Fourier Multipliers on a Vector-Valued Function Space

We study multiplier theorems on a vector-valued function space, which is generalization of the results Calderón and Torchinsky, Grafakos, He, Honzík, Nguyen, an improvement result Triebel. For $$0\frac{d}{s-(d/\min {(1,p,q)}-d)}$$ , then $$\begin{aligned} \big \Vert \{\big ( m_k \widehat{f_k}\big )^{\vee }\big \}_{k\in {\mathbb {Z}}}\big _{L^p(\ell ^q)}\lesssim _{p,q} \sup _{l\in {Z}}}{\big m_l(2^l\cdot )\big _{L_s^r({\mathbb {R}}^d)}} \{f_k\big ^q)}, ~~f_k\in {\mathcal {E}}(A2^k), \end{aligned}$$ under condition $$\max {(|d/p-d/2|,|d/q-d/2|)}<s<d/\min {(1,p,q)}$$ . An extension to $$p=\infty will be additionally considered in scale Triebel–Lizorkin space. Our sharp sense Sobolev space above estimate cannot replaced by spaces $$L_s^r$$ with $$r\le \frac{d}{s-(d/\min