz) C_ BMOFact(Tz) c_ BMOn~ct(~). Knowing which of these inclusions is proper would entail the characterization of the symbols of the little Hankel operators and give an answer to the weak factorization problem for H 1(~).

REFERENCES

[Aron] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337--404. [Arv] W. Arveson,Interpolation problems in nest algebras, J. Funct. Anal. 20 (1975), 208-233. [BaTi] M. Bakonyi and D. Timotin, On a conjecture of Cotlar and Sadasky on multidimensional Hankeloperators, C.R. Acad. Sci. Paris, Series 1325, No. 10 (1997), 1071-1075. [Bo] A. Bonami, Some comments on factorization in the bidisc, personal communication. [Bon] F. E Bonsall, Boundedness of Hankel Matrices, J. London Math. Soc. (2) 29 (1984), 289--300. [Car] L. Carleson, lnterpolation by boundedfunctions and the corona problem, Ann. of Math. (2) 76 (I 962), 547-559. [Carl] L. Carleson, A counterexamplefor measures bounded on HP for the bi-disc, Mittag Leffier Report No. 7, t974. [ChJ S.-Y. A. Chang, Carleson measure on the bi-disc, Ann. of Math. (2) 109 (1979), 613--620. [ChFeq S.-Y. A. Chang and R. Fefferman,A continuous version of duality of H 1 with BMO on the bi-disc, Ann. of Math. (2) 112 (1980), 179-201. BOUNDED MEAN OSCILLATION ON THE POLYDISK 267

[ChFefl ] S.-Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and HP theory onproduct domains, Bull. Amer. Math. Soc. 12 (1985), 1-43. [CoSa] M. Cotlar and C. Sadosky, The Helson-~Szeg6 theorem in LP of the bidimensional toms, in Harmonic Analysis and Partial Differential Equations (Boca Raton, EL 1988) ( M. Milman and T. Schonbek, eds.), Contemporary Mathematics 107, Amer. Math. Soc., Providence, RI, 1990. [CoSa I ] M. Cotlar and C. Sadosky, Nehari and Nevalinna-Pickproblems and holomorphic extensions in the polydisk in terms of restricted BMO, J. Funct. Anal. 121 (1994), 205-210. [CoSa2] M. Cotlar and C. Sadosky, Two distinguished subspaces of product BMO and Nehari--AAK theory for Hankel operators on the toms, Integral Equations and Operator Theory 26 (1996), 273--304. [CRW] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), 61 I--635. [FeSt] C. Fefferman and E. M. Stein, Hp spaces ofseveral variables, Acta Math. 129 (1972), 137-193. [Fe] C. Fefferman, Characterization of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587-588. [Fef] R. Fefferman, Bounded mean oscillation on thepolydisc, Ann. of Math. (2) 110 (1979), 395--406. [Fefl] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. (2) 126 (1987), 109--130. [F] S.H. Ferguson, The Nehari Problem for the Hardy space of the torus, J. Operator Theory 40 (I 998), 309--321. [GR] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Notas de Matemfitica (L. Nachbin, ed.), 116, North-Holland, Amsterdam, 1985. [Gar] J. B. Gamett, Bounded Analytic Functions, Academic Press, New York, 1981. [JoNi] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415---426. [KrLi] S.G. Krantz and S.-Y. Li, Factorization offunctions in subspaces of L 1 and applications to the corona problem, Indiana Univ. Math. J. 45 (1996), 83-102. [Lin] lng-Jer Lin, Factorization theorems for Hardy spaces of the bidisc, 0 < p < 1, Prec. Amer. Math. Soc. 124 (1996), 549-560. [Ne] Z. Nehari, On bounded bilinearforms, Ann. of Math. (2) 65 (I 957), 153-162. [Nik] N. Nikol'skit, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986. [Pal L. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970), 341-372. [Sal] N. Salinas, Products of kernel functions and module tensor products, Oper. Theory Adv. Appl. 32 (1988), 219-241. [Sar] D. E. Saras•n• Functi•n The•ry •n the Unit Circle• N•tes f•r lectures given at Virginia P•lytechnic Institute, Blacksburg, Virginia, 1978.

Sarah H. Ferguson DEPARTMENT OF MATHEMATICS WAYNE STATEUNIVERSITY DETROIT, M148202, USA email: [email protected] u

Cora Sadosky DEPARTMENTOF MATHEMATICS HOWARDUNIVERSITY WASHr~OTON,DC 20059, USA email: [email protected] (Received June 6, 1999)