Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces

Abstract

Motivated by the recent paper [X. Zhu, Products of differentiation composition and multiplication from Bergman type spaces to Bers spaces, Integral Transform. Spec. Funct. 18 (3) (2007) 223-231], we study the boundedness and compactness of the weighted differentiation composition operator Dφ, un (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 ∈ N0, from the mixed-norm space H(p, q, φ{symbol}), where p,q > 0 and φ{symbol} is normal, to the weighted-type space Hμ∞ or the little weighted-type space Hμ, 0∞. For the case of the weighted Bergman space Aαp, p > 1, some bounds for the essential norm of the operator are also given.