On dilation operators in Triebel-Lizorkin spaces

Abstract

We consider dilation operators $T_k:f\rightarrow f(2^k\cdot)$ in the framework of Triebel-Lizorkin spaces $F^s_{p,q}(\mathbb{R}^n)$. If $s>n\max\big(\frac 1p -1,0\big)$, $T_k$ is a bounded linear operator from $F^s_{p,q}(\mathbb{R}{^n)$ into itself and there are optimal bounds for its norm. We study the situation on the line $s=n\max\big(\frac 1p -1,0\big)$, an open problem mentioned in [ET96, 2.3.1]. It turns out that the results shed new light upon the diversity of different approaches to Triebel-Lizorkin spaces on this line, associated to definitions by differences, Fourier-analytical methods and subatomic decompositions.