Traces of Besov, Triebel-Lizorkin and Sobolev spaces on metric spaces

TL;DR

This paper proves trace theorems for various function spaces on Ahlfors regular metric spaces, extending classical Euclidean results to more general settings using hyperbolic fillings.

Contribution

It introduces new trace theorems for Besov, Triebel-Lizorkin, and Sobolev spaces on metric spaces, generalizing Euclidean results with a novel hyperbolic filling approach.

Findings

Trace theorems for Triebel-Lizorkin and Besov spaces on metric spaces

Extension of classical Euclidean trace results to Ahlfors regular spaces

Use of hyperbolic fillings to define and analyze function spaces

Abstract

We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces $Z$. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices $s<1,$ as well as the first order Haj{\l}asz-Sobolev space $M^{1,p}(Z)$. They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset $F \subset Z$ are Besov spaces defined intrinsically on $F$. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.