Operator theoretic differences between Hardy and Dirichlet-type spaces

For $0<p<\infty$, the Dirichlet-type space $\Dp$ consists of those analytic functions $f$ in the unit disc $\D$ such that $\int_\D|f'(z)|\sp p(1-|z|)^{p-1}\,dA(z)<\infty$. Motivated by operator theoretic differences between the Hardy space $H^p$ and $\Dp$, the integral operator {displaymath} T_g(f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ,\quad z\in\D, {displaymath} acting from one of these spaces to another is studied. In particular, it is shown, on one hand, that $T_g:\Dp\to H^p$ is bounded if and only if $g\in\BMOA$ when $0<p\le 2$, and, on the other hand, that this equivalence is very far from being true if $p>2$. Those symbols $g$ such that $T_g:\Dp\to H^q$ is bounded (or compact) when $p<q$ are also characterized. Moreover, the best known sufficient $L^\infty$-type condition for a positive Borel measure $μ$ on $\D$ to be a $p$-Carleson measures for $\Dp$, $p>2$, is significantly relaxed, and the established result is shown to be sharp in a very strong sense.

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