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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

Citations:

Zbl 0385.32014
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References:

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[2] Axler, Canad. Math. Bull. 28 pp 237– (1985) · Zbl 0578.32010
[3] DOI: 10.1307/mmj/1029003075 · Zbl 0589.46042
[4] Amar, Canad. J. Math. 30 pp 711– (1978) · Zbl 0385.32014
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[10] DOI: 10.1007/BF01099672 · Zbl 0278.46032
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