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Pointwise multipliers from the Hardy space to the Bergman space. (English) Zbl 0936.30038
The author studies the problem of characterizing regions $G$ with the property that the Hardy space $H^2(G)$ is contained in the Bergman space $L^2_a(G)$. For example, he shows the following. When $G$ is a simply connected region, $D$ is an open unit disc and $\tau:D\to G$ is a Riemann map, he shows that $H^2(G)\subseteq L^2_a(G)$ if and only if $\tau$ is Lipschitz of order $1/2$. When $\varphi$ is analytic on $D$ and $G=\varphi(D)$, $H^2(G) \subseteq L^2_a(G)$ if $\varphi'$ is a multiplier. The author also gives examples of multipliers and raises three questions.

30H05Bounded analytic functions
46E15Banach spaces of continuous, differentiable or analytic functions