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 $0<p<\infty$ and $0<q\leq \infty$ we obtain that if $r>\frac{d}{s-(d/\min{(1,p,q)}-d)}$, then $$\big\Vert \big\{\big( m_k \hat{f_k}\big)^{\vee}\big\}_{k\in\mathbb{N}}\big\Vert_{L^p(l^q)}\lesssim_{p,q} \sup_{l\in\mathbb{N}}{\big\Vert m_l(2^l\cdot)\big\Vert_{L_s^r(\mathbb{R}^d)}} \big\Vert \big\{f_k\big\}_{k\in\mathbb{N}}\big\Vert_{L^p(l^q)}, ~~f_k\in\mathcal{E}(A2^k),$$ 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 $r\leq \frac{d}{s-(d/\min{(1,p,q)}-d)}$.