Hankel Operators on Bergman Spaces of Annulus Induced by Regular Weights
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Hankel Operators on Bergman Spaces of Annulus Induced by Regular Weights Xiao Feng WANG1)
Li Hong YANG
Jin XIA
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China E-mail : [email protected] [email protected] [email protected] Abstract This paper is devoted to studying Bergman spaces Apω1,2 (M ) (1 < p < ∞) induced by regular-weight ω1,2 on annulus M . We characterize the function f in L1ω1,2 (M ) for which the induced Hankel operator Hf is bounded (or compact) from Apω1,2 (M ) to Lqω1,2 (M ) with 1 < p, q < ∞. Keywords
Hankel operators, regular-weight, annulus, Bergman spaces
MR(2010) Subject Classification
1
47B35
Introduction
Let D denote the open unit disk in the complex plane C, and let M = {z ∈ D : r0 < |z| < 1} 0 be the annulus in D where 0 < r0 < 1. For convenience, let M1 = {z ∈ D : 1+r < |z| < 1} and 2 1+r0 M2 = {z ∈ D : r0 < |z| ≤ 2 }, then M = M1 ∪ M2 . z−a | is the pseudo-hyperbolic metric on D, then for any a ∈ D and Suppose ρ(a, z) = | 1−az r ∈ (0, 1) the pseudo-hyperbolic disk (a, r) = {z ∈ D | ρ(a, z) < r} with center a and the radius r. Let {ai }∞ i=1 be some (or any) r-lattice of M1 under the pseudo-hyperbolic metric ρ(z, w), r0 ∞ and { bj }j=1 be some (or any) r-lattice of M2 under the pseudo-hyperbolic metric ρ( rz0 , rw0 ). ∞ ∞ For any z ∈ M1 , z ∈ i=1 (ai , r), and for any z ∈ M2 , rz0 ∈ j=1 ( rbj0 , r). We use the notation A B if there exists a positive constant C such that A ≤ CB for two quantites A and B. Moreover, write A B if A B and B A. ˆ for the family of radial weights Suppose ω ∈ L1 [1, 0) is a radial weight, we will denote ω ∈ D 1 such that ω ˆ (z) = |z| ω(s)ds is doubling, i.e., there exists some constants C = C(ω) ≥ 1 such ˆ satisfies that ω ˆ (r) ≤ C ω ˆ ( 1+r ) for any 0 ≤ r < 1. Futhermore, if ω ∈ D 2
1
ω(r)
r
ω(s)ds , 1−r
0 ≤ r < 1,
then we call ω is regular, denoted by ω ∈ R. If ω ∈ R, then there exists a positive constant C depending on r ∈ (0, 1), such that −1 C ω(z) < ω(ξ) < Cω(z), whenever ξ ∈ (z, r). In other words, ω ∈ R is equivalent to Received August 12, 2019, accepted June 5, 2020 Supported by NNSF of China (Grant Nos. 11971125, 11471084) 1) Corresponding author
Yang L. H. et al.
2
ω(z) ω(ξ) on (z, r), see [16]. From [22], if ξ ∈ (z, r), then 1 − |ξ| 1 − |z|, and |(z, r)| (1 − |z|2 )2 . And several examples of weighs R are given by [13, (4.4)–(4.6)]. For the study of the regular weighted Bergman spaces, we can see [5, 13–16]. Suppose ω1 (z) and ω2 (z) are non-negative integrable functions on D, let r0 ω1,2 (z) = ω1 (z)χM1 (z) + ω2 χM2 (z), z ∈ D. z For 0 < p < ∞, define Lpω1,2 (M ) to be the space of all Lebesgue measurable functions f satisfying the following condition |f (z)|p ω1,2 (z)dA(z) < ∞, (1.1) f pp = M
dxdy π
is the normalized Lebesgue area measure. where dA(z) = It is easy to know that Lpω1,2 (M ) is a Banach space when 1 ≤ p < ∞. In particular, L2ω1,2
