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Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. (English) Zbl 1165.30029
Summary: Motivated by the recent paper [X. Zhu, Integral Transform. Spec. Funct. 18, No. 3, 223–231 (2007; Zbl 1119.47035)], we study the boundedness and compactness of the weighted differentiation composition operator D φ,u n (f)(z)=u(z)f (n) (φ(z)), where u is a holomorphic function on the unit disk 𝔻,φ is a holomorphic self-map of 𝔻 and n 0 , from the mixed-norm space H(p,q,ϕ), where p,q>0 and ϕ is normal, to the weighted space H μ or the little weighted space H μ,0 . For the case of the weighted Bergman space A α p ,p>1, some bounds for the essential norm of the operator are also given.
MSC:
30H05Bounded analytic functions
References:
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