Inner function in Dirichlet type spaces

Let X and Y be two spaces of analytic functions in the unit disc D with X subset of Y. For an inner function theta, theta is said to be (X, Y)-improving if f theta is an element of X whenever f is an element of X and f theta is an element of Y. In this note, for a certain nonnegative and increasing function K defined in [0; infinity), we consider the space of Dirichlet type D-K consisting of those functions f analytic in D such that integral(D)vertical bar f'(z)vertical bar K-2(1 - vertical bar z vertical bar(2))dA(z) < infinity. We prove that an inner function belongs to D-K if and only if it is (D-K boolean AND BMOA; BMOA)-improving. We discuss also inner functions in D-K as multipliers of D-K boolean AND BMOA. (C) 2014 Elsevier Inc. All rights reserved.