Lipschitz Type Characterizations for Bergman-Orlicz Spaces and Their Applications
- PDF / 230,312 Bytes
- 14 Pages / 612 x 792 pts (letter) Page_size
- 99 Downloads / 181 Views
Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
LIPSCHITZ TYPE CHARACTERIZATIONS FOR BERGMAN-ORLICZ SPACES AND THEIR APPLICATIONS∗
êT)
Rumeng MA (
Mµ¢)
Jingshi XU (
†
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China E-mail : [email protected]; [email protected] Abstract We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk. Key words
Bergman-Orlicz space; Lipschitz condition; hyperbolic metric
2010 MR Subject Classification
1
32A36
Introduction
The Bergman spaces were first introduced by Bergman in [1]. Since then, the theory of Bergman spaces has attracted the attention of many authors, due to its connection with harmonic analysis, approximation theory, hyperbolic geometry, potential theory, and partial differential equations; see [2, 4, 7, 15] and references therein. In particular, Bergman spaces can be characterized by derivatives and Lipschitz type conditions (see [13, 15]). We remark here that similar characterizations for Hardy-Sobolev spaces have appeared in the literature before; see [3, 5, 6, 8, 9, 14] for details. Nam in [10] gave Lipschitz type characterizations of harmonic Bergman spaces on the upper half-space in Rn . Peng, Xing and Jiang characterized the boundedness and compactness of multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball of Cn in [11]. Recently, Sehba, in [12], obtained derivatives characterizations of Bergman-Orlicz spaces which generalized those of Bergman spaces. Motivated by these results, in this article we shall consider the characterizations of Bergman-Orlicz spaces by Lipschitz type conditions. Before stating our results, we firstly recall some important notions. We let Bn = {z ∈ Cn : |z| < 1} be the open unit ball in Cn . We also call it the disk of Cn . For any α > −1, let dνα (z) = Cα (1 − |z|2 )α dν(z), where dν is the normalized volume measure on Bn and Cα is a positive constant such that να (Bn ) = 1. ∗ Received
August 23, 2018; revised May 29, 2020. The work was supported by Hainan Province Natural Science Foundation of China (2018CXTD338), the National Natural Science Foundation of China (11761026 and 11761027), and Guangxi Natural Science Foundation (2020GXNSFAA159085). † Corresponding author: Jingshi XU.
1446
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
Given a convex function Φ : [0, +∞) → [0, +∞), we say that Φ is a growth function if it is a continuous and non-decreasing function. For a growth function Φ, the Orlicz space LΦ (Bn , dνα ) is the space of functions f such that Z Φ(|f (z)|/λ)dνα (z) < ∞, for some λ > 0. Bn
Let H(Bn ) denote the space of holomorphic functions on Bn . The weighted Bergman-Orlicz Φ space AΦ α (Bn ) is the
Data Loading...
