Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings

Abstract

We give a new characterization of (homogeneous) Triebel-Lizorkin spaces ${\dot{\mathcal{F}}}_{p,q}^s(Z)$ 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.

arXiv comment32 pages