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 Dφ,un(f)(z)=u(z)f(n)(φ(z)), where u   is a holomorphic function on the unit disk DD, φ   is a holomorphic self-map of DD and n∈N0n∈N0, from the mixed-norm space H(p, q, ϕ), where p,q > 0 and ϕ   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.