Fourier multiplier theorems for Triebel-Lizorkin spaces

Abstract

In this paper we study sharp generalizations of ${\dot{F}}_p^{0,q}$ multiplier theorem of Mikhlin-Hörmander 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 ${\sup }_{k\in \mathbb{Z}}\Vert (m(2^k\cdot )\phi {)}^{\vee }{\Vert }_{K_u^{s,t}}<\infty$ implies ${\dot{F}}_p^{0,q}$ boundedness of multiplier operator $T_m$ where $\phi$ is a cutoff function. Our result also covers the $BMO$-type space ${\dot{F}}_{\infty }^{0,q}$.

arXiv commentto appear in Math. Z