Weighted Composition Operators from Generalized Weighted Bergman Spaces to Weighted-Type Spaces
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Research Article Weighted Composition Operators from Generalized Weighted Bergman Spaces to Weighted-Type Spaces Dinggui Gu Department of Mathematics, JiaYing University, Meizhou, GuangDong 514015, China Correspondence should be addressed to Dinggui Gu, [email protected] Received 3 November 2008; Revised 22 November 2008; Accepted 24 November 2008 Recommended by Kunquan Lan Let ϕ be a holomorphic self-map and let ψ be a holomorphic function on the unit ball B. The boundedness and compactness of the weighted composition operator ψCϕ from the generalized weighted Bergman space into a class of weighted-type spaces are studied in this paper. Copyright q 2008 Dinggui Gu. 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.
1. Introduction Let B be the unit ball of Cn and let HB be the space of all holomorphic functions on B. For f ∈ HB, let Rfz
n ∂f zj z ∂z j j1
1.1
represent the radial derivative of f ∈ HB. We write Rm f RRm−1 f. For any p > 0 and α ∈ R, let N be the smallest nonnegative integer such that pN α > p −1. The generalized weighted Bergman space Aα is defined as follows: p Aα
f ∈ HB | fApα
f0
N R fzp 1 − |z|2 pNα dvz
1/p
−1, the Besov space A−n1 , and the Hardy space H 2 . See 1, 2 for some basic facts on the weighted Bergman space. Let μ be a positive continuous function on 0, 1. We say that μ is normal if there exist positive numbers α and β, 0 < α < β, and δ ∈ 0, 1 such that see 3 μr 0; 1 − rα μr lim ∞. r → 1 1 − rβ
μr is decreasing on δ, 1, 1 − rα μr is increasing on δ, 1, 1 − rβ
lim
r →1
1.3
An f ∈ HB is said to belong to the weighted-type space, denoted by Hμ∞ Hμ∞ B, if fHμ∞ sup μ |z| fz < ∞,
1.4
z∈B
∞ where μ is normal on 0, 1. The little weighted-type space, denoted by Hμ,0 , is the subspace ∞ ∞ of Hμ consisting of those f ∈ Hμ such that
lim μ |z| fz 0.
1.5
|z| → 1
See 4, 5 for more information on Hμ∞ . Let ϕ be a holomorphic self-map of B. The composition operator Cϕ is defined as follows: Cϕ f z f ◦ ϕz,
f ∈ HB.
1.6
Let ψ ∈ HB. For f ∈ HB, the weighted composition operator ψCϕ is defined by ψCϕ f z ψzf ϕz ,
z ∈ B.
1.7
The book 6 contains a plenty of information on the composition operator and the weighted composition operator. In the setting of the unit ball, Zhu studied the boundedness and compactness of the weighted composition operator between Bergman-type spaces and H ∞ in 7. Some extensions of these results can be found in 8. Some necessary and sufficient conditions for the weighted composition operator to be bounded or compact between the Bloch space and H ∞ are given in 9. In the setting of the unit polydisk, some necessary and sufficient conditions for a weighted composition operator to be bounded and compact between
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