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Weighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ball. (English) Zbl 1250.47030

Necessary and sufficient conditions for boundedness and for compactness of weighted composition operators μC φ from the weighted Bergman-Privalov-type space AN p,α to a weighted type space or a little weighted-type space on the unit ball in the n-dimensional complex vector space n are given. The spaces and the composition operator mentioned above are defined in the following way. Let B be the unit ball in n , H(B) be the space of all holomorphic functions on B, dV(z) be the normalized Lebesgue measure on B and dV α (z)=c α (1-|z| 2 ) α dV(z), where α>-1 and c α is a normalization constant, that is, V α (B)=1. For each p1 and α>-1, the weighted Bergman-Privalov-type space AN p,α is defined in the following way:

AN p,α ={fH(B): B ln p (1+|f(z)|)dV α (z)<}·

The weighted-type space H μ is defined as the space that consists of all fH(B) such that sup zB μ(z)|f(z)|<, where μ(z) is a positive continuous function (a weight) on B. The little weighted-type space H μ,o consists of all fH(B) such that

lim |z|1 μ(z)|f(z)|=0·

For fH(B), the corresponding weighted composition operator is defined by

(μC φ )(f)(z)=μ(z)f(φ(z)),zB,

where φ is a holomorphic self-map of B and μH(B) is fixed.

47B33Composition operators
32A37Spaces of holomorphic functions (several variables)
46E15Banach spaces of continuous, differentiable or analytic functions
47B38Operators on function spaces (general)
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