Operators induced by radial measures acting on the Dirichlet space

Abstract

Let D be the unit disc in the complex plane. Given a positive finite Borel measure μ on the radius [0, 1), we let μn denote the n-th moment of μ and we deal with the action on spaces of analytic functions in D of the operator of Hibert-type Hμ and the operator of Cesàro-type Cμ which are defined as follows: If f is holomorphic in D, f(z)=∑∞n=0anzn (z∈D), then Hμ(f) is formally defined by Hμ(f)(z)=∑∞n=0(∑∞k=0μn+kak)zn (z∈D) and Cμ(f) is defined by Cμ(f)(z)=∑∞n=0μn(∑nk=0ak)zn (z∈D ). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in D. In this paper we study the action of the operators Hμ and Cμ on the Dirichlet space D and, more generally, on the analytic Besov spaces Bp (1≤p<∞).