Superposition operators, Hardy spaces, and Dirichlet type spaces

For 0 < p < oo and alpha > 1 the space of Dirichlet type D-alpha(p): consists of those functions f which are analytic in the unit disc D and satisfy f(D) (1 - |z|)(alpha)|f'(z)|(P) dA(z) < infinity. The space D-p(p-1) is the closest one to the Hardy space H-P among all the D-alpha(p) Our main object in this paper is studying similarities and differences between the spaces HP and D-p-1(p) (0 < p < infinity) regarding superposition operators. Namely, for 0 < p < infinity and 0 < a < infinity, we characterize the entire functions phi such that the superposition operator S-phi. with symbol p maps the conformally invariant space Q. into the space D-p-1(p), and, also, those which map D-p-1(p) into Q. and we compare these results with the corresponding ones with HP in the place of D-p-1(p). We also study the more general question of characterizing the superposition operators mapping vg, into Q. and Q. into D-alpha(p), for any admissible triplet of numbers (p, alpha, s). (C) 2018 Elsevier Inc. All rights reserved.