Fourier multipliers on a vector-valued function space

We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calder'on-Torchinsky and Grafakos-He-Honz'ik-Nguyen, and an improvement of the result of Triebel. For $0<p<\mathrm{\infty }$ and $0<q\le \mathrm{\infty }$ we obtain that if $r>\frac{d}{s-(d\mathrm{/}\min (1,p,q)-d)}$, then $$\parallel \left\{\right(m_k\widehat{f_k}{)}^{\vee }{\}}_{k\in \mathbb{N}}{\parallel }_{L^p(l^q)}{\lesssim }_{p,q}\underset{l\in \mathbb{N}}{\sup }\parallel m_l(2^l\cdot ){\parallel }_{L_s^r({\mathbb{R}}^d)}\parallel \{f_k{\}}_{k\in \mathbb{N}}{\parallel }_{L^p(l^q)},f_k\in \mathcal{E}(A{2}^k),$$ under the condition $\max (\mathrm{\mid }d\mathrm{/}p-d\mathrm{/}2\mathrm{\mid },\mathrm{\mid }d\mathrm{/}q-d\mathrm{/}2\mathrm{\mid })<s<d\mathrm{/}\min (1,p,q)$. An extension to $p=\mathrm{\infty }$ 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 $r\leq \frac{d}{s-(d/\min{(1,p,q)}-d)}$.