Hankel operators induced by radial Bekoll\'e-Bonami weights on Bergman spaces

We study big Hankel operators $H_f^\nu:A^p_\omega \to L^q_\nu$ generated by radial Bekoll\'e-Bonami weights $\nu$, when $1<p\leq q<\infty$. Here the radial weight $\omega$ is assumed to satisfy a two-sided doubling condition, and $A^p_\omega$ denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of $H_f^\nu$ and $H_{\overline{f}}^\nu$ is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau, Zhao and Zhu (Indiana Univ. Math. J. 2016), the respective spaces depend on the weights $\omega$ and $\nu$ in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.


Introduction and main results
Let HpDq denote the space of analytic functions in the unit disc D " tz P C : |z| ă 1u. A function ω : D Ñ r0, 8q, integrable over the unit disc D, is called a weight. It is radial if ωpzq " ωp|z|q for all z P D. For 0 ă p ă 8 and a weight ω, the Lebesgue space L p ω consists of (equivalence classes of) complex-valued measurable functions f in D such that where dApzq " dx dy{π denotes the normalized Lebesgue area measure on D. The weighted Bergman space A p ω is the space of analytic functions in L p ω . As usual, A p α denotes the weighted Bergman space induced by the standard radial weight pα`1qp1´|z| 2 q α . If ν is a radial weight then A 2 ν is a closed subspace of L 2 ν and the orthogonal projection from L 2 ν to A 2 ν is given by where B ν z are the reproducing kernels of A 2 ν ; f pzq " xf, B ν z y A 2 ν for all z P D and f P A 2 ν . The study of the boundedness of weighted Bergman projections on L p -spaces is a compelling topic that has attracted a considerable amount of attention during the last decades. A well known result due to Bekollé and Bonami [4,5] describes the weights ω such that the Bergman projection P η , induced by the standard weight pη`1qp1´|z| 2 q η , is bounded on L q ω for 1 ă q ă 8. We denote this class of weights by B q pηq, and write B q " Y ηą´1 B q pηq for short. In the case of a standard weight, the Bergman reproducing kernels are given by the neat formula p1´zζq´p 2`ηq . However, for a general radial weight ν the Bergman reproducing kernels B ν z may have zeros [19] and such explicit formulas for the kernels do not necessarily exist. This is one of the main obstacles in dealing with P ν [9,17]. Nonetheless, we shall prove in Proposition 6 below that if ν P B q is radial, then P ν : L q ν Ñ L q ν is bounded for each 1 ă q ă 8. The proof of this relies on accurate estimates for the integral means of B z ν recently obtained in [17,Theorem 1], and the result itself plays an important role in the proof of the main discovery of this paper.
All the above makes the class of radial weights in B q an appropriate framework for the study of the big Hankel operator H ν f pgqpzq " pI´P ν qpf gqpzq, f P L 1 ν , z P D, on weighted Bergman spaces. For an analytic function f , the Hankel operator H β f , induced by a standard projection, has been widely studied on Bergman spaces since the pioneering work of Axler [3], which was later extended in [1]. In the case of a rapidly decreasing weight ν and f P HpDq, Galanopoulos and Pau [11] did an extensive research on H ν f on A 2 ν ; see [2] for further results. For general symbols, Zhu [22] was the first to build up a bridge between Hankel operators and the mean oscillation of the symbols in the Bergman metric, and this idea has been further developed in several contexts [6,7,8,23]; see [24] and the references therein for further information on the theory of Hankel operators. More recently, Pau, Zhao and Zhu [13] described the complex valued symbols f such that the Hankel operators H β f and H β f are simultaneously bounded from A p α to L q β , provided 1 ă p ď q ă 8. Our primary aim is to extend this last-mentioned result to the context of radial B q -weights. To do this, some definitions are needed. For a radial weight ω, we assume throughout the paper that p ωpzq " ş 1 |z| ωpsq ds ą 0 for all z P D, for otherwise the Bergman space A p ω would contain all analytic functions in D. A radial weight ω belongs to the class p D if there exists a constant C " Cpωq ą 1 such that p ωprq ď C p ωp 1`r 2 q for all 0 ď r ă 1. Moreover, if there exist K " Kpωq ą 1 and C " Cpωq ą 1 such that then ω P q D. We write D " p D X q D for short. For basic properties of these classes of weights and more, see [14,15] and the references therein. Let βpz, ζq denote the hyperbolic distance between z, ζ P D, ∆pz, rq the hyperbolic disc of center z and radius r ą 0, and Spzq the Carleson square associated to z. For 0 ă p, q ă 8 and radial weights ω, ν, define γpzq " γ ω,ν,p,q pzq " p νpzq Further, for f P L 1 ν,loc , write p f r,ν pzq " ş ∆pz,rq f pζqνpζq dApζq νp∆pz,rqq and MO ν,q,r pf qpzq "˜1 νp∆pz, rqq ż ∆pz,rq |f pζq´p f r,ν pzq| q νpζq dApζq¸1 q for all z P D. It is worth noticing that for prefixed r ą 0, the quantity νp∆pz, rqq may equal to zero for some z arbitrarily close to the boundary if ν P p D. However, if ν P D, then there exists r 0 " r 0 pνq ą 0 such that νp∆pz, rqq -νpSpzqq ą 0 for all z P D if r ě r 0 . The space BMOp∆q ω,ν,p,q,r consists of f P L q ν,loc such that }f } BMOp∆qω,ν,p,q,r " sup zPD pMO ν,q,r pf qpzqγpzqq ă 8.
We will show that if ν P D, then BMOp∆q ω,ν,p,q,r does not depend on r for r ě r 0 . In this case, we simply write BMOp∆q ω,ν,p,q . The main result of this study reads as follows and it will be proved in Section 5. Theorem 1. Let 1 ă p ď q ă 8, ω P D, ν P B q a radial weight and f P L q ν . Then The approach employed in the proof of this result follows the guideline of [13,Thorem 4.1], however a good number of steps cannot be adapted straightforwardly and need substantial modifications. In Section 2 we prove some results concerning the classes of weights involved in this work and the boundedness of the Bergman projection P ν , while in Section 3 we introduce and study two spaces of functions on D. One of them is denoted as BAp∆q ω,ν,p,q , and although its initial definition depends on r, it can be described in terms of an appropriate Berezin transform or simply observing that f P BAp∆q ω,ν,p,q if and only the multiplication operator M f pgq " f g is bounded from A p ω to L q ν [16]. The second one, denoted by BOp∆q ω,ν,p,q , consists of continuous functions on D such that the oscillation in the Bergman metric is bounded in terms of the auxiliary function γ given in (1.2). We show that f P BOp∆q ω,ν,p,q if and only if |f pzq´f pζq| À }f } BOp∆qω,ν,p,q p1`βpz, ζqqΓ τ pz, ζq z, ζ P D, for an appropriate (small) constant τ " τ pω, νq ą 0. If ω and ν are standard weights, then Γ τ does not coincide with the function playing the corresponding role in [13,Lemma 3.2]; in the latter case the function is simpler in many aspects and does not depend on the additional parameter τ . Then, we show that BMOp∆q ω,ν,p,q " BAp∆q ω,ν,p,q`B Op∆q ω,ν,p,q . (1.3) In order to prove this decomposition, due to the complex nature of Γ τ pz, ζq, we are forced to split D into several regions depending on z, establish sharp estimates for Γ τ pz, ζq in each region and then apply properties of weights in D. The identity (1.3) together with a description of the boundedness of the integral operator and its maximal counterpart from A p ω to L q ν , see Section 4 below, are key tools to prove that each f P BMOp∆q ω,ν,p,q induces a bounded Hankel operator from A p ω to L q ν . Theorem 1 will be proved in Section 5.
Finally, in Section 6, as a byproduct of Theorem 1, we describe the analytic symbols such that H f : A p ω Ñ L q ν is bounded. The space B dγ consists of f P HpDq such that where γ is given by (1.2).
Throughout the paper 1 p`1 p 1 " 1 for 1 ă p ă 8. Further, the letter C " Cp¨q will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. We will use the notation a À b if there exists a constant C " Cp¨q ą 0 such that a ď Cb, and a Á b is understood in an analogous manner. In particular, if a À b and a Á b, then we will write ab.
Throughout the proofs we will repeatedly use several basic properties of weights in the classes p D and q D. For a proof of the first lemma, see [14,Lemma 2.1]; the second one can be proved by similar arguments.
Two more results on weights of more general nature than Lemmas A and B are also needed.

Lemma 3.
Let ω be a radial weight. Then the following statements are equivalent: (i) ω P p D; (ii) For some (equivalently for each) ν P D there exists a constant C " Cpω, νq ą 0 such that ż 1 r ωptqp νptq p ωptq dt ď C p νprq, 0 ď r ă 1; (iii) For some (equivalently for each) ν P D there exists a constant C " Cpω, νq ą 0 such that Proof. Let first ω P p D and 0 ď r ă 1, and consider ρ r n " ρ r n pω, Kq for all n P N Y t0u. Then Lemma B, applied to ν P D Ă q D, and Lemma A(v), applied to ω, imply ż 1 r ωptqp νptq p ωptq dt " 1 pC β q n " p νprq log K C β C β´1 , 0 ď r ă 1, for a suitably fixed K " Kpωq ą 1, and thus (ii) is satisfied. Conversely, (ii) implies and since ν P D Ă p D by the hypothesis, we deduce p ωprq À p ω`1`r 2˘f or all 0 ď r ă 1. Thus ω P p D. Let ω P p D and 0 ď r ă 1, and consider ρ n " ρ n pω, Kq for all n P N Y t0u. Fix k " kpω, Kq P N Y t0u such that ρ k ď r ă ρ k`1 . Then where, by Lemma B, applied to ν P D Ă q D, and Lemma A(v), applied to ω, for some α " αpνq ą 0 and for a suitably fixed K " Kpωq ą 1, and similarly, The statement (iii) follows from these estimates. Conversely, by replacing r by 1`r 2 in (iii) we obtain and since ν P D Ă p D by the hypothesis, we deduce p ωprq À p ω`1`r 2˘f or all 0 ď r ă 1. Thus ω P p D.
The argument used to prove (i)ñ(ii) in the said lemma shows that p σ Á p ν on r0, 1q, provided ω P q D and ν P D. Thus p σ -p ν, and σ P D by Lemmas A(ii) and B.
The next lemma says that in many instances concerning A p -norms we may replace ω by r ω " p ω{p1´|¨|q if ω P D. This result has the flavor of radial Carleson measures and indeed can be established by appealing to the characterization of Carleson measures for the Bergman space A p ω induced by ω P p D given in [16]. That approach requires showing that the involved weights belong to p D, which is of course the case, and thus involves more calculations than the simple proof given below.
Proof. The function p1´|¨|q κ´1 p ω is a weight for each κ ą´α by Lemma B. Therefore an integration by parts shows that (2.1) is equivalent to Another integration by parts reveals that both integrals from r to 1 above are bounded by a constant times p by Lemma 4. The assertion follows.
The last auxiliary results shows that each radial weight in the Bekollé-Bonami class B q belongs to D, and for each ν P D the maximal Bergman projection is bounded on L q ν . It is worth noticing that obviously D Ć Y 1ăqă8 B q because ν P D may vanish on a set of positive measure. Proposition 6. Let 1 ă q ă 8 and ν P B q a radial weight. Then ν P D. Moreover, Pν : L q ν Ñ L q ν is bounded for all ν P D.
Proof. If ν P B q , then by [5] there exists β ą´1 such that Since ν is radial, this condition easily implies ν P D.
Let now 1 ă q ă 8 and ν P D, and define h " p ν´1 qq 1 . Then Moreover, by symmetry, (2.2) with q 1 in place of q is satisfied. Since ν P p D, we may apply [17, and ż D |B ν z pζq|h p pzqνpzq dApzq À h p pζq, ζ P D.
The following lemma is easy to establish; see [13, Lemma 3.1] for a similar result.
and therefore f P L q ν belongs to BMOp∆q if and only if for each z P D there exists λ z P C such that sup zPD˜γ pzq q νp∆pz, rqq ż ∆pz,rq |f pζq´λ z | q νpζq dApζq¸ă 8.
Since ω P p D by the hypothesis, and p ωptq " p ωp0q for t ă 0, Lemma A(ii) implies for some C " Cpωq ą 0 and β " βpωq ą 0. Further, p νpzq -p νpζq and p ωpzq -p ωpζq if βpz, ζq ď r by Lemma A(ii). The assertion follows from these estimates.
For continuous f : D Ñ C and 0 ă r ă 8, define Ω r f pzq " supt|f pzq´f pζq| : βpz, ζq ă ru, z P D, and let BOp∆q " BOp∆q ω,ν,p,q,r denote the space of those f such that Lemma 9 shows that the space BOp∆q " BOp∆q ω,ν,p,q,r is independent of r.
Therefore (ii) is satisfied.
For 0 ă p, q ă 8, 0 ă r ă 8 and radial weights ω, ν, the space BAp∆q " BAp∆q ω,ν,p,q,r consists of f P L q ν,loc such that For c, σ P R and a radial weight ν, the general Berezin transform of ϕ P L 1 νp1´|¨|q σ is defined by The next lemma shows, in particular, that the space BAp∆q " BAp∆q ω,ν,p,q,r is independent of r as long as r is sufficiently large depending on ν P D.
Proof. It is obvious that (ii), (iii) and (iv) are equivalent by the definitions. Assume (ii) is satisfied, that is,ˆż D |gpζq| q |f pζq| q νpζq dApζq˙1 q À }g} A p ω , g P A p ω . (3.5) For z P D, let g z pζq "´1´| z| 1´zζ¯λ`1 p , where λ " λpωq ą 0 is that of Lemma A(iv). Further, since ν P q D by the hypothesis, there exists r ν P p0, 8q such that νp∆pz, rqq ą 0 for all r ě r ν . For g " g z and r ě r ν , (3.5) yields But since ν P D, applications of Lemmas A(ii) and B show that if r is sufficiently large. It follows that f P BAp∆q " BAp∆q ω,ν,p,q,r for all such r, and thus (i) is satisfied. Conversely, if (i) is satisfied, then by using (3.6) we deducẽ ż ∆pz,rq |f pζq| q νpζq dApζq¸1 Therefore |f | q νdA is a q-Carleson measure for A p ω by [18, Theorem 3]. By integrating only over ∆pz, rq in (v) and using (3.6) we obtain (i) from (v). To complete the proof of the lemma, it remains to show the converse implication. To do this, pick up a sequence ta j u and 0 ă r ă 8 in accordance with [24,Lemma 4.7], and observe that p ω is essentially constant in each hyperbolically bounded region by Lemma A(ii). Then by using (3.6), the hypothesis (i), the election of c and σ, and finally Lemmas A(ii) and B, we deduce and thus (v) is satisfied.
With these preparations we are ready to show that BMOp∆q " BAp∆q`BOp∆q. This follows from the case (ii) of the next theorem.
Proof. Obviously, (iii) implies (iv). Next assume (iv). The relation (3.6) shows that there exists r 0 " r 0 pνq ą 0 such that which together with Lemma 7 shows that (i) is satisfied. Assume now (i), and let f 2 " p f r,ν . Since f P L q ν , q ě 1 and r ě r ν , the function f 2 is well defined and continuous. Since ω, ν P D by the hypothesis, one may use Lemmas A(ii) and B together with the argument in [13,[1651][1652] with minor modifications to show that f 2 " p f r,ν P BOp∆q and f 1 " f´p f r,ν P BAp∆q. Thus (ii) is satisfied.
To complete the proof it suffices to show that (ii) implies (iii), so assume f " f 1`f2 , where f 1 P BAp∆q and f 2 " p f r,ν P BOp∆q. Since p f r,ν " p f 1r,ν`p f 2r,ν , it suffices to prove the condition in (iii) for f 1 and f 2 separately. First observe that by Lemma A(iii) the constant function 1 satisfies because c ą maxtγpνq, σu´1 by the hypothesis. This together with Hölder's inequality and Lemma 10 yields B´ˇˇf 1´p f 1r,ν pzqˇˇq¯γpzq q À´Bp|f 1 | q qpzq`| p f 1r,ν pzq| q¯γ pzq q ď´Bp|f 1 | q qpzq`z |f 1 | q r,ν pzq¯γpzq q À 1, z P D, and thus (iii) for f 1 P BAp∆q is satisfied.

Hence (3.8) and Lemmas A(iii) and 3(ii) yield
Since D " Y 4 j"1 D j pzq for each z P D, by combining the four cases we obtain (3.7). Thus (ii) implies (iii), and the proof is complete.

Boundedness of integral operators
In order to deal with the boundedness of Hankel operators, we need a technical result concerning certain integral operators. For f P L 1 b and b, c P R, define In the analytic case the operator T b,c can be interpreted as a fractional differentiation or integration depending on the parameters b and c [21]. The boundedness of these operator between L p spaces induced by standard weights has been characterized in [20].
Lemma A(ii) shows that for η P p D there exists a constant c 0 " c 0 pσq ą 1 such that hypotheses (i) and (ii) of the next lemma are satisfied for all c ě c 0 . Lemma 12. Let 1 ă p ď q ă 8, b ą´1, c ą 1 and σ, η P D such that , 0 ď r ă 1.

Therefore (2) yields
Lemma B together with the assumption (3) yields since η P D Ă q D by the hypothesis. In a similar fashion, (3) together with the hypothesis (i) gives and hence I 2 pzq À p ηpzq´1 q for all z P D. This estimate and Minkowski's integral inequality (Fubini's theorem in the case q " p) now yield , ζ P D, we deduce by the assumption (3). It follows that }S b,c pf q} L q η À }f } A p r σ . This finishes the proof because }f } A p r σ -}f } A p σ for all f P HpDq by Lemma 5 provided σ P D.

Proof of Theorem 1
In order to prove the sufficiency part of Theorem 1 we shall use the next result which follows from the argument used in the proof of [13,Lemma 4.5].
Lemma 13. Let 1 ă q ă 8 and ν, ω weights such that P ω : L q ν Ñ L q ν is bounded. Then Proposition 14. Let 1 ă p ď q ă 8, ν P B q a radial weight and ω P D. If f P BOp∆q, then H ν f : A p ω Ñ L q ν is bounded.