Can someone tell me whether there is a characterization of boundedness of multiplication operators on the Dirichlet space of the unit disc? These are holomorphic functions $f$ on the unit disk $D$ such that $$\iint_D |f'|^2<\infty$$

For the Hardy space, the multipliers are precisely the bounded analytic functions on the disc. Thanks in advance!


Unfortunately, the multipliers of the Dirichlet space are not as easy to describe as of Hardy and Bergman spaces. The characterization was obtained by Stegenga in Multipliers of the Dirichlet space (free access). In addition to boundedness, it requires a Carleson-type condition in terms of capacity: $$ \iint_{\bigcup S(I_j)} |f'|^2 \le A \operatorname{cap}(\bigcup I_j) $$ for all finite disjoint collections of subarcs $I_j$ of the unit circle.

Some easy-to-verify sufficient conditions were subsequently obtained Univalent multipliers of the Dirichlet space by Axler and Shields.

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