Generalized Cesàro operators on weighted Dirichlet spaces

Abstract

Given a complex Borel measure η∈M(D), we study the boundedness of the Cesàro-type operator Cη given byCη(f)(z)=∑n=0∞(∫Dwndη(w))(∑k=0nak)zn where f(z)=∑n=0∞anzn, acting on the weighted Dirichlet spaces Dρ consisting in functions f∈H(D) with ∑n=0∞|an|2(n+1)ρn<∞ where (ρn) is a sequence of positive real numbers. We get new estimates on Taylor coefficients of functions in the Dirichlet space D0 and extend the results known for Dα and positive measures ν supported in [0,1) to more general weights and general measures.