Operator theoretic differences between Hardy and Dirichlet-type spaces

For 0 < p < ∞ , the Dirichlet-type space D p − 1 p consists of the analytic functions f in the unit disc D such that ∫ D | f ′ ( z ) | p ( 1 − | z | ) p − 1 d A ( z ) < ∞ . Motivated by operator theoretic differences between the Hardy space H p and D p − 1 p , the integral operator T g ( f ) ( z ) = ∫ 0 z f ( ζ ) g ′ ( ζ ) d ζ , z ∈ D , acting from one of these spaces to another is studied. In particular, it is shown, on one hand, that T g : D p − 1 p → H p 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 T g : D p − 1 p → H q 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 D p − 1 p , p > 2 , is significantly relaxed, and the established result is shown to be sharp in a very strong sense.