Composition operator for functions of bounded variation

TL;DR

This paper characterizes the precise conditions under which a homeomorphism ensures that composition with functions of bounded variation remains within that space, focusing on measure-theoretic properties and set mappings.

Contribution

It establishes necessary and sufficient conditions for homeomorphisms to preserve BV functions under composition, including measure and perimeter mapping properties.

Findings

Characterizes when $u o u \

f$ preserves BV functions via measure conditions.

Identifies conditions for homeomorphisms to map finite perimeter sets to finite perimeter sets.

Abstract

We study the optimal conditions on a homeomorphism $f:\Omega\subset \R^n\to \R^n$ to guarantee that the composition $u\circ f$ belongs to the space of functions of bounded variation for every function $u$ of bounded variation. We show that a sufficient and necessary condition is the existence of a constant $K$ such that $|Df|(f^{-1}(A))\leq K\Ln(A)$ for all Borel sets $A$. We also characterize homeomorphisms which maps sets of finite perimeter to sets of finite perimeter. Towards these results we study when $f^{-1}$ maps sets of measure zero onto sets of measure zero (i.e. $f$ satisfies the Lusin $(N^{-1})$ condition).