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A limit case of the Hörmander multiplier theorem. (English) Zbl 0639.42012
Let m be a multiplier in \({\mathbb{R}}^ n \)satisfying \(\sup_{t>0}\| \phi m(t\cdot)\|_{B^ 2_{n/2,1}}<\infty,\) where \(B^ 2_{n/2,1}\) is a Besov-space, \(\phi\) a bump-function in (1/2,2). It follows from E. M. Stein’s proof [Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. 30 (1970; Zbl 0207.135)] of the Hörmander multiplier criterion, that the associated convolution operator \(T_ m\) is bounded on \(L^ p\), \(1<p<\infty\). We prove that \(T_ m: H^ 1\to L^{1r}\) is bounded (\(H^ 1\) Hardy-space, \(L^{1r}\) Lorentz-space). The result is almost sharp in the sense, that \(L^{1r}\) cannot be replaced by \(L^{1s}\), \(s<2\). Essentially used is a variant of Calderón-Zygmund-theory involving sequence-valued maximal-functions.
Reviewer: A.Seeger

MSC:
42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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