Fourier Multipliers from $L^p$-spaces to Morrey Spaces on the Unit Circle

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]).