Trace theorems for functions of bounded variation in metric spaces

TL;DR

This paper establishes the existence and estimates of boundary traces for functions of bounded variation in metric measure spaces, without requiring exterior information, and proves a Maz'ya-type inequality for such functions.

Contribution

It introduces a new approach to trace theorems in metric spaces that does not depend on exterior function values, expanding the theoretical framework.

Findings

Existence of traces with $L^1$ estimates in metric measure spaces.

Trace theorems without exterior function knowledge.

A Maz'ya-type inequality for BV functions vanishing on positive capacity sets.

Abstract

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality, and obtain $L^1$ estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'ya-type inequality for functions of bounded variation that vanish on a set of positive capacity.