Fourier multiplier theorems for Triebel-Lizorkin spaces

In this paper we study sharp generalizations of ${\dot{F}}_p^{0,q}$ multiplier theorem of Mikhlin-H"ormander type. The class of multipliers that we consider involves Herz spaces $K_u^{s,t}$. Plancherel’s theorem proves $\widehat{L_s^2}=K_2^{s,2}$ and we study the optimal triple $(u,t,s)$ for which $\sup_{k\in$\mathbb{Z}$}{\big\Vert \big( m(2^k\cdot)\varphi\big)^{\vee}\big\Vert_{K_u^{s,t}}}<\infty$implies$\dot{F}_p^{0,q}$boundednessofmultiplieroperator$T_m$where$\varphi$ is a cutoff function. Our result also covers the $BMO$-type space ${\dot{F}}_{\mathrm{\infty }}^{0,q}$.