Weighted Bergman spaces induced by doubling weights in the unit ball of $$\mathbb {C}^n$$ C n
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Weighted Bergman spaces induced by doubling weights in the unit ball of Cn Juntao Du1 · Songxiao Li2
· Xiaosong Liu3 · Yecheng Shi4
Received: 25 October 2019 / Revised: 25 October 2019 / Accepted: 15 October 2020 © Springer Nature Switzerland AG 2020
Abstract p This paper is devoted to the study of the weighted Bergman space Aω in the unit ball n B of C with doubling weight ω satisfying
1
ω(t)dt < C
r
1 1+r 2
ω(t)dt, 0 ≤ r < 1.
p
The q-Carleson measures for Aω are characterized in terms of a neat geometric condip tion involving Carleson block. Some equivalent characterizations for Aω are obtained by using the radial derivative and admissible approach regions. The boundedness and p q compactness of Volterra integral operator Tg : Aω → Aω are also investigated in this paper with 0 < p ≤ q < ∞, where
1
Tg f (z) = 0
f (t z)g(t z)
dt , t
f ∈ H (B), z ∈ B.
Keywords Weighted Bergman space · Carleson measure · Volterra integral operator · Doubling weight Mathematics Subject Classification 32A36 · 47B38
1 Introduction Let B be the open unit ball of Cn and S the boundary of B. When n = 1, then B is the open unit disk in complex plane C and always denoted by D. Let H (B) denote the space of all holomorphic functions on B. For any two points
This project was funded by the Science and Technology Development Fund, Macau SAR (File No. 186/2017/A3), NNSF of China (Nos. 11720101003, 11701222, 11901271), China Postdoctoral Science Foundation (No. 2018M633090) and a grant of Lingnan Normal University (No. 1170919634). Extended author information available on the last page of the article 0123456789().: V,-vol
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z = (z 1 , z 2 , . . . , z n ) and w = (w1 , w2 , . . . , wn ) in Cn , we define z, w = z 1 w1 + · · · + z n wn and |z| =
z, z =
|z 1 |2 + · · · + |z n |2 .
Let dσ and d V be the normalized surface and volume measures on S and B, respectively. For 0 < p ≤ ∞, the Hardy space H p (B)(or H p ) is the space consisting of all functions f ∈ H (B) such that f H p := sup M p (r , f ), 0 0 determined by ω, such that ω(r ˆ ) 1 < < C, C (1 − r )ω(r )
when 0 ≤ r < 1.
ω is called a rapidly increasing weight, denote by ω ∈ I, if lim
r →1
ω(r ˆ ) = ∞. (1 − r )ω(r )
ˆ See [8,9] for more details about I, R, D. ˆ After a calculation, we see that I ∪ R ⊂ D. p In [9], J. Peláez and J. Rättyä introduced a new class function space Aω (D), the weighted Bergman space induced by rapidly increasing weights in D. In [9], they investigated some basic properties of ω with ω ∈ R ∪ I, described the q-Carleson p p measure for Aω (D), gave equivalent characterizations of Aω (D), characterized the boundedness, compactness and Schatten classes of Volterra integral operator Jg on p ˆ See [8–14] Aω (D). In [8], J. Peláez extended many results from ω ∈ R ∪ I to ω ∈ D. p ˆ for many results on Aω (D) with ω ∈ D. p Motivated by [9], we extend the Bergman space Aω (D) with ω ∈ Dˆ to the unit ball. p p ˆ Let ω ∈ D and 0 < p < ∞. The weighted Bergman space Aω = Aω (B) is the space of all f ∈ H (B)
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