Fourier multiplier theorems for Triebel-Lizorkin spaces

TL;DR

This paper extends Fourier multiplier theorems for Triebel-Lizorkin spaces using Herz spaces, identifying optimal conditions for boundedness of multiplier operators, including cases involving BMO spaces.

Contribution

It generalizes classical multiplier theorems by incorporating Herz spaces and determines optimal conditions for boundedness on Triebel-Lizorkin spaces.

Findings

Established sharp multiplier theorems involving Herz spaces.

Identified optimal parameters for multiplier boundedness.

Extended results to include BMO-type Triebel-Lizorkin spaces.

Abstract

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 $\dot{F}_{\infty}^{0,q}$.