Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings

TL;DR

This paper introduces a new way to characterize Triebel-Lizorkin spaces on metric spaces using hyperbolic fillings, proving invariance under quasisymmetric maps and exploring interpolation properties.

Contribution

It provides a novel characterization of Triebel-Lizorkin spaces on general metric measure spaces via hyperbolic fillings, and establishes their invariance and interpolation properties.

Findings

New characterization of Triebel-Lizorkin spaces using hyperbolic fillings.

Proof of quasisymmetric invariance for certain metric spaces.

Initial results on complex interpolation of these spaces.

Abstract

We give a new characterization of (homogeneous) Triebel-Lizorkin spaces $\dot{\mathcal F}^{s}_{p,q}(Z)$ in the smoothness range $0 < s < 1$ for a fairly general class of metric measure spaces $Z$. The characterization uses Gromov hyperbolic fillings of $Z$. This gives a short proof of the quasisymmetric invariance of these spaces in case $Z$ is $Q$-Ahlfors regular and $sp = Q > 1$. We also obtain first results on complex interpolation for these spaces in the framework of doubling metric measure spaces.