A note on BV and 1-Sobolev functions on the weighted Euclidean space

TL;DR

This paper explores the equivalence of different definitions of functions of bounded variation and 1-Sobolev functions in Euclidean spaces with arbitrary Radon measures, clarifying their relationships.

Contribution

It establishes the equivalence between multiple notions of BV and 1-Sobolev functions in weighted Euclidean spaces with Radon measures, unifying existing concepts.

Findings

Proves equivalence of BV notions under Radon measures

Analyzes relationships between definitions of 1-Sobolev functions

Provides a unified framework for BV and Sobolev functions in weighted spaces

Abstract

In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various definitions of 1-Sobolev function.