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 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 ∈ N 0 , from the mixed-norm space H( p, q, ϕ), where p, q > 0 and ϕ is normal, to the weighted-type space H μ ∞ or the little weighted-type 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.