## 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.
Reviewer: T.Nakazi (Sapporo)

### MSC:

 30H05 Spaces of bounded analytic functions of one complex variable 46E15 Banach spaces of continuous, differentiable or analytic functions

### Keywords:

embedding; Hardy space; Bergman space