Abstract
The boundedness and compactness of the weighted differentiation composition operator from the mixed‐norm space to the nth weighted‐type space on the unit disk are characterized.
Highlights
Throughout this paper D will denote the open unit disk in the complex plane C, H D the class of all holomorphic functions on D, and H∞ H∞ D the space of all bounded holomorphic functions on D with the norm f ∞ supz∈D|f z |.The mixed norm space Hp,q,γ Hp,q,γ D, 0 < p, q < ∞, −1 < γ < ∞, consists of all f ∈ H D such that fq Hp,q,γMpq f, r 1 − r γ dr < ∞, Mp f, r 2π f reiθ p dθ 1/pAbstract and Applied AnalysisA positive continuous function on D is called weight
The quantity bWμn D f is a seminorm on the nth weighted-type space Wμn D and a norm on Wμn D /Pn−1, where Pn−1 is the set of all polynomials whose degrees are less than or equal to n − 1
Assume that m ∈ N0, n ∈ N, p, q > 0, γ > −1, φ is an analytic self-map of D and u ∈ H D
Summary
Throughout this paper D will denote the open unit disk in the complex plane C, H D the class of all holomorphic functions on D, and H∞ H∞ D the space of all bounded holomorphic functions on D with the norm f ∞ supz∈D|f z |. The mixed norm space Hp,q,γ Hp,q,γ D , 0 < p, q < ∞, −1 < γ < ∞, consists of all f ∈ H D such that fq
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