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June 2014 Fourier Multipliers from $L^p$-spaces to Morrey Spaces on the Unit Circle
Takashi IZUMI, Enji SATO
Tokyo J. Math. 37(1): 199-209 (June 2014). DOI: 10.3836/tjm/1406552439

Abstract

Let $p$, $\lambda$ be real numbers such that $1\leq p\leq \infty$, and $0\leq\lambda\leq1$. Also we let $L^p(\mathbb{T})$ be the $L^p$-spaces on the unit circle $\mathbb{T}$, $L^{p,\lambda}(\mathbb{T})$ Morrey spaces on $\mathbb{T}$ (cf.~[14]), and $M(L^p,L^{p,\lambda})$ the set of all translation invariant bounded linear operators from $L^p(\mathbb{T})$ to $L^{p,\lambda}(\mathbb{T})$. Figa-Talamanca and Gaudry~[2] showed $M(L^p,L^p)\neq M(L^q,L^q)\ (1<p<q\leq2)$. In this paper, we generalize Gaudry's result. Our main results are $M(L^p,L^{p,\lambda})\neq M(L^q,L^{q,\nu})\ {\rm for}\ \lambda/p\neq\nu/q$ $(1<p, q<\infty,\ 0<\lambda,\nu<1)$, and $M(L^p,L^{p,\lambda})\neq M(L^q,L^{q,\nu})$ for $2<p<q$ and $\lambda/p=\nu/q$\ $(0<\lambda,\nu<1)$. Moreover, we show a relation between $M(L^p,L^{p,\lambda})$ and the measure whose distribution function satisfies a Lipschitz condition (cf.~[4]).

Citation

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Takashi IZUMI. Enji SATO. "Fourier Multipliers from $L^p$-spaces to Morrey Spaces on the Unit Circle." Tokyo J. Math. 37 (1) 199 - 209, June 2014. https://doi.org/10.3836/tjm/1406552439

Information

Published: June 2014
First available in Project Euclid: 28 July 2014

zbMATH: 1303.42003
MathSciNet: MR3264522
Digital Object Identifier: 10.3836/tjm/1406552439

Subjects:
Primary: 42A45
Secondary: 42A55

Rights: Copyright © 2014 Publication Committee for the Tokyo Journal of Mathematics

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Vol.37 • No. 1 • June 2014
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