A limit case of the Hörmander multiplier theorem
type: Item
instance of: scholarly article
title: A limit case of the Hörmander multiplier theorem
zbMATH Open document ID: 0639.42012
DOI: 10.1007/BF01501167
author: Andreas Seeger
published in: Monatshefte für Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178370
review text: 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 \textit{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.
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 4042511
zbMATH Keywords: Hörmander multiplier criterion
zbMATH Keywords: Hardy-space
zbMATH Keywords: Lorentz-space
zbMATH Keywords: Calderón- Zygmund-theory
zbMATH Keywords: maximal-functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4042511