## Multipliers of Bergman spaces into Lebesgue spaces.(English)Zbl 0587.30048

Let U be the open unit disk in the complex plane and let $$L^ p_ a$$ denote the space of analytic functions f on U such that $$\int_{U}| f|^ pdm<+\infty$$, where dm denotes ordinary 2-dimensional Lebesgue measure. In this paper are given necessary and sufficient conditions on a positive measure $$\mu$$ in order that $(\int | f|^ qd\mu)^{1/q}\leq C(\int_{U}| f|^ pdm)^{1/p}\quad for\quad all\quad f\in L^ p_ a,$ with the constant C independent of f. The only previously unknown case was $$0<q<p<+\infty$$. To describe the conditions we consider for $$0<\epsilon <1$$ and $$a\in U$$ the pseudo- hyperbolic disk $D_{\epsilon}(a)=\{z\in U:\quad | z-a| /| 1-\bar az| <\epsilon \}.$ If $$\mu$$ is a positive measure on U, associate with it a function $$k(a)=\mu (D_{\epsilon}(a))/m(D_{\epsilon}(a))$$. The necessary and sufficient condition, then, is that $$\int | k|^ sdm<+\infty$$ where s is the conjugate index to p/q, i.e. $$1/s+q/p=1.$$
The sufficiency of this condition is a simple consequence of the subharmonicity of $$| f|^ q$$ and Hölder’s inequality. The proof of necessity uses results of E. Amar [Can. J. Math. 30, 711- 737 (1978; Zbl 0385.32014)] on interpolation sequences for $$L^ p_ a$$.

### MSC:

 30H05 Spaces of bounded analytic functions of one complex variable 46E10 Topological linear spaces of continuous, differentiable or analytic functions 30E99 Miscellaneous topics of analysis in the complex plane

### Keywords:

Bergman space; Carleson measure; multipliers

Zbl 0385.32014
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### References:

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