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Journal of Function Spaces
Volume 2018, Article ID 8751849, 8 pages
https://doi.org/10.1155/2018/8751849
Research Article

Derivative and Lipschitz Type Characterizations of Variable Exponent Bergman Spaces

1School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
2School of Science, Guangxi University for Nationalities, Nanning 530006, China

Correspondence should be addressed to Jingshi Xu; moc.621@uxihsgnij

Received 5 April 2018; Revised 20 June 2018; Accepted 16 July 2018; Published 1 August 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 Rumeng Ma and Jingshi Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give derivative and Lipschitz type characterizations of Bergman spaces with log-Hölder continuous variable exponent.

1. Introduction

The Bergman spaces were introduced in [1]. Since then, the theory of Bergman spaces has grown quickly, due to its connection with harmonic analysis, approximation theory, hyperbolic geometry, potential theory, and partial differential equations; see [25]. In particular, they can be characterized by derivatives and Lipschitz type conditions. Indeed, Zhu in [5] gave the derivatives characterizations of Bergman spaces. Wulan and Zhu in [6] gave Lipschitz type characterizations for Bergman spaces. We remark here that Lipschitz type characterizations for Sobolev spaces were considered in [711].

Recently, in [12], Chacón and Rafeiro introduced variable exponent Bergman spaces on the open unit ball of the plane and obtained that the Bergman projection and the Berezin transform are bounded and polynomials are dense in these spaces. Then in [13], Chacón, Rafeiro, and Vallejo gave a characterization of Carleson measures for variable exponent Bergman spaces. These results are generalizations of constant exponent Bergman spaces. The theory of variable function spaces has attracted many authors’ attention for four decades. Since there are huge literatures, we only recommend [1416]. Motivated by those papers, in this paper, we shall extend the derivatives characterizations in [5] and Lipschitz type characterizations in [6] to variable exponent Bergman spaces on the open unit ball of for any integer To state our results, we firstly recall some definitions.

We denote the Euclidean norm on by Then we let , the open unit ball in Let be the normalized volume measure on For any , let , where is a positive constant such that In this paper, we only consider the case of

For a measurable function , we call it a variable exponent and denote Denote by the set of all variable exponents with Let For a complex-valued measurable function on , we define the modular of by and the Luxemburg-Nakano norm by The variable Lebesgue space is the set of all complex-valued measurable functions on such that It is a Banach space equipped with the Luxemburg-Nakano norm.

If , then the variable exponent Bergman space is the class of all holomorphic functions on which belong to the variable exponent Lebesgue space It is easy to show that is a closed subspace of When is a constant, these spaces are called weighted Bergman space with standard weights; see [3, 4] for details. As usual, we denote by the space of holomorphic functions on

Given , the radial derivative of at is defined by The complex gradient of at is defined by And the invariant complex gradient of at is given by where is the automorphism of mapping to

For any , let be a biholomorphic map on such that and The explicit formulas are available for (see [5]).

Let be the Bergman metric on , namely, for , It was known that is also a distance function on . is called the pseudohyperbolic metric on For any and , we let , the pseudohyperbolic ball centered at with radius . is Euclidean ball with In particular, if is fixed, then the volume of is comparable to

For any and , we let , the hyperbolic ball centered at with radius If , then withConsequently, if is fixed, then the volume of is also comparable to

Definition 1. A function is said to be log-Hölder continuous or satisfy the Dini-Lipschitz condition on if there exists a positive constant such thatfor all such that We will denote by the set of all log-Hölder continuous functions on .

Now, the main result of the paper is the following.

Theorem 2. Suppose , and is holomorphic in Then the following conditions are equivalent.
(a)
(b)
(c)
(d)
(e) There exists a continuous function in such that, for all ,(f) There exists a continuous function in such that, for all ,(g) There exists a continuous function such that in and for all ,

In Section 2, we shall collect some results which we shall need in the paper. The proof of Theorem 2 will be given in Section 3. Finally, we claim that the notation means there exists a constant such that , and means and

2. Preliminaries

In this section, we recall some preliminary results that we shall need in our paper.

Lemma 3 ([5], Lemma 2.20). Let be a positive number. Then there exists a positive constant such that for all with Moreover, if is bounded above, then we may choose to be independent of

The following Jensen type inequality was proved in [17] in the context of spaces of homogeneous type (SHT).

Lemma 4. Suppose that Thenfor all , , provided that

Remark 5. Usually, Lemma 4 holds for Euclidean balls. We shall use it in pseudohyperbolic metric. Since each pseudohyperbolic ball is actually Euclidean ball, we have the above form. And .

Definition 6. Given a function , the Hardy-Littlewood maximal function of , denoted by , is defined for any by

Lemma 7 ([14], Theorem 3.16). Let Then the Hardy-Littlewood maximal function is bounded in , and it means that there exists a positive constant such that for each

Let denote a family of pairs of nonnegative measurable function and denote the Muckenhoupt weight.

Lemma 8 ([14], Theorem 5.24). Suppose that for some the family is such that, for all ,Given , if and the maximal operator is bounded on , then there is a positive constant independent of such that

For , we define the following radial test function: where is the normalizing constant in the sense For , we define Notice that is a function supported on the set (where stands for the closed unit ball with radius ) and

Definition 9. Given a function , we will define its mollified dilation as where stands for the complex ball with radius

The following lemma is the counterpart of Theorem 3.5 in [12] for any integer

Lemma 10. Let and Then there exists a positive constant for , such that Moreover, as

Proof. Let , , Then by making change of variable and using Fubini’s theorem, we obtain Since , we have Therefore, considering the maximal function , and using Hölder’s inequality and Lemma 7 we have that which shows the first part of the lemma.
For the second part, fix and choose a function with compact support on , such that (see [14] Theorem 2.72). Then by the previous part of the proof, Therefore, we only need to prove the convergence in norm for the compactly supported function Defining , we obtain that And Thus, and consequently Therefore, we have reduced the convergence to the case of a constant exponent.
In this case, from [3] we know that radial dilations converge in -norm to Consequently, if we define the translation operator , then by Minkowski’s inequality and the result follows from the continuity of on the space .

From the above lemma, we have the following lemma.

Lemma 11. Let Then the set of holomorphic polynomials is dense in

Lemma 12 ([5], Proposition 1.13). If is real and is in , then where is any automorphism of and

Lemma 13 ([5], Lemma 2.14). If is holomorphic in , then for all

Lemma 14 ([5], Theorem 1.12). Let and For , define When , then and are bounded in
When , then Finally, The notation means that the ratio has a positive finite limit as

The Bergman projection operator is defined for functions on by

To proceed, we need the class for Let be a positive measurable function, and is called belonging to the class if there exists a constant such that, for every pseudo-ball , where

Lemma 15 ([18], Theorem 1). Let A necessary and sufficient condition for the Bergman projection to be bounded on is that belongs to the class for

Lemma 16. Let Then the Bergman projection operator is bounded from onto

Proof. We follow the proof of Theorem 4.4 in [12]. It is clear that is holomorphic function on , so we only need to prove that is bounded and surjective from to By the property of the Muckenhoupt (see [19]), if then for Now we pick such that By the remark 1 in [18], we have that By Lemma 15 we conclude that the family satisfies (17). By the fact that and Lemmas 7 and 8, then there exists a constant such that This shows that is bounded in
In order to prove that is surjective, we use the fact that for every In particular, this equality holds for any polynomial. Thus, if , we use Lemma 11 to find a sequence of polynomials converging in to But since in then we have that

Lemma 17 ([5], Lemma 2.24). Suppose , , and Then there exists a constant such that for all and all

The last result deals with the limits of and as tends in the radial direction.

Lemma 18 ([6], Lemma 5.2). Suppose and , where is a scalar. Then

3. Proof of Theorem 2

Proof of Theorem 2. We shall divide the proof into 8 steps. In Step 1, by Lemma 13 we obtain that (b) implies (c) and (c) implies (d).
In Step 2, we prove (a) implies (b). We firstly consider that such that We follow the idea of the proof of Theorem 2.16 in [5] and make some crucial modifications where needed. Fix It follows from Lemma 2.4 in [5] that for fixed there exists a constant such that for all holomorphic in
Now, for each , let be the biholomorphic mapping of which interchanges to , and let Then by making an obvious change of variables according to Lemma 12, we obtain Since the volume of is comparable to , then we have Then by Lemma 4 we have Integrating both sides of the above inequality over with respect to and using Fubini’s theorem, we have that By Lemma 14 we have that Thus using Lemma 3 and the above result we obtain that Therefore
Finally, for a general , define , and we apply the previous result for and conclude that
In Step 3, to prove (d) implies (a), we assume that is a holomorphic function in such that the function Let be a sufficiently large positive constant. Then by the proof of Theorem 2.16 in [5] we have that where is the Bergman projection operator. By Lemma 16 we know that Therefore
In Step 4, we prove (a) implies (e). If and , we have Fix It follows that, for , we have Since in the relatively compact set the Euclidean metric is comparable to the pseudohyperbolic metric, then is comparable to in the relatively compact set Thus, there is a constant , which depends only on , such that for all Replacing by , and by , respectively, and using the Möbius invariance of the pseudohyperbolic metric and the invariant gradient, we obtain for all and in with
Let where Then it is easy to see that the function is continuous on and (10) holds. Now we only need to prove that Since is already in , the remainder is to show that
Recall that if and satisfy (8), then So we can choose such that for all For any , any , we have Then we have due to By the triangle inequality, we have So That is, Equivalently, we have whenever
By Lemmas 17 and 3 we obtain that where we used that the volume of is comparable to Without loss of generality, we assume Therefore, using Lemma 4 we obtain that Integrating both sides of the above inequality over with respect to and using Fubini’s theorem, we have that Due to , by Step 2 we have that Thus
In Step 5, it is easy to see that (e) implies (f), according to , for all
In Step 6, we prove that (f) implies (a). Assume that there exists a continuous function such that (11) holds for all We fix and let , where is a scalar. Then for all in Let approach and apply Lemma 18; we obtain that for all Since , then , and by Step 3 we have that
In Step 7, we prove that (g) implies (a). If there exists a continuous function such that in and for all , then Let and fix For each , let Let tend to , and we obtain Therefore for all , we obtain Thus Since , thus Therefore, we have by the equivalency of (c) and (a) which we have proved.
In Step 8, we prove that (e) implies (g). If Condition (e) holds in this theorem, Lemma 1.2 in [5] says that Thus By the Cauchy inequality, , we obtain that, for all , By condition (e), there exists a continuous function in such that, for all , Therefore, Since we have Write , then Since , we have so Since , we have

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Jingshi Xu is supported by Hainan Province Natural Science Foundation of China (2018CXTD338). Rumeng Ma and Jingshi Xu are supported by the National Natural Science Foundation of China (Grant nos. 11761026 and 11761027).

References

  1. S. Bergmann, “Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I,” Journal Für Die Reine Und Angewandte Mathematik, vol. 169, pp. 1–42, 1933 (German). View at Publisher · View at Google Scholar · View at MathSciNet
  2. S. Bergman, The Kernel Function and Conformal Mapping, American Mathematical Society, New York, NY, USA, 1950. View at MathSciNet
  3. P. Duren and A. Schuster, “Bergman spaces,” in Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, USA, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  4. H. Hedenmalm, B. Korenblum, and K. Zhu, “Theory of bergman spaces,” in Graduate Texts in Mathematics, vol. 199, Springer, New York, NY, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  5. K. Zhu, “Spaces of holomorphic functions in the unit ball,” in Graduate Texts in Mathematics, vol. 226, Springer, New York, NY, USA, 2004. View at Google Scholar · View at MathSciNet
  6. H. Wulan and K. Zhu, “Lipschitz type characterizations for Bergman spaces,” Canadian Mathematical Bulletin, vol. 52, no. 4, pp. 613–626, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  7. B. Bojarski and P. Hajlasz, “Pointwise inequalities for Sobolev functions and some applications,” Studia Mathematica, vol. 106, pp. 77–92, 1993. View at Google Scholar · View at MathSciNet
  8. P. Hajłasz, “Sobolev spaces on an arbitrary metric space,” Potential Analysis, vol. 5, no. 4, pp. 403–415, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. P. Hajłasz and P. Koskela, “Sobolev met Poincaré,” Memoirs of the American Mathematical Society, vol. 145, no. 688, pp. 1–101, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  10. P. Koskela and E. Saksman, “Pointwise characterizations of Hardy-Sobolev functions,” Mathematical Research Letters, vol. 15, no. 4, pp. 727–744, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. D. Yang, “New characterizations of Hajłasz-Sobolev spaces on metric spaces,” Science China Mathematics, vol. 46, no. 5, pp. 675–689, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  12. G. R. Chacón and H. Rafeiro, “Variable exponent Bergman spaces,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 105, pp. 41–49, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  13. G. R. Chacón, H. Rafeiro, and J. C. Vallejo, “Carleson measures for variable exponent Bergman spaces,” Complex Analysis and Operator Theory, vol. 11, no. 7, pp. 1623–1638, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser, Basel, Switzerland, 2013. View at Publisher ·