Operator theoretic differences between Hardy and Dirichlet-type spaces

Abstract

For 0<p<∞, the Dirichlet-type space Dp−1p consists of the analytic functions f in the unit disc D such that ∫D|f′(z)|p(1−|z|)p−1dA(z)<∞. Motivated by operator theoretic differences between the Hardy space Hp and Dp−1p, the integral operatorTg(f)(z)=∫0zf(ζ)g′(ζ)dζ,z∈D, acting from one of these spaces to another is studied. In particular, it is shown, on one hand, that Tg:Dp−1p→Hp is bounded if and only if g∈BMOA when 0<p⩽2, and, on the other hand, that this equivalence is very far from being true if p>2. Those symbols g such that Tg:Dp−1p→Hq is bounded (or compact) when p<q are also characterized. Moreover, the best known sufficient L∞-type condition for a positive Borel measure μ on D to be a p-Carleson measure for Dp−1p, p>2, is significantly relaxed, and the established result is shown to be sharp in a very strong sense.