Pointwise multipliers from the Hardy space to the Bergman space

Abstract

For which regions $G$ is the Hardy space $H^{2}(G)$ contained in the Bergman space $L^{2}_{a}(G)$? This paper relates the above problem to that of finding the multipliers of $H^{2}(\mathbb{D})$ into $L^{2}_{a}(\mathbb{D})$. When $G$ is a simply connected region this leads to a solution of the above problem in terms of Lipschitz conditions on the Riemann map of $\mathbb{D}$ onto $G$. For arbitrary regions $G$, it is shown that if $G$ is the range of a function whose derivative is a multiplier from $H^{2}(\mathbb{D})$ to $L^{2}_{a}(\mathbb{D})$, then $H^{2}(G)$ is contained in $L^{2}_{a}(G)$. Also, if $G$ has a piecewise smooth boundary, then it is shown that $H^{2}(G)$ is contained in $L^{2}_{a}(G)$ if and only if the angles at all the “corner” points are at least $\pi/2$. Examples of multipliers from $H^{2}(\mathbb{D})$ to $L^{2}_{a}(\mathbb{D})$ are given; and in particular, every Bergman inner function is such a multiplier.