Fourier multipliers on a vector-valued function space

Abstract

We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calderón-Torchinsky and Grafakos-He-Honzík-Nguyen, and an improvement of the result of Triebel. For $0<p<\infty$ and $0<q\le \infty$ we obtain that if $r>\frac{d}{s-(d/\min (1,p,q)-d)}$, then $$\Vert \left\{\right(m_k\widehat{f_k}{)}^{\vee }{\}}_{k\in \mathbb{N}}{\Vert }_{L^p(l^q)}{\lesssim }_{p,q}\underset{l\in \mathbb{N}}{\sup }\Vert m_l(2^l\cdot ){\Vert }_{L_s^r({\mathbb{R}}^d)}\Vert \{f_k{\}}_{k\in \mathbb{N}}{\Vert }_{L^p(l^q)},f_k\in \mathcal{E}(A{2}^k),$$ under the condition $\max (|d/p-d/2|,|d/q-d/2|)<s<d/\min (1,p,q)$. An extension to $p=\infty$ will be additionally considered in the scale of Triebel-Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by a smaller Sobolev space $L_s^r$ with $r\le \frac{d}{s-(d/\min (1,p,q)-d)}$.

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