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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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References:
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