A trace theorem for Besov functions in spaces of homogeneous type

TL;DR

This paper extends the trace theorem for Besov functions from Euclidean spaces to metric spaces of homogeneous type, establishing conditions under which Besov functions can be restricted to subsets.

Contribution

It generalizes the Euclidean trace theorem to metric spaces, providing an extension, interpolation, and restriction framework for Besov functions in this setting.

Findings

Proved that Besov functions in a subset are restrictions of higher-regularity functions in the larger space.

Showed that interpolation between potential spaces yields Besov spaces.

Established restriction theorems for Besov functions in metric measure spaces.

Abstract

The aim of this paper is to prove a trace theorem for Besov functions in the metric setting, generalizing a known result from A. Jonsson and H. Wallin in the Euclidean case. We show that the trace of a Besov space defined in a `big set' $X$ is another Besov space defined in the `small set' $F\subset X$. The proof is divided in three parts. First we see that Besov functions in $F$ are restrictions of functions of the same type (but greater regularity) in $X$, that is we prove an Extension theorem. Next, as an auxiliary result that can also be interesting on its own, we show that the interpolation between certain potential spaces gives a Besov space. Finally, to obtain that Besov functions in $X$ can in fact be restricted to $F$, a Restriction theorem, we first prove that this result holds for functions in the potential space, and then by the interpolation result previously shown, it must hold in the Besov case. For the interpolation and restriction theorems, we make additional assumptions on the spaces $X$ and $F$, and on the order of regularity of the functions involved.