Zero sequences, factorization and sampling measures for weighted Bergman spaces

The zero sets of the Bergman space $A^p_\omega$ induced by either a radial weight $\omega$ admitting a certain doubling property or a non-radial Bekoll\'e-Bonami type weight are characterized in the spirit of Luecking's results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz's factorization result from 1977 for functions in $A^p_\omega$. The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space $A^p_\omega$ induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.


Introduction and main results
Let HpDq denote the space of analytic functions in the unit disc D " tz P C : |z| ă 1u of the complex plane C. A function ω : D Ñ r0, 8q, integrable over 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 weighted Bergman space A p ω consists of f P HpDq such that where dApzq " dx dy π is the normalized Lebesgue area measure on D. As usual, A p α stands for the classical weighted Bergman space induced by the standard radial weight ωpzq " p1´|z| 2 q α , where´1 ă α ă 8. and M 8 pr, f q " max |z|"r |f pzq|. For 0 ă p ď 8, the Hardy space H p consists of f P HpDq such that }f } H p " sup 0ără1 M p pr, f q ă 8.
In this paper we are mainly interested in zero-sequences, factorization, dominating sets and sampling measures for the Bergman space A p ω induced by either a non-radial weight belonging to a kind of Bekollé-Bonami class or a radial weight admitting a certain doubling property. Our studies of zeros and factorization go hand-in-hand and dominating sets and sampling measures are almost equally interrelated. While our results on zeros and factorization improve and generalize certain results in the literature, our findings on dominating sets and sampling measures are less complete but offer a somewhat new approach to these topics and also give arise to several open problems.
We begin with zeros and factorization. Let ω : D Ñ r0, 8q be a weight, 0 ă p ă 8 and f P A p ω such that f ı 0. Then a sequence Z Ă D is called the zero set (or sequence) of f , denoted by Zpf q, if f paq " 0 for all a P Z, counting multiplicities, and f paq ‰ 0 for all a P DzZ. A set Z is called a zero set for A p ω if there exists a nonzero function f P A p ω such that Z " Zpf q. The set of all zero sets of A p ω is denoted by ZpA p ω q. Zeros of functions in Hardy spaces are neatly characterized by the Blaschke condition and each Hardy-function f admits the well-known inner-outer factorization, where the inner part is a product of a singular inner function and a Blaschke product containing all the zeros of f and the non-vanishing outer part has the same norm as f [9]. The situation of Bergman spaces is completely different because the distribution of zeros of functions in Bergman spaces is not that well understood neither such an efficient factorization as in the Hardy space case is known. Even if the geometric distribution of zeros is not completely described, the difference between known sufficient and necessary conditions for a sequence to be a zero set for A p α is small. Probably the most commonly known results in this context are due to Luecking, Korenblum, Hedenmalm, Horowitz and Seip. Horowitz [12,14,15] studied unions, subsets and dependence on p of the zero sets of functions in A p α and obtained accurate information on certain sums involving the moduli of zeros. Some of these results were generalized to certain A p ω in [28]. The studies by Korenblum [18], Hedenmalm [11] and Seip [38,39] employ methods based on the use of densities defined in terms of partial Blaschke sums, Stolz star domains and Beurling-Carleson characteristic of the corresponding boundary set, and yield more complete results. Luecking [23] gave a description of zero sets of A p α in terms of certain auxiliary functions induced by the zeros. Even if this characterization does not reveal the geometric distribution very transparently, it yields accurate information on the subsets of zero sets. Horowitz [13] also established a useful factorization theorem for functions in A p α , but this factorization does not allow to take one of factors non-vanishing, and thus does not behave equally well as the inner-outer factorization in Hardy spaces. This factorization result was generalized to some A p ω in [16,28]. We start with employing Luecking's [23] approach to describe A p ω zero sets when ω belongs to a kind of Bekollé-Bonami class. For 1 ă q ă 8, write ω P B q if the weight ω is (almost everywhere) strictly positive and where the supremum is taken over all Carleson squares S Ă D, and |S| denotes the Euclidean area of S. Denote B 8 " Ť qą1 B q for short. Recall that each Carleson square is of the form Spzq " " re iθ P D : |z| ă r ă 1, | arg ze´i θ | ă 1´|z| 2 * , z P Dzt0u.
In the beginning of Section 2 we briefly analyze the classes B q . In particular, we show that B p Ĺ B q for 1 ă p ă q ă 8 and discuss their Kerman-Torchinsky properties. Following Luecking [23], for a sequence Z Ă D, we use the notation The first of our main results on Bergman zero sets is a generalization of [23, Theorem 3] and reads as follows.
Theorem 1. Let 0 ă p ă 8, ω P B 8 and Z be a sequence in D. Then the following statements are equivalent: (a) Z P ZpA p ω q; (b) Any subsequence of Z belongs to ZpA p ω q; (c) ř aPZ p1´|a|q 2 ă 8 and there is a nowhere zero function F P HpDq such that F W Z P L p ω ; (d) ř aPZ p1´|a|q 2 ă 8 and there is a nonzero function F P HpDq such that F W Z P L p ω .
Moreover, if (a) is true, then the mapping f Þ Ñ f {ψ Z is a continuous isomorphism from tf P A p ω : Z Ă Zpf qu onto tF P HpDq : F W Z P L p ω u. If F is that of Theorem 1 (c), and h "´p log |F |, then h is harmonic and Z " e p log |F | e pk Z " expppk Z´h q. Therefore the equivalence of (a) and (c) in Theorem 1 leads to the following result.
Corollary 2. Let 0 ă p ă 8, ω P B 8 and Z be a sequence in D. Then Z P ZpA p ω q if and only if there exists a harmonic function h such that ż D expppk Z pzq´hpzqqωpzq dApzq ă 8.
By performing a certain perturbation on a zero set, in this case moving the points closer to the boundary, it becomes a zero set for some other weighted Bergman space. A sequence pz n q in D is separated or equivalently uniformly discrete if inf k‰n ρ p pz k , z n q " inf k‰n |ϕ z k pz n q| ą 0, where ϕ a pzq " a´z 1´az is the standard automorphism of the unit disc. Corollary 3. Let 0 ă p ă 8 and ω P B 8 . Let Z be a zero set for A p ω such that it is a finite union of separated sequences. Let 0 ă γ ă 1 and suppose that there exists another set Z 1 and a one-to-one correspondence σ : Z Ñ Z 1 such that 1´|σpaq| 2 " γ`1´|a| 2˘a nd ρ p pa, σpaqq is uniformly bounded away from 1 on Z. Then Z 1 is a zero set for A p{γ ω . To deduce Corollary 3, note first that since Z is a zero set for A p ω by the hypothesis, there exists a harmonic function h such that expppk Z´h q is integrable with respect to ωdA by Corollary 2. Therefore it suffices to find a harmonic majorant g of k Z 1 {γ´k Z . Indeed, if such g exists, then pk Z 1 {γ´ppg`hq ď pk Z´h , and hence expp p γ k Z 1´ppg`hqq is integrable with respect to ωdA, and consequently Z 1 is a zero set for A p{γ ω by Corollary 2. A function g with desired properties is constructed in the proof of [24,Theorem 4].
Horowitz [12,13] also obtained some results in the spirit of Corollary 3 describing how the zero sets depend on the parameter p. Some of those results were generalized for certain A p ω in [28], and can be further improved by applying Luecking's approach in studying zero sets. In particular, we note that, by applying Proposition 13 below instead of [13, Lemma 3], Theorem 4 and Corollary 2 in [13] can be generalized to the case ω P B 8 with only minor modifications to the original proofs.
We now turn to consider factorization of functions in A p ω . By refining Horowitz' original probabilistic argument by Luecking's method to study the zero sets, and then adopting the whole reasoning to the class of weights we are interested in we derive the following factorization result.
Theorem 4. Let 0 ă p ă 8 and ω P B 8 such that the polynomials are dense in A p ω . Let f P A p ω and 0 ă p 1 , p 2 ă 8 such that p´1 " p´1 1`p´1 2 . Then there exist f 1 P A p 1 ω and f 2 P A p 2 ω such that f " f 1 f 2 and for some constant C " Cpωq ą 0.
The density of polynomials is needed as a hypothesis only to guarantee the existence of a dense family of functions with finitely many zeros. Any other requirement implying this property would suffice here. The question of when polynomials are dense in A p ω is an old problem and remains unsolved in general, see for example [28,Section 1.5] for basic information and relevant references. However, in certain special cases the density of polynomials can be deduced from an atomic decomposition. For example, [22,Theorem 4.1] offers such a decomposition for functions in certain weighted Bergman spaces induced by non-radial weights in terms of the kernel functions of the standard weighted Bergman spaces. Now that these kernels are analytic beyond the boundary of the unit disc, this in turn yields the density of polynomials. Another relevant reference regarding atomic decomposition in the non-radial case is [5]. The argument used in these studies does not work for the class B 8 , and that is understandable because B 8 contains weights that induce very small Bergman spaces.
We next discuss a specific step in the proofs of the results above and then turn to consider zeros and factorization for A p ω induced by a radial weight. The key ingredient in the proofs of Theorems 1 and 4 is Proposition 13 which states that The proof of this fact eventually boils down to showing that is a bounded operator from L q ω into itself for some q ą 1. This is in turn equivalent to the boundedness of the Bergman projection P α pf qpzq " ż D f pwq p1´zwq 2`α`1´| w| 2˘α dApwq, z P D, for α " 2 on certain L q -spaces, and therefore this step is done at once by using the classical characterization of the one-weight inequality for the Bergman projection by Bekollé and Bonami [3,4]. This yields the hypothesis ω P B 8 in Theorems 1 and 4, the proofs of which are presented in Section 2.
The defect in the hypothesis ω P B 8 is that it does not allow ω to vanish in a set of positive measure, neither ω may be small in a relatively large part of each outer annulus of D. Our next goal is to show that by restricting our consideration to radial weights we can do better and no longer need to require such smoothness. To do this, let ω be a radial weight such that p ωpzq " ş 1 |z| ωpsq ds ą 0 for all z P D, for otherwise A p ω " HpDq. A radial weight ω belongs to the class p D if there exists a constant C " Cpωq ě 1 such that the doubling inequality p ωprq ď C p ωp 1`r 2 q is valid for all 0 ď r ă 1. If there exist K " Kpωq ą 1 and C " Cpωq ą 1 such that p ωprq ě C p ω`1´1´r K˘f or all 0 ď r ă 1, then ω P q D. Additionally, we write D " p D X q D. The classes of weights p D and D emerge from fundamental questions in operator theory: recently Peláez and Rättyä [32] showed that the weighted Bergman projection P ω , induced by a radial weight ω, is bounded from L 8 to the Bloch space B " tf P HpDq : sup zPD |f 1 pzq|p1´|z|q ă 8u if and only if ω P p D, and further, it is bounded and onto if and only if ω P D. For further information on these classes of weights, see [27,28,29,30]. Moreover, since weights in p D may very well vanish in a set of positive measure, p D is not contained in B 8 .
from which the well known inequality M 8 pr, f q À M p p 1`r 2 , f qp1´rq´1 {p yields Now that ω P p D, there exist C " Cpωq ą 0 and β " βpωq ą 0 such that p ω prq ě C p ωp0q p1´rq β by Lemma A(ii) below. Hence |f pzq| À }f } A p ω p1´|z|q´β`1 p for all z P D, and it follows that log |f | P L 1 . Therefore the reasoning to be used in the proof of Proposition 13 can be applied in this case also. However, as explained above, the argument relies on the characterization of the one-weight inequality for the Bergman projection by Bekollé and Bonami, which guarantees the boundedness of the auxiliary operator R : L q ω Ñ L q ω under the hypothesis ω P B q Ă B 8 . But what is actually needed here is to show that ż DˆżD |f pζq| p q p1´|z| 2 q 2 |1´zζ| 4 dApζq˙q ωpzq dApzq À }f } p for a sufficiently large q " qpωq ą p. The functions involved being analytic, this inequality is better controllable by using Carleson embedding theorems [29] rather than weight inequalities for L p -functions [3,4]. By using this approach we will prove the statement of Proposition 13 for ω P p D, and deduce the following result.
Theorem 5. The statements in Theorems 1 and 4 and Corollaries 2 and 3 are valid when the hypothesis ω P B 8 is replaced by ω P p D.
Details of the deduction yielding this theorem are given in Section 3. The reason why this approach does not yield a better result in the non-radial case is that known Carleson embedding theorems impose growth and/or smoothness restrictions to the weight. In particular, [6, Theorem 3.1] states that if Pὴ : L q ω Ñ L q ω is bounded for some q ą 1 and η ą´1, then A p ω is continuously embedded into L p µ if and only if µp∆pz, rqq À ωp∆pz, rqq for all z P D. Each radial weight p1´|z|q´1´log e 1´|z|¯´β with β ą 1 belongs to B 8 , but does not satisfy the Bekollé-Bonami condition, and thus this approach does not give us anything better than what we know already. This and many other possible applications, some of which will appear later in this paper, suggest that it would be desirable to obtain new information on the embedding A p ω Ă L p µ when ω is non-radial and µ is a positive Borel measure on D. In the radial case the hypothesis on the density of polynomials in Theorem 4 is of course always satisfied and can thus be omitted.
The statement in Theorem 4 for ω P B 8 Y p D significantly improves the main result in [28, Chapter 3] and Horowitz' original result as well because now the factorization is available for the whole class p D and the constant C appearing in the inequality for the norms is independent of the parameters p, p 1 and p 2 .
We mention three immediate consequences of our results so far. The factorization given in Theorem 5 shows that the statement in [28, Theorem 4.1(iv)] concerning the integration operator T g pf qpzq " ş z 0 f pζqg 1 pζq dζ induced by g P HpDq is valid for ω P p D. The theorem states that T g : A p ω Ñ A q ω is bounded in the upper triangular case 0 ă q ă p ă 8 if and only if g P A s ω with 1 s " 1 q´1 p . Also the Aleman-Sundberg question, discussed in more detail in [28, p. 38], on subsets of zero-sets has an affirmative answer when ω P p D by Theorem 1 because in this case each subset of a zero set is always a zero set. Moreover, the statement in [27, Proposition 7.5] concerning bounded and compact composition operators acting between different weighted Bergman spaces is valid under the hypotheses ω P B 8 Y p D and the density of polynomials in A p ω by Theorems 4 and 5. The proposition says that if 0 ă p, q ă 8, n P N and ν is any weight, then the composition operator C ϕ , defined by C ϕ pf q " f˝ϕ, is bounded (resp.compact) from A p ω to A q ν if and only if C ϕ : A np ω Ñ A nq ν is bounded (resp.compact). Therefore one may assume that the both parameters p and q are greater than two and this often simplifies some arguments.
To finish our consideration of zeros and factorization, we give an application of the obtained factorization in the study of the small Hankel operator induced by a conjugate analytic symbol in the upper triangular case. Let ω be a weight such that the reproducing kernels B ω z of A 2 ω exist, that is, f pzq " xf, B ω z y A 2 ω for all z P D and f P A 2 ω . For f P HpDq consider the small Hankel operator Corollary 6. Let 1 ă q ă p ă 8 and 1 ă s ă 8 such that 1 s " 1 q´1 p , and let ω P p D. Then This corollary combined with our earlier observation on T g acting from A p ω to A q ω confirms the well known phenomenon that in many cases the boundedness of the integration operator and the small Hankel operator are characterized by the same condition. Corollary 6 extends the results of Pau and Zhao [25] to the case of Bergman spaces of one complex variable induced by doubling weights.
Hankel, integration and composition operators are among the most studied objects in operator theory of analytic function spaces and the literature concerning the subject is vast. Since none of these operators is in the main focus of the present paper, we invite the reader to see [7,40] for the theory of composition operators, and [31] for the case p D, [1,2] for integration, and [35,41] for Hankel. For some recent developments on Hankel operators in Bergman spaces, see [26] and references therein.
Our next goal is to study dominating sets for A p ω in order to obtain a sufficient condition for a positive Borel measure µ on D to be a sampling measure for If this inequality is valid for all f P A p ω , then G is called a dominating set for A p ω . To state the results, some more notation is needed. The reproducing kernels of the Hilbert space A 2 ω induced by a radial weight ω are given by where ω x " ş 1 0 s x ωpsq ds for all´1 ă x ă 8, and each f P A 1 ω satisfies Our first result on dominating sets reads as follows.
Theorem 7. Let 0 ă q ă p ă 8 and ω P p D. Then there exists a constant C " Cpω, qq ą 0 such that Moreover, if f P A p ω and E " Epε, q, f q is the set of points z P D for which Therefore ε ą 0 may be chosen such that G " DzE satisfies ż G |f pzq| p ωpzq dApzq ě and thus G is a dominating set for f P A p ω if ε ą 0 is sufficiently small.
The inequality (1.3) with C " 1 is easy to establish if q ě 1. Namely, an application of the reproducing formula (1.2) This gives the assertion for q " 1, which together with Hölder's inequality can be used to establish the claim in the case q ą 1. This kind of reasoning does not seem to work for 0 ă q ă 1, and we will argue differently; we first use the subharmonicity of |f | q together with the fact that |B ω z pζq| -B ω z pzq for all ζ P ∆pz, rq and z P D if r " rpωq P p0, 1q is sufficiently small [33,Lemma 8], and then employ a proof of a Carleson embedding theorem for specific subharmonic functions [27,Theorem 3.3].
The estimate (1.5) can be obtained quite easily for ω P D Ă p D by using the boundedness of the maximal Bergman projection Pὼ on L p ω for each p ą 1, but the general case ω P p D is more laborious and relies on the L p -estimates of the kernel functions B ω z given in [30]. The special case of Theorem 7 concerning standard weighted Bergman spaces can be found in [24,Lemma 2]. The proof there is different and does not carry over to the situation of Theorem 7.
The proof of (1.5) in Theorem 7 does not work for p " q. In that case, it is natural to replace the the right hand side of (1.4) by an average over a subset of a pseudohyperbolic disc centered at z. To state the result some notation is needed. Let r P p0, 1q, ν a positive Borel measure on D and Epzq Ă ∆pz, rq such that νpEpzqq ą 0 for all z P D. Define and E " Epf, νq " tz P D : |f pzq| p ď εQpf qpzqu.
and thus DzE is a dominating set for f P A p ω if ε " εpr, ωq ą 0 is sufficiently small.
Special cases of Theorem 8 with ν being the Lebesgue measure can be found in [19, Lemmas 2 and 3]. Our proof is different from these results and relies on Carleson measures.
Since our main results so far concern also certain non-radial weights, it is reasonable to discuss that aspect of Theorems 7 and 8 as well. The first obstruction in the proof of Theorem 7 for non-radial weights is the pointwise estimate |B ω z pζq| -B ω z pzq which does not have a known sufficiently general non-radial extension. The second problem arises with Carleson measures, and finally the lack of satisfactory norm estimates for kernel functions prevents our reasoning from carrying over to the non-radial case all together. The situation of Theorem 8 is better because of [20, Lemmas 1 and 2] and the proof of [22,Theorem 3.9], though the weight ω in [20] is rather particular (but the domain lies in C n and is quite general). A careful inspection of the proof of [29,Theorem 9] shows that the argument used to obtain Theorem 8 works for weights ω that are doubling in Carleson squares, denoted by ω P p DpDq and defined in detail below, if N : A p ω Ñ L p ω is bounded. This raises the question if the maximal operator N pf qpzq " sup ζPΓpzq |f pζq|, where Γpζq " are non-tangential approach regions with vertexes inside the disc [28, Chapter 4.1], is bounded from A p ω to L p ω when ω P p DpDq. Unfortunately, we do not know the answer to this question. We next proceed to study dominating sets for the whole space A p ω . For 1 ă q ă 8 and a (almost everywhere) positive weight ω, write ω P C q if for some some (equivalently for each) r P p0, 1q, there exists a constant C " Cpq, r, ωq ą 0 such that ď C|∆pz, rq|, z P D, and set C 8 " Y qą1 C q . Luecking [22,Theorem 3.9] showed that if G Ă D is measurable, 0 ă p ă 8 and ω P C 8 such that |G X ∆pz, rq| ě δ|∆pz, rq|, z P D, (1.6) for some δ ą 0 and r P p0, 1q, then G is a dominating set for A p ω . This condition is equivalent to the existence of a constant δ 0 " δ 0 ą 0 such that |GXS| ą δ 0 |S| for all Carleson squares S. One can also replace the pseudohyperbolic disc ∆pa, rq by a suitable Euclidean disc, for example, Dpa, ηp1´|a|qq for a fixed 0 ă η ă 1 would work here. For the proofs of these equivalences, see [19]. The condition (1.6) is known to be also a necessary condition for G to be a dominating set for A p ω if ω is one of the standard weights by [19]. The existing literature does not offer results concerning the converse statement of the abovementioned result in the non-radial case. We next turn our attention to this matter, and to do it we write ω P p DpDq if there exists C " Cpωq ą 0 such that ωpSpaqq ď CωpSp 1`|a| 2 e i arg a qq for all a P Dzt0u. It is easy to see that each ω P p DpDq satisfies ωpSpa 1 qq ď CpC`1qωpSpaqq for all a, a 1 P Dzt0u with |a 1 | " |a| and arg a 1 " arg a˘p1´|a|q. Therefore ωpSpaqq À ωpSpbqq whenever |b| " 1`|a| 2 and Spbq Ă Spaq. Moreover, it is obvious that radial weights in p DpDq form the class p D.
Theorem 9. Let 0 ă p ă 8 and ω P p DpDq. If G is a dominating set for A p ω , then there exists a constant δ ą 0 such that ωpG X Sq ą δωpSq (1.7) for all Carleson squares S.
The proof of Theorem 9 is based on characterizations of weights in p DpDq given in Lemma 14 below and appropriately chosen test functions. By combining Theorem 9 with [22, Theorem 3.9] and imposing severe additional hypothesis on ω one can certainly obtain a characterization of dominating sets in the non-radial case but because of these extra assumptions the resulting description is far from being satisfactory. The approach involving the Lebesgue measure and yielding (1.6) is natural and has been efficiently used in [19], [20] and [22], but it seems that the arguments used there are not adoptable as such to prove (1.7) to be a sufficient condition. It is of course equally natural to measure the set G as in (1.7) by using the weight ω itself that induces the space. Moreover, the studies on Carleson measures [27], [28], [29], [34] strongly support the use of Carleson squares instead of pseudohyperbolic discs as testing sets, at least when ω induces a very small weighted Bergman space. It is also worth noticing that the hypothesis ω P p DpDq allows ω to vanish in a relatively large part of each outer annulus of D, meanwhile weights in C 8 may not have this property because of the negative power´q 1 q appearing in the definition of C q . Therefore a complete solution to the question of when a set G is a dominating set for A p ω remains as an open problem in both non-radial weight classes C 8 and p DpDq. Finally, we turn our attention to sampling. A positive Borel measure on D is a sampling The measures µ satisfying the inequality "À" are the p-Carleson measures for A p ω . These measures in the case ω P p D have been studied in [28,29,34], and can be characterized in terms of the weighted maximal function µ is a p-Carleson measure for A p ω if and only if M ω pµq P L 8 , and }Id} p Therefore these measures are independent of p for each ω P p D. Before stating our result, some more notation is needed. For a positive Borel measure µ on D and r P p0, 1q, let k r pzq " µp∆pz, rqq{ωp∆pz, rqq for all z P D. The next theorem describes a condition sufficient to guarantee that a positive Borel measure µ is a sampling measure for A p ω . Recall that the hypothesis µp∆pa, rqq À ωp∆pa, rqq characterizes Carleson measures for certain A p ω as mentioned in the paragraph just after Theorem 5.
Theorem 10. Let 0 ă p ă 8, ε ą 0 and either ω P p D such that ω ą 0 almost everywhere on D, and µ a p-Carleson measure for A p ω , or ω P C 8 and µp∆pa, rqq À ωp∆pa, rqq for all a P Dzt0u. Then there exists an r P p0, 1q such that µ is a sampling measure for A p ω whenever the set G " tz P D : k r pzq ą ε}M ω pµq} L 8 u is a dominating set for A p ω .
The proof of Theorems 10 follows the ideas of Luecking [21], but a crucial step in the case of p D relies on the characterization of Carleson measures. Additionally, the presence of the weight ω also makes the use of convenient changes of variables and automorphisms difficult and thus forces us to make some more delicate observations.
One can readily see from the proof that in the case of p D one may omit the extra hypothesis on the positivity of ω by replacing k r by k ‹ r pzq " µp∆pz, rqq{ωpSpzqq. In this case the set G may become essentially smaller and thus it being dominating set would be a stronger hypothesis.
Let pµ n q be a sequence of measures on D. We say that pµ n q converges weakly to a measure µ, denoted by µ n á µ, if ż Theorem 11. Let 0 ă p ă 8 and ω P p D, and let pµ n q be a sequence of p-Carleson measures for A p ω such that sup n }M ω pµ n q} L 8 ă 8. Then pµ n q has a weakly convergent subsequence. Further, if µ n á µ, then lim nÑ8 ż D |f pzq| p dµ n pzq "

8)
and µ is a p-Carleson measure for A p ω with }Id} A p ω ÑL p µ ď lim inf nÑ8 }Id} A p ω ÑL p µn . Furthermore, if µ n are sampling measures with sampling constants at most Λ ą 0, then µ is also a sampling measure with a sampling constant at most Λ.
The proof of Theorem 11 follows the lines of that of [24, Theorem 1], but the crucial step which differs from the original argument relies on Carleson embedding for tent spaces given in [29,Theorem 9]. These tent spaces of measurable functions are defined by using the maximal function N pf q.
Luecking [24] characterized the sampling measures for Bergman spaces induced by standard weights. However, the methods used there do not generalize for weights in the class p D because the weights in p D can be such that compositions of functions in A p ω with Möbius transformations cannot be controlled in norm. Thus, the problem of characterizing sampling measures for A p ω when ω P p D remains open. However, the closely related sampling sequences were characterized by Seip [37] in small weighted Bergman spaces induced by weights admitting a pointwise doubling condition in ω instead of p ω.

Zeros and factorization when ω P B 8
We begin with briefly analyzing the classes B q and B 8 . For´1 ă α ă 8, write dA α pzq " p1´|z| 2 q α dApzq for short. Bekollé and Bonami [3,4] showed that for 1 ă p ă 8, P α : L p ω rαs Ñ L p ω rαs is bounded if and only if where A α pEq " ş E dA α for each measurable set E Ă D. Denote by BB p,α the set of these weights, and write BB 8,α " Y 1ăpă8 BB p,α . By comparing this to the definition of B q , we see that ω P B q if and only if ω r2q´2s P BB q,2 . Moreover, it is known that ω P BB 8,α if and only if there exist δ P p0, 1q and C ą 0 such that for all Carleson squares S. This condition corresponds to the characterization of the restricted weak-type inequality for the Hardy-Littlewood maximal operator by Kerman and Torchinsky [17]. For more on A 8 -conditions, see [8] and the references therein.

Proposition 12.
Let ω be an almost everywhere strictly positive weight. Then the following assertions hold: (iv) If there exist q ą 1, δ P p 1 q , 1q and C ą 0 such that
Proof. Hölder's inequality and the inequality p1´|z|q 2 À |S| for z P S imply B q pωq À B p pωq for 1 ă p ă q ă 8, and thus B p Ă B q . The inclusion is seen to be strict by considering standard power weight p1´|z|q α with p ă α`1 ă q. Thus (i) is satisfied. Moreover, since W q,ω pzqp1´|z| 2 q 2q 1 " ωpzq´1 q´1 and W q,ω pzq´1 q 1´1 " ωpzqp1´|z| 2 q 2q , the assertion (ii) follows by the definition of B q .
To see (v), let q ě α 2 ě 0. Then Proof. Since 1´x 2 ă 2 log 1 x for x P p0, 1q, each factor in the denominator of h is less than 1. Thus the "only if" part along with the first inequality is trivial, and for the converse, it suffices to consider the case where Z " Zpf q.
To see the second inequality, we start by constructing the denominator of h. For f P HpDq with f p0q ‰ 0 and the zero sequence Zpf q, Jensen's formula gives log |f p0q|`ÿ aPZpf q log r |a| χ r|a|,1q prq " 1 2π If f P A p ω with ω P B 8 , then log |f | is area integrable. Indeed, since ω P B 8 , there exists q " qpωq ą 1 such that ω P B q , and hence ż D log |f pzq| dApzq " because log x ď 1 δ x δ for all x, δ ą 0. But the last integral is convergent because ω P B q , and hence log |f | P L 1 . Therefore, as in [23, p. 348], an integration of (2.2) with respect to 2r dr now gives log |f p0q|`ÿ aPZpf qˆl and applying this to w Þ Ñ f pϕ z pwqq yields for z R Zpf q. Exponentiating this and applying Jensen's inequality then gives for any δ ą 0. We next consider the linear integral operator appearing on the right-hand side of (2.3), and will show its boundedness on L q ω . To do this, write f´2pzq " f pzq`1´|z| 2˘´2 and ω rxs pzq " ωpzq`1´|z| 2˘x for short. Then }f } L q ω " }f´2} L q ω r2qs and }Rpf q} L q ω " is the maximal Bergman projection. Hence, R : L q ω Ñ L q ω is bounded if and only if P2 : L q ω r2qs Ñ L q ω r2qs is bounded, and the corresponding operator norms are equal. By the characterization of Bekollé and Bonami [3,4], this is equivalent to ż Spaq ω r2q´2s pzqp1´|z|q 2 dApzq˜ż Spaq ω r2q´2s pzq´1 q´1 p1´|z|q 2 dApzq¸q´1 À p1´|a|q 4q , a P Dzt0u, Thus, R : L q ω Ñ L q ω is bounded if and only if ω P B q . Moreover, }R} L q ω ÑL q ω ď CB q pωq maxt1, 1 q´1 u , where C " Cpω, qq, by [36,Theorem 1.5].
Let now Z " Zpf q. Clearly, f P A p ω implies |f | δ P L p{δ ω . Choose δ " p{q, so that p{δ " q. By the boundedness of R and (2.3), we obtain where C " Cpq, ωq. Since q " qpωq, the norm estimate we are after follows. l Proof of Theorem 1. We start the proof by using the function h in Proposition 13 to find the analytic function F in cases (c) and (d) of the theorem. To do this, we first make some observations about the functions ψ, k and W defined in Section 1. Standard estimates using the power series of the exponential show thaťˇˇˇa uniformly on compact subsets of D, and thus ψ P HpDq if ř aPZ p1´|a|q 2 ă 8. This sum converges for Z Ă Zpf q whenever log |f | is integrable on D. This, in turn, is true for any f P A p ω with ω P B 8 by the proof of Proposition 13. A direct calculation shows that and therefore see [23] for details. The last exponential is now exactly the function W Z defined in the first section, and the second product is the denominator of h in Proposition 13. The function h can thus be written as where C " ś aPZ |a|e p1´|a| 2 q{2 . We will soon see that f {ψ Z is the function F we are after. We first show the equivalence between (a) and (c). The calculations above together with Proposition 13 show that if f P A p ω and Z " Zpf q, then ř aPZ p1´|a|q 2 ă 8 and F W Z " C´1h P L p ω , where F " f {ψ Z has no zeros. Conversely, if F is a nowhere zero analytic function with F W Z P L p ω , ř aPZ p1´|a|q 2 ă 8, then f " F ψ Z P A p ω because |ψ Z pzq| ď W Z pzq for all z, and clearly Zpf q " Z.
Since considering a subsequence Z 1 of Z instead of Z will only decrease the values of k Z and W Z , it is clear that (a) and (b) are equivalent; (a) ñ (c) ñ (b) ñ (a). Now, if F is any (nonzero) analytic function with F W Z 1 P L p ω , ř aPZ 1 p1´|a|q 2 ă 8, then f " F ψ Z 1 P A p ω and Z 1 Ă Zpf q. The equivalence of (a) and (b) now gives Z 1 P ZpA p ω q, and thus (d) implies (a). Since (c) is a special case of (d), the equivalence part of the theorem is proved.
Finally, because of the part of the proof considering the function h, it is clear that the last statement of the theorem is simply a restatement of Proposition 13. l Proof of Theorem 4. The first and the last inequality are obvious, so it suffices to prove the middle one. For 0 ă p ă q ă 8, ω P B 8 and f P A p ω , consider the function gpzq " |f pzq| p ź z k PZpf q 1´p q`p q |ϕ z k pzq| q |ϕ z k pzq| p , z P D, defined in [13,Lemma 2], and let h be as in Proposition 13 with Z " Zpf q. Let npr, f q denote the number of zeros of f in Dp0, rq, counted according to multiplicity, and N pr, f q " ż r 0 npsq´np0, f q s ds`np0, f q log r be the integrated counting function. Two integrations by parts and Jensen's formula show that is a positive measure of unit mass on D. Hence Replacing now f by f˝ϕ z and using [28, (3.9)] to pass from dσ to dA, we obtain logpgpzqq " Thus g ď h p , from which Proposition 13 gives }g} L 1 ω ď C}f } p A p ω for some constant C " Cpωq ą 0. This is the statement of [28,Lemma 3.3] with the difference that now we have better control over the constant appearing on the right-hand side of the inequality and ω is only required to belong to B 8 . By following the proof of [28, Theorem 3.1], which in turn follows Horowitz' original probabilistic argument, now gives the assertion of Theorem 4 for functions f with finitely many zeros. To complete the argument used in the said proof, it suffices to show that every norm-bounded family in A p ω with ω P B 8 is a normal family of analytic functions. To see this, let x ą 1 such that ω P B x . Then the subharmonicity and Hölder's inequality yield where the constant of comparison depends only on the fixed r P p0, 1q. Since ω P B x , the last integral is finite, and Montel's theorem shows that every norm-bounded family in A p ω is a normal family of analytic functions. With this guidance we consider Theorem 4 proved. l

Zeros and factorization when ω P p D
We will need the following technical auxiliary result [29, Lemma 1].
Proof of Theorem 5. As explained in the introduction we begin with modifying the proof of Proposition 13 so that it covers the case ω P p D. This boils down to showing (1.1) for sufficiently large q " qpωq ą maxtp, 1u. To prove (1.1), let q ě 2 and kpζq " p1´|ζ|q ε , where ε ă 1´1{q is fixed. Writing Ipf q for the left-hand side of (1.1) and using Hölder's inequality and Fubini's theorem, we have and therefore Ipf q À }f } p To establish (1.1), it now suffices to show that µ is a p-Carleson measure for A p ω , that is, µpSq À ωpSq for all Carleson squares S by [29,Theorem 1]. By Fubini's theorem and the inequality εq´2q`1 ă´q ă´1, the term corresponding to the second summand satisfies Proof of Corollary 6. For each p ą 1 and ω P D, the dual of A p ω can be identified with A p 1 ω via the A 2 ω -pairing xf, gy A 2 ω " lim rÑ1´ş D f r g r ω dA, where f r pzq " f przq, by the proofs of [30,Theorem 6 and Corollary 7]. Therefore the boundedness of h ω f : A p ω Ñ A q ω is equivalent to Now that there exist C " Cprq ą 0 and R " Rpr, Cq P p0, 1q such that tz : Spaq X ∆pz, rq ‰ Hu Ă Spbq for some b " bpaq P Dzt0u with arg b " arg a and 1´|b| " Cp1´|a|q and for all |a| ě R, we deduce µpSpaqq ď ωpSpbqq ď Cp1´|a|q by the hypothesis ω P p D and Lemma A. Consequently, µ is a p-Carleson measure for A p ω by [29,Theorem 1], and the lemma is proved. l We next proceed towards the proof of Theorem 9.
By repeating the method used in the first part of the proof, it is easy to see that (iii) in fact implies (ii). Therefore (iv) follows by choosing η ą log K`1 C sufficiently large so that ωpSpaqqp1´|a|q´η is essentially increasing.
Finally, assume (iv) and denote a ‹ " 1`|a| 2 e i arg a for a P Dzt0u. Then ωpSpaqq and it follows that ω P p DpDq.
Proof of Theorem 9. Let G be a dominating set for A p ω . It suffices to show (1.7) for points a P D close to the boundary. Let β " βpωq ą 0 and η " ηpωq ą 0 be those in Lemma 14. For each a P Dzt0u, define for all a P Dzt0u. For given a close to the boundary, consider the points a n defined in the proof of Lemma 14. Then the proof of (iii) implies (iv) with K " 1 in the said lemma gives ż for all |a| sufficiently close to the boundary and m sufficiently large. It follows that ż DzSpanq |f a pzq| p ωpzq dApzq À 1 2 mn , n P N.
Since G is a dominating set for A p ω by the hypothesis, there exists δ " δpGq ą 0 such that Fix n P N sufficiently large such that ż DzSpanq |f a pzq| p ωpzq dApzq ă δ 2 .
Since ω P p DpDq, we have ωpSpaqq " ωpSpa 0 qq ě C´nωpSpa n qq, and now that n is fixed, we deduce ωpSpa n qq À ωpG X Spa n qq for all a P D sufficiently close to the boundary. The claim (1.7) follows from this estimate. l

Sampling measures
A positive Borel measure µ on D is a q-Carleson measure for A p ω if A p ω is continuously embedded into L q µ . In order to prove Theorem 10 we need the following lemma which is a generalization of [21,Theorem 2.3]. It readily follows from the proof that if the hypothesis on ν is replaced by νp∆pa, rqq À ωp∆pa, rqq, then on the left in both occasions Spζq must be replaced by ∆pζ, rq. Luecking [22,Lemma 3.10] showed this for ν " ω P C 8 under the hypothesis µp∆pa, rqq À ωp∆pa, rqq. Recall that, as discussed after Theorem 5, this last requirement characterizes p-Carleson measures for A p ω if ω satisfies the the Bekollé-Bonami condition by [6, Theorem 3.1].
Lemma 15. Let 0 ă p ă 8, 0 ă r ă R 2 ď 1 4 and ω a weight. Let µ and ν be positive Borel measures on D such that dr µpzq " µp∆pz,Rqq p1´|z|q 2 dApzq is a p-Carleson measure for A p ω and νpSpaqq À ωpSpaqq for all a P Dzt0u. Then there exists a constant C " Cpp, R, ω, µ, νq ą 0 such that ż In particular, if ω P p D and µ is a p-Carleson measure for A p ω , then the statement is valid. Proof. First, it is easy to show that there exists a constant C " Cpp, Rq ą 0 such thaťˇˇˇf where b " bpa, Rq P D is such that arg b " arg a and 1´|b| -1´|a| for all a P D. Now that µ is a p-Carleson measure for A p ω by the hypothesis, [ ď rC 1{p }f } A p ω . Since by the hypothesis G is dominating set, there exists a constant α ą 0 such that ş G |f pzq| p ωpzq dApzq ě α}f } p A p ω for all f P A p ω . Consequently, choosing r such that r p C ă C 1 εα}M ω pµq} L 8 yields and thus the proof is complete when 1 ď p ă 8.
If p ă 1, then one can simply apply the inequality |x´y| p ě |x| p´| y| p to the left hand side of (5.1) to obtain ż Proof of Theorem 11. The first statement can be established by following the proof of [24,Theorem 1]. Namely, since µ n pSpaq X Dp0, rqq ď }M ω pµ n q} L 8 ωpDq for each a P D, the hypothesis sup n }M ω pµ n q} L 8 ă 8 implies that for each r P p0, 1q, the sequence`µ n | Dp0,rq˘n PN is bounded in C 0 pDp0, rqq˚. Hence there is a subsequence that converges in the weak˚-topology by the Banach-Alaoglu theorem. By diagonalization we may extract a subsequence pµ n j q such that, for each r P p0, 1q, pµ n j | Dp0,rq q converges to a measure µ r supported in Dp0, rq. Since clearly µ s " µ r on Dp0, rq for all r ă s, we may define a measure µ as the limit lim rÑ1´µr . If now h P C c pDq, then it's support is in some Dp0, rq, and thus ż D hpzq dµ n j pzq Ñ ż D hpzq dµpzq, j Ñ 8. For the converse inequality, let ε ą 0 and take r " rpf q P p0, 1q such that ż DzDp0,rq |f pzq| p ωpzq dApzq ă ε.
Let h P C c pDq such that hpzq ď 1 for all z P D and h " 1 on Dp0, rq. Then where the last inequality follows from the fact that the classical non-tangential maximal function is a bounded operator from H p to L p of the boundary [10, Theorem 3.1 on p. 57]. Therefore there exists a constant C " Cppq ą 0 such that ż