Bergman projection induced by radial weight
Abstract.
We establish characterizations of the radial weights on the unit disc such that the Bergman projection , induced by , is bounded and/or acts surjectively from to the Bloch space , or the dual of the weighted Bergman space is isomorphic to the Bloch space under the pairing. We also solve the problem posed by Dostanić in 2004 of describing the radial weights such that is bounded on the Lebesgue space , under a weak regularity hypothesis on the weight involved. With regard to LittlewoodPaley estimates, we characterize the radial weights such that the norm of any function in is comparable to the norm in of its derivative times the distance from the boundary. This lastmentioned result solves another wellknown problem on the area. All characterizations can be given in terms of doubling conditions on moments and/or tail integrals of , and are therefore easy to interpret.
Key words and phrases:
Bergman space, Bergman projection, Bloch space, bounded mean oscillation, doubling weight, LittlewoodPaley formulaContents
1. Introduction and main results
Projections are essential building blocks of the concrete operator theory on spaces of analytic functions. One cornerstone is to characterize their boundedness which together with its multiple consequences makes it a pivotal topic in the theory [3, 6, 7, 18]. Indeed, bounded analytic projections can be used to establish duality relations and to obtain useful equivalent norms in spaces of analytic functions [2, 15, 21, 23, 25, 35].
The question of when the Bergman projection induced by a radial weight on the unit disc is a bounded operator from one space into another is of primordial importance in the theory of Bergman spaces. The longstanding problem of describing the radial weights such that is bounded on the Lebesgue space had been known to experts since decades before it was formally posed by Dostanić in 2004 [10]. A natural limit case of this setting is when acts from to the Bloch space. The surjectivity of the operator becomes another relevant question in this limit case.
The main findings of this study are shortly listed as follows. We establish characterizations of the radial weights on the unit disc such that is bounded and/or acts surjectively, or the dual of the weighted Bergman space is isomorphic to the Bloch space under the pairing. We also solve the problem posed by Dostanić under a weak regularity hypothesis on the weight involved. With regard to LittlewoodPaley estimates, we describe the radial weights such that the norm of any function in the weighted Bergman space is comparable to the norm in of its derivative times the distance from the boundary. This lastmentioned result solves another wellknown problem on the area. All characterizations can be given in terms of doubling conditions on moments and/or tail integrals of , and are therefore easy to interpret.
Let denote the space of analytic functions in the unit disc . For a nonnegative function , the extension to , defined by for all , is called a radial weight. For and such an , the Lebesgue space consists of measurable functions such that
where is the normalized area measure on . The corresponding weighted Bergman space is . Throughout this paper for all , for otherwise .
For a radial weight , the orthogonal Bergman projection from to is
where are the reproducing kernels of . As usual, stands for the classical weighted Bergman space induced by the standard radial weight , are the kernels of , and denotes the corresponding Bergman projection.
It is wellknown that the standard weighted Bergman projection is bounded and onto from the space of bounded measurable functions to the Bloch space , which consists of such that . This raises the question of which properties of a radial weight are determinative for to be bounded? And what makes the embedding continuous? These questions maybe paraphrased in terms of the dual space of because the dual can be identified with under the pairing
It is known that and under the pairing. Here stands for the little Bloch space which consists of such that , as .
The first main result of this paper characterizes the class of radial weights such that is bounded, and establishes the corresponding natural duality relations, under our initial hypothesis on the positivity of . To state the result, some more notation is needed. A radial weight belongs to the class if the tail integral satisfies the doubling property for some constant and for all . We give a glimpse of how weights in the class behave. If tends to zero too fast, for example exponentially, when approaching the boundary, then obviously cannot belong to . However, the growth of to infinity is not restricted because each increasing weight belongs to , and hence, in particular, may tend to zero much slower than any positive power of the distance to the boundary. The containment in also restricts the oscillation of , yet permits it to vanish identically in a relatively large part of each outer annulus of . Concrete examples illustrating some characteristic properties and/or strange behavior of weights in will be represented in the forthcoming sections, see also [24, Chapter 1].
Theorem 1.
Let be a radial weight. Then the following statements are equivalent:

is bounded;

There exists such that for each there is a unique such that for all with , that is, via the pairing;

There exists such that for each there is a unique such that for all with , that is, via the pairing;

.
One of the main obstacles in the proof of Theorem 1 and throughout this work is the lack of explicit expressions for the Bergman reproducing kernel . For a radial weight , the kernel has the representation for each orthonormal basis of , and therefore we are basically forced to work with the formula induced by the normalized monomials. Here are the odd moments of , and in general from now on we write for all . Therefore the influence of the weight to the kernel is transmitted by its moments through this infinite sum, and nothing more than that can be said in general. This is in stark contrast with the neat expression of the standard Bergman kernel which is easy to work with as it is zerofree and its modulus is essentially constant in hyperbolically bounded regions. In general, the Bergman reproducing kernel induced by a radial weight may have a wild behavior in the sense that for a given radial weight there exists another radial weight such that , but have zeros, see [8, Proof of Theorem 2] and also [30]. The proof of Theorem 1 draws strongly on [25, Theorem 1], which says, in particular, that
(1.1) 
for . It is of course the special case that plays a role in the proof of Theorem 1. For we have , and therefore, as the moments are involved in the kernel, the appearance of the tail integrals on the right hand side of (1.1) is natural. This also implies if and only if there exists a constant such that for all . Thus the containment in can be equally well stated in terms of the tail integrals as the moments.
As for characterizing the continuous embedding of into , we write if there exist constants and such that for all .
Theorem 2.
Let be radial weight. Then the following statements are equivalent:

There exists such that for each there is such that and , that is, is continuously embedded into ;

There exists such that for all and , that is, each induces an element of via the pairing;

There exists such that for all and , that is, each induces an element of via the pairing;

.
In view of Theorems 1 and 2, it is tempting to think that a kind of reverse doubling condition for tail integrals would characterize the continuous embedding , and thus the class as well. But unfortunately this is not the case. Namely, by writing if there exist constants and such that for all , we have by Proposition 14 below. Therefore it is not enough to consider the tail integrals but we must indeed work with a true moment condition, and that is of course more difficult. The reason why contains in a sense much worse weights than is that the global moment integral may easily eat up irregularities of the weight that make the tail condition of to fail. That is how a family of counter examples is constructed to get the proposition.
To prove Theorem 2 we first show that (iii) implies (iv) by testing with two appropriate families of analytic (lacunary) polynomials. Then, assuming , for each we construct an explicit preimage such that , and thus obtain (i). In order to do this, we use properties of smooth universal Cesáro basis of polynomials [21, Section 5.2], a description of the coefficient multipliers of the Bloch space and several characterizations of the class . In particular, we will show that if and only if for some (equivalently for each) , there exists a constant such that
(1.2) 
a description of which will be useful for our purposes also in other instances. The proof is then completed by passing from (i) to (ii), which is easy, and then observing that (ii) trivially implies (iii).
With Theorems 1 and 2 in hand, we characterize the radial weights such that is bounded and onto, or equivalently can be identified with the Bloch space via the pairing. To this end we write .
Theorem 3.
Let be a radial weight. Then the following statements are equivalent:

is bounded and onto;

via the pairing with equivalence of norms;

via the pairing with equivalence of norms;

;

.
The proof of Theorem 3 boils down to showing that . While is proved with the aid of characterizations of classes of weights obtained en route to Theorem 2, the proof of the opposite inclusion is more involved and will be actually established by showing the inequality in Theorem 2(ii) under the hypothesis .
It is worth underlining that , despite the fact that is a proper subclass of . Roughly speaking the phenomenon behind this identity is that the severe oscillation that is possible for weights in is ruled out by intersecting with , and that makes the sets of weights equal.
We will show in Theorem 15 below that for each there exists a weight which induces a weighted Bergman space only slightly larger than but the weight itself lies outside of . Therefore is neither bounded nor onto by Theorems 1 and 2. In particular, for given one may pick up a weight such that , but is neither bounded nor onto meanwhile of course and both have these properties. This shows in very concrete terms that being bounded and/or onto depends equally well on the growth/decay of the inducing weight as on its regularity.
Theorem 3 raises the problem of finding a useful description of the dual of when . To deal with this question we will need some more notation. For a radial weight and with Maclaurin series and , denote
whenever the limit exists. For and , set
and . For , the Hardy space consists of such that , and is the space of functions in that have bounded mean oscillation on the boundary. We define
where . Our next result describes the dual of when . We will discuss the case when on is considered. We also note that it is not hard to identify the dual of with the Bloch space under an appropriate paring when and , see [32, Theorem 1] for details.
Theorem 4.
Let . Then can be identified with via the pairing with equivalence of norms.
The proof of Theorem 4 is quite involved, and requires a large number of technical lemmas and known results from the existing literature. The first observation is that the dual of can be identified with the space of coefficient multipliers from to . Second, by using [19, Theorem 2.1], it is shown that there exists and a sequence of polynomials such that and , where denotes the convolution. These are the new norms that we will work with. Then, it is shown that the operator satisfies . In this step we strongly use the hypothesis and the fact that the Cesáro means are uniformly bounded on [19, Theorem 2.2]. Now that Fefferman’s duality relation allows us to identify the coefficient multipliers from to with , the operator finally gives the isomorphism we are after. One of the crucial steps in the argument is the norm estimate of on stated above, and it deserves special attention for a reason. Namely, a part of the proof of this relies on the technical fact that if and only if for some (equivalently for each) , there exists a constant such that
(1.3) 
This analogue of the characterization (1.2) of the class for is obtained as a consequence of the LittlewoodPaley estimates to be discussed next.
It is well known that bounded Bergman projections on weighted spaces can be used to establish equivalent norms on weighted Bergman spaces in terms of derivatives [2, 35]. Before we proceed to consider the weighted inequalities for the Bergman projection , which is one of the main topics of the paper, we discuss such norm estimates, usually called the LittlewoodPaley formulas. The arguments we employ do not appeal to bounded Bergman projections, but when we will return to , we will also explain how projections can be used to obtain certain norm estimates in our setting.
Probably the most known LittlewoodPaley formula states that for each standard radial weight we have
where . Different extensions of this equivalence as well as related partial results can be found in [2, 4, 5, 22]. The question for which radial weights the above equivalence is valid has been a known open problem for decades, and gets now completely answered by our next result.
Theorem 5.
Let be a radial weight, and . Then
(1.4) 
if and only if .
We can actually do a bit better than what is stated in Theorem 5. Namely, on the way to the proof we will establish the following result.
Theorem 6.
Let be a radial weight, and . Then there exists a constant such that
(1.5) 
if and only if .
The proof of (1.5) for is based on standard techniques, but the other implication is much more involved. In particular, the proof reveals that (1.5) holds if and only if the inequality there holds for all monomials only, and that (1.3) is a characterization of . Thus, in our approach, this latter consequence of the proof of Theorem 6 is used to achieve Theorem 4.
As for the proof of Theorem 5, we will show that
is satisfied whenever . Since implies that , , this is obtained by reducing the question to the case , using an appropriate exponential partition of depending on the weight and the inequality [17, Lemma 2] which states that
(1.6) 
The last step in the proof of Theorem 5 consists of testing with monomials in (1.4), and then using (1.2), (1.3) and Theorem 3 to deduce . It is also worth mentioning that unlike , the class is not closed under multiplication by for any [29, Theorem 3].
A description of the radial weights such that
remains an open problem for with . This matter will be briefly discussed at the end of the paper. For LittlewoodPaley estimates in the case of nonradial weight under additional regularity hypotheses, see [2, 4, 5].
The question of describing the radial weights such that
(1.7) 
was formally posed by Dostanić [10, p. 116]. In the recent years there has been an increasing activity concerning this norm inequality [9, 10, 11, 25, 34]. Our next result provides a sufficient condition, much weaker than known conditions so far and of correct order of magnitude, for (1.7) to hold.
Theorem 7.
Let and . Then is bounded.
The proof of Theorem 7 draws on the fact that for any radial weight and , the Bergman projection is bounded on if and only if with equivalence norms under the pairing. We will prove this duality relation by following a similar scheme to that in the proof of Theorem 4. However, in this case the preparations are easier because of our previous result concerning equivalent norms on in terms of type norm of the Hardy norms of blocks of the Maclaurin series, whose size depend on the weight , see [26, Theorem 3.4] and also [23, Theorem 4].
In the opposite direction to Theorem 7, Dostanić [10] showed that for a radial weight and , the reverse Hölder’s inequality
(1.8) 
is a necessary condition for to be bounded. We will offer two simple proofs of this fact as well as two equivalent statements in Proposition 17 below. The natural question now is that does the Dostanić’ condition (1.8) imply ? Unfortunately, we do not know an answer to this, but we can certainly say that these conditions are the same if sufficient regularity on is required. To state the result, write for all . The proof of the following result and more can be found in Section 8.
Corollary 8.
Let be a radial weight and such that either
(1.9) 
for some , or for each there exists such that
(1.10) 
for some constant . Then is bounded if and only if .
The innocent looking condition (1.9) is not easy to work with in the case of irregularly behaving weights because it requires precise control over the moments. We have failed to construct a weight which does not satisfy this condition. With regard to (1.10), we mention here that the auxiliary function is an increasing unbounded concave function, and it is easy to see that if is essentially increasing, then (1.10) is trivially satisfied.
Concerning the maximal Bergman projection
we will show the following result.
Theorem 9.
Let and . Then is bounded if and only if .
Theorem 9 is of its own interest, but we underline that it combined with Theorem 7 shows that the cancellation of the kernel plays an essential role in the boundedness of when . In particular, we have the following immediate consequence of the previous two theorems.
Corollary 10.
Let and . Then is bounded but is not.
We also prove that the boundedness of is equally much related to the regularity of the weight as to its growth, see the note after Proposition 18 below.
The inequality (1.7) can of course also be interpreted as . In view of the LittlewoodPaley formulas, it is natural to ask for which radial weights , the Bergman projection is bounded from to the Dirichlettype space . Here and from now on, for and , denotes the space of such that
The following result gives an answer this question. Its proof uses Theorem 6 and the norm estimate (1.1).
Theorem 11.
Let be a radial weight, and . Then the following statements are equivalent:

is bounded;

There exists such that for each there is a unique such that and . That is, is continuously embedded into ;

is bounded;

.
In the proof we show that is bounded whenever . Since this trivially implies (iv), we deduce that (1.5) is satisfied when . This gives an example of how bounded projections can be used to get LittlewoodPaley estimates in our context.
Concerning Theorem 11, we can actually take a step further and characterize the radial weights such that the dual of can be identified with under the pairing.
Theorem 12.
Let be a radial weight, and . Then the following statements are equivalent:

is bounded and onto;

via the pairing with equivalence of norms;

with equivalence of norms;

.
The proof of Theorem 12 is strongly based on Theorems 5, 6 and 11. In addition, to show that (ii) implies (iv), two appropriate families of lacunary series are constructed to deduce .
Before we move on to norm inequalities involving different weights, we point out two things. First, all the results presented so far show that each of the classes of weights , and appear in a very natural manner in the operator theory of Bergman spaces induced by radial weights. Second, the conditions describing the radial weights such that either (1.4) or (1.5) is satisfied, or is bounded and/or onto do not depend on .
Our next goal is to study the question of when for given radial weights and we have
(1.11) 
The most commonly known result concerning the one weight inequality (1.11) for the Bergman projection is undoubtedly due to Bekollé and Bonami [6, 7], and concerns the case when is an arbitrary weight and the inducing weight is standard. See [1, 28, 31] for recent extensions of this result. Moreover, the main result in [25] characterizes (1.11) under the hypothesis that both and are regular weights i.e. they satisfy for all . Regular weights form a subclass of whose elements do not have zeros and are pointwise comparable to radial weights for . Therefore these weights are really smooth and not that hard to work with.
Our next result substantially improves [25, Theorem 3]. In particular, it characterizes the pairs such that (1.11) holds under the assumption and . To state the result, some more notation is needed. For and radial weights and , let , and define for all
(1.12) 
and
(1.13) 
Theorem 13.
Let , and a radial weight. Then the following statements are equivalent:

is bounded;

is bounded and ;

and ;

and .
Moreover,
(1.14) 
The proof of Theorem 13 is divided into several steps, and many auxiliary results interesting in themselves are used. We first establish by a simple testing a necessary condition for to be bounded, provided all the three weights involved are radial. This is a natural generalization of the one weight case given by the Dostanić condition (1.8), and is stated as Proposition 19 below. In the case it shows that and cannot be so different from each other neither in growth/decay nor in regularity if is bounded. The second auxiliary result is Proposition 20 below and it states that if is bounded. This says that and cannot induce very small Bergman spaces when is bounded. This is one of the significant differences between and . Proposition 20 is achieved with the aid of Hardy’s inequality and appropriate testing. The most involved and technical part of the proof of Theorem 13 consists of showing that (iv) implies (i) via an instance of Shur’s test. This step relies on a suitable choice of auxiliary function and the norm estimate (1.1). Actually more is true than what is stated in the theorem because we will show in Proposition 21 that the condition describes the pairs such that is bounded provided one of the weights involved belongs to . Finally, we deduce (iii) from (ii) via standard methods, and we observe that (iii) yields (iv) for any radial weight.
The two weight inequality with arbitrary weights for the Bergman projection remains an open problem even in the case . However, it was recently characterized in [14] under the hypotheses that all the three weights involved are radial and satisfy certain regularity hypotheses. We note that the method of proof used there does not help us to prove the most difficult part of Theorem 13 which consists of showing that (iv) implies (i).
The rest of the paper is organized as follows. Theorems 1, 2 and 3 are proved in Section 2, but the proof of Theorem 4 is postponed to Section 7. In Section 3 we prove Theorems 5 and 6, while Section 4 contains the proofs of Theorems 11 and 12. Section 5 is devoted to the study of the inequality (1.7). It contains the proofs of Theorems 7 and 9 and Corollary 10. Theorem 13 is proved in Section 6. Finally, in Section 8 we discuss two open problems cited in the introduction and pose conjectures for them.
For clarity, a word about the notation already used in this section and to be used throughout the paper. The letter 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 if there exists a constant such that , and is understood in an analogous manner. In particular, if and , then we write and say that and are comparable.
2. plus dualities and
Proof ofTheorem 1. First observe that if is a weight such that is continuously embedded into , then
(2.1) 
To prove that (i) and (iv) are equivalent, assume first . By using and [25, Theorem 1] we deduce
and hence is bounded by (2.1). Conversely, if is bounded, then (2.1) and the Hardy’s inequality [12, Chapter 3] yield
For , let be defined by . Since
there exists a constant such that