  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 $r\leq
\frac{d}{s-(d/\min{(1,p,q)}-d)}$.
