Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball

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disk into functions of two weighted Bergman spaces with the same weight (again using Blaschke products). These strong factorization results are no longer possible to obtain [11] in the setting of Hardy and Bergman spaces in the unit ball of the complex euclidian space C n of dimension n when n ≥ 2, but it is still possible to obtain some "weak" factorizations for functions in these spaces.
For two Banach spaces of functions, A and B, defined on the same domain, the weakly factored space A B is defined as the completion of finite sums with the following norm: When 0 < p ≤ 1, weak type factorizations for the Hardy spaces H p and the weighted Bergman spaces A p α on the unit ball of C n are well known (see [6,9] for Hardy spaces; and [5,20], or [24, Corollary 2.33] for Bergman spaces). However, when 1 < p < ∞, even for unweighted Bergman spaces the problem was still open (see, for example [4]).
In this paper we completely solve the above problem for Bergman spaces by establishing weak factorizations for a weighted Bergman space A q β , with 1 < q < ∞ and β > −1, into two weighted Bergman spaces with non necessarily the same weight, on the unit ball B n of C n . The following is our main result.
Theorem 1 Let 1 < q < ∞ and β > −1. Then for any p 1 , p 2 > 0 and α 1 , α 2 > −1 satisfying (1. 1) In this context, by "=" we mean equality of the function spaces and equivalence of the norms. The inclusion A is a direct consequence of Minkowski and Hölder inequalities, so that the interesting part is the other inclusion with the corresponding estimates for the norms. Now we are going to recall the definition of the weighted Bergman spaces. First we need some notations. For any two points z = (z 1 , . . . , z n ) and w = (w 1 , . . . , w n ) in C n , we use z, w = z 1w1 + · · · + z nwn to denote the inner product of z and w, and |z| = z, z = |z 1 | 2 + · · · + |z n | 2 to denote the norm of z in C n . Let B n = {z ∈ C n : |z| < 1} be the unit ball in C n and S n = {z ∈ C n : |z| = 1} be the unit sphere in C n . Let H (B n ) be the space of all analytic functions on B n . We use dv to denote the normalized volume measure on B n and dσ to denote the normalized area measure on S n . For For 0 < p < ∞ and −1 < α < ∞, let L p (B n , dv α ) be the weighted Lebesgue space which contains measurable functions f on B n such that

Denote by
, the weighted Bergman space on B n , with the same norm as above. If α = 0, we simply write them as L p (B n , dv) and A p respectively and f p for the norm of f in these spaces.
It is a well-known fact that to obtain weak factorization results is equivalent to give a "good" description of the boundedness of certain Hankel forms. A Hankel form is a bilinear form B on a space of analytic functions such that for any f and g, B( f, g) is a linear function of f g. These forms have been extensively studied on Hardy spaces and on Bergman spaces. For the case of the Hardy space on the unit disk, a classical result by Nehari [18] says that the Hankel form (under the usual integral pair for Hardy spaces) with an analytic symbol b is bounded on H 2 × H 2 if and only if b ∈ BMOA, the space of analytic functions of bounded mean oscillation. The proof used the fact that a function in H 1 can be factored into product of two functions in H 2 . Unfortunately, such strong factorization is not possible (see [11]) for Hardy spaces in the unit ball B n of C n . However, Coifman, Rochberg and Weiss [6] were able to generalize Nehari's result to the unit ball B n by using a weak factorization of H 1 . Namely, they proved that Our approach to the problem for weighted Bergman spaces on the unit ball is the opposite to the one of Coifman et al. [6]. We first characterize boundedness of the Hankel forms on weighted Bergman spaces, and with this result the weak factorization easily follows.
Given α > −1 and a holomorphic symbol function b we define the associated Hankel type bilinear form T α b for polynomials f and g by where the integral pair , α is defined as Since the polynomials are dense in the weighted Bergman spaces, the Hankel form α 2 for any p 1 , p 2 > 0 and any α 1 , α 2 > −1. We The norm of T α b is given by The next result characterizes boundedness of the Hankel form T α b acting on A p 1 We will see in Sect. 3 that this implies the weak factorization in Theorem 1.
Theorem 2 Let 1 < p 1 , p 2 < ∞, and α, α 1 , α 2 > −1 satisfy where q and β are real numbers satisfying (1.1), and q and β are determined by the condition Remark Note that, condition (1.3) guarantees that q > 1 and β > −1. When q and β satisfy condition (1.1), automatically we would have β > −1 (to see this, simply add two equations in (1.1) together). By a general duality theorem for weighted Bergman spaces (see Theorem A in Sect. 2), the condition b ∈ A q β means that the symbol b belongs to the dual space of A q β under the pairing given by (1.2). It turns out that boundedness of the Hankel form T α b is equivalent to boundedness of a (small) Hankel operator, which we are going to introduce in a moment. Let α > −1. It is well-known that, the integral operator is the orthogonal projection from L 2 (B n , dv α ) onto the weighted Bergman space A 2 α . The above formula can be used to extend P α to a linear operator from Denote by A p α the conjugate analytic functions f on B n that are in L p (B n , dv α ). Clearly, Given f ∈ L 1 (B n , dv α ) and a polynomial g, the weighted (small) Hankel operator is defined by Due to the density of polynomials, the small Hankel operator h α f is densely defined on the weighted Bergman space A p α for 1 ≤ p < ∞. We will study boundedness of the small Hankel operator with conjugate analytic symbols, that is, Let f ∈ H (B n ) and α such that where q and β are real numbers such that α 2 if and only if (1.5) is true (see [22,Theorem 70]). Hence the above result concerns the boundedness of h α f from a smaller space to a larger space. Also, by [24, Theorem 2.11], condition (1.6) means that the integral operator P α is a bounded projection from L p 2 If one considers the operator Hence, for , with equivalent norms. Here, the numbers α 2 and p 2 are defined by the relation Comparing Theorem 2 with Theorem 3, notice that the first inequality in (1.3) is equivalent to condition 1 < p 2 < p 1 < ∞. Also, when p 2 and α 2 are replaced by p 2 and α 2 , condition (1.6) turns out to be equivalent to α 2 > −1, and therefore is always satisfied; and the second inequality in (1.3) is equivalent to condition (1.5). Therefore, Theorem 3 implies Theorem 2.
In the case of the same weights, that is, when α 1 = α 2 = β = α, all the restrictions in Theorem 3 reduces to p 2 > 1. We isolate this case here, since it proves a conjecture in [4].
The paper is organized as follows: in Sect. 2 we give some necessary concepts and recall some key results which are needed in our proof of the main result. In Sect. 3 we give in detail the connection between weak factorizations and Hankel forms. The proof of Theorem 3 is given in Sect. 4. In the following, the notation A B means that there is a positive constant C such that A ≤ C B, and the notation A B means that both A B and B A hold.

Preliminaries
We need the following duality theorem. In this generality the result is due to Luecking [16] (see also, Theorem 2.12 in [24]).
We need the following well known integral estimate that can be found, for example, in [24,Theorem 1.12].
For any a ∈ B n with a = 0, we denote by ϕ a (z) the Möbius transformation on B n that exchanges 0 and a. It is known that, for any z ∈ B n where s a = 1−|a| 2 , P a is the orthogonal projection from C n onto the one dimensional subspace [a] generated by a, and Q a is the orthogonal projection from C n onto the orthogonal complement of [a]. When a = 0, ϕ a (z) = −z. The function ϕ a has the following properties: ϕ a • ϕ a (z) = z, and For z, w ∈ B n , the pseudo-hyperbolic distance between z and w is defined by and the hyperbolic distance on B n between z and w induced by the Bergman metric is given by For z ∈ B n and r > 0, the Bergman metric ball at z is given by It is known that, for a fixed r > 0, the weighted volume v α (D(z, r )) (1 − |z| 2 ) n+1+α .
We refer to [24] for all of the above facts.
A sequence {a k } of points in B n is a separated sequence (in Bergman metric) if there exists a positive constant δ > 0 such that β(z i , z j ) > δ for any i = j. We need a well-known result on decomposition of the unit ball B n . The following version is Theorem 2.23 in [24].
Lemma C There exists a positive integer N such that for any 0 < r < 1 we can find a sequence {a k } in B n with the following properties: r ). Any sequence {a k } satisfying the conditions of the above lemma is called a lattice (or an r -lattice if one wants to stress the dependence on r ) in the Bergman metric. Obviously any r -lattice is separated.
For convenience, we will denote by D k = D(a k , r ) andD k = D(a k , 4r ). Then Lemma C says that B n = ∪ ∞ k=1 D k and there is an positive integer N such that every point z in B n belongs to at most N of setsD k .
We also need the following atomic decomposition theorem for weighted Bergman spaces. This turns out to be a powerful theorem in the theory of Bergman spaces. The result is basically due to Coifman and Rochberg [5], and can be found in Chapter 2 of [24].
Theorem D Suppose p > 0, α > −1, and b > n max 1, Then we have (i) For any separated sequence {a k } in B n and any sequence λ = {λ k } ∈ p , the function belongs to A p α and f p,α λ p .

(ii) There is an r -lattice {a k } in B n such that, for any f ∈
and λ p f p,α .
In the proof given in [24], part (i) requires that the sequence {a k } is an r -lattice for some r ∈ (0, 1], but it is well known that only the separation of the sequence {a k } is needed.

Weak factorizations and Hankel forms
It is well known to specialists, but difficult to find in the literature, that the obtention of weak factorizations is equivalent to estimates for small Hankel operators or Hankel forms (in our case, estimates with loss). The equivalence between boundedness of the bilinear Hankel form T α b and weak factorization can be formulated as the following result. Since this is the basis for our obtention of the weak factorization for Bergman spaces, for completeness, we offer the proof here of the implication that gives the factorization.
Proof We will prove (ii) implies (i). The other implication is easier, and the interested reader can follow the argument in [1, Corollary 1.2] for a proof. By the atomic decomposition in Theorem D, we have the inclusion In order to have the corresponding estimate for the norms, we will show that, for any bounded linear functional F on A Thus, suppose F ∈ (A Hence F ∈ (A p 2 α 2 ) * , and so, by Theorem A, there is an unique function where p 2 and α 2 satisfy Now, for polynomials g and h we have The proof is complete.

Proof of Theorem 3
In this section we prove Theorem 3, from which Theorem 2 follows. Before that, for s ≥ 0 and α > −1, let R α,s denote the unique continuous linear operator on H (B n ) satisfying for all w ∈ B n . If f ∈ A 1 α , then R α,s f is given by the following integral expression More properties of the "differential type" operators R α,s can be found in [24,Section 1.4]. Now we are ready to prove Theorem 3.

Proof of Theorem 3
As we noticed before, we just need to prove that S α f : A α 2 is bounded, and then from Hölder's inequality the result follows. Indeed, α 2 is bounded, we are going to show that f ∈ A q β . We begin with using an argument of Luecking (see, e.g., [17]). Let r k (t) be a sequence of Rademacher functions (see [8,Appendix A]). Let b be large enough so that Fix any r > 0, and let {a k } be an r -lattice and {D k } be the associated sets in Lemma C. By Theorem D, we know that, for any sequence of real numbers λ = {λ k } ∈ p 1 , the function Since S α f : A for almost every t in (0, 1). Integrating both sides with respect to t from 0 to 1, and using Fubini's Theorem and Khinchine's inequality (see [21, p.12]), we get If p 2 ≥ 2, then 2/ p 2 ≤ 1, and from (4.4) we have If 1 < p 2 < 2, then 2/ p 2 > 1, from (4.4), by Hölder's inequality we get since each z ∈ B n belongs to at most N of the setsD k . Combining the above two inequalities, and applying (4 By subharmonicity, we know that

From this we obtain
Let R α,b be the integral operator defined in (4.1). Then Then, we can apply (4.8) with the r -lattice {a k } for which the atomic decomposition in Theorem D for A q β holds. That is, for any h ∈ A q β , there exists a sequence μ = {μ k } ∈ q with μ q h q ,β such that Since then (4.9), Hölder's inequality and (4.8) gives This finishes the proof.

Compactness
Under the assumptions of Theorem 3, actually one has that the small Hankel operator α 2 is bounded if and only if it is compact. This is from a general result of Banach space theory. It is known that, for 0 < p 2 < p 1 < ∞, every bounded operator from p 1 to p 2 is compact (see, for example Theorem I.2.7, p. 31 in [15]). Since the weighted Bergman space A p α is isomorphic to p (see, Theorem 11, p. 89 in [21], note that the same proof there works for weighted Bergman spaces on the unit ball B n ), we get directly the above result.

Small Hankel operators with the same weights
Concerning the boundedness of the small Hankel operator h α α (the case when all the weights are the same) for all possible choices of 0 < p 1 , p 2 < ∞, we mention here that the case p 1 = p 2 > 1 is by now classical (see [4,13,23]), and in this case the boundedness is equivalent to the symbol f being in the Bloch space B, that consists of those holomorphic functions f on B n with Here, R f denotes the radial derivative of f , that is, The Bloch space also admits an equivalent norm in terms of the invariant gradient The case 0 < p 1 ≤ p 2 is completely settled in [4] (actually the results are stated for the unweighted Bergman spaces A p , but the proofs works also for the weighted case). The description for the case p 1 = p 2 = 1 is that f must belong to the so called logarithmic Bloch space, a result that goes back to the one dimensional result of Attele [2]. Concerning estimates with loss, Bonami and Luo [4] obtained a description for the case 0 < p 2 < p 1 with p 2 < 1 (again the result in [4] is stated for the unweighted Bergman spaces). Thus, in view of Corollary 4, to complete the picture it remains to deal with the case p 1 > p 2 = 1 (this problem is also open for the unit disk).
In that case, also in [4], some partial results are obtained (again for the unweighted case). Mainly, they provide a pointwise estimate that is necessary for the small Hankel operator to be bounded, and they show that the condition is sufficient. Moreover, they conjecture that the previous condition is also necessary. We have not been able to prove the conjecture, but we are going to shed some light on that problem. Before going to the proof we need first some preparation. First of all, recall that the Bloch space is the dual of A 1 α under the integral pairing , α (see [24,Theorem 3.17]). We also need the following lemma, whose one dimensional analogue is essentially proved in [3].
for any f ∈ H (B n ) and a ∈ B n .
Proof We are going to prove first that, for 0 ≤ t < n + 1 + σ , From [24, p. 51], for β big enough, say β ≥ 1 + σ , we have Take a small number ε > 0 with σ − ε max( p, p ) > −1, where p denotes the conjugate exponent of p, and t < n + 1 + σ − εp. An application of Hölder's inequality and Lemma B yields This together with Fubini's theorem and [19,Lemma 2.5] gives proving (5.2). Now, a change of variables z = ϕ a (ζ ) gives (see [24,Proposition 1.13]) is in the Bloch space with g z B ≤ C with the constant C independent of the point z. Therefore, from the pointwise estimate for functions in Bergman spaces, we get and (5.8) follows due to Theorem 6. The necessity of (5.9) is also a consequence of Theorem 6. Indeed, clearly M f : B → A p 1 α is bounded if and only if the measure dμ f (z) = | f (z)| p 1 dv α (z) is a p 1 -Carleson measure for the Bloch space (see [7,10] for the definition); and by Proposition 1.4 in [7] (the one dimensional case appears in [10] and [14]) this implies (5.9), finishing the proof.
The established connection between Hankel operators on Bergman spaces and Carleson measures for the Bloch space makes even more interesting the problem (as far as we know, still open) of describing those measures.