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 \(L^ p_ a\).


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


Zbl 0385.32014
Full Text: DOI


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