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 $\dot{F}_p^{0,q}$ boundedness of multiplier operator $T_m$ where $\varphi$ is a cutoff function. Our result also covers the $BMO$-type space $\dot{F}_{\infty}^{0,q}$.