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

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-phi u(n)(f) (z) = u(z)f((n)) (phi(z), where u is a holomorphic function on the unit disk D, phi is a holomorphic self-map of D and n is an element of N-0, from the mixed-norm space H(p,q,phi), where p,q > 0 and phi is normal, to the weighted-type space H-u(infinity) or the little weighted-type space H-mu,0(infinity). For the case of the weighted Bergman space A(alpha)(p), p > 1, some bounds for the essential norm of the operator are also given. (C) 2009 Elsevier Inc. All rights reserved.