Carleson measures, multipliers and integration operators for spaces of Dirichlet type

Abstract

For 0 < p < ∞ and α > − 1 , we let D α p denote the space of those functions f which are analytic in the unit disc D = { z ∈ C : | z | < 1 } and satisfy ∫ D ( 1 − | z | 2 ) α | f ′ ( z ) | p d x d y < ∞ . In this paper we characterize the positive Borel measures μ in D such that D α p ⊂ L q ( d μ ) , 0 < p < q < ∞ . We also characterize the pointwise multipliers from D α p to D β q ( 0 < p < q < ∞ ) if p − 2 < α < p . In particular, we prove that if ( 2 + α ) p − ( β + 2 ) q > 0 the only pointwise multiplier from D α p to D β q ( 0 < p < q < ∞ ) is the trivial one. This is not longer true for ( 2 + α ) p − ( β + 2 ) q ⩽ 0 and we give a number of explicit examples of functions which are multipliers from D α p to D β q for this range of values.