Functions of bounded fractional variation and fractal currents

TL;DR

This paper introduces a new class of functions called bounded fractional variation, extending classical bounded variation to fractional orders, and explores their properties, representations, and applications to fractal boundaries and degree theory.

Contribution

It characterizes bounded fractional variation functions via Whitney's flat chains and Ambrosio-Kirchheim currents, and applies these to fractal boundary problems and degree functions.

Findings

Characterization of fractional variation functions as flat chains and currents

Extension to H"older differential forms and higher integrability

Sharp integrability results for Brouwer degree functions

Abstract

Extending the notion of bounded variation, a function $u \in L_c^1(\mathbb R^n)$ is of bounded fractional variation with respect to some exponent $\alpha$ if there is a finite constant $C \geq 0$ such that the estimate \[ \biggl|\int u(x) \det D(f,g_1,\dots,g_{n-1})_x \, dx\biggr| \leq C\operatorname{Lip}^\alpha(f) \operatorname{Lip}(g_1) \cdots \operatorname{Lip}(g_{n-1}) \] holds for all Lipschitz functions $f,g_1,\dots,g_{n-1}$ on $\mathbb R^n$. Among such functions are characteristic functions of domains with fractal boundaries and H\"older continuous functions. We characterize functions of bounded fractional variation as a certain subspace of Whitney's flat chains and as multilinear functionals in the setting of Ambrosio-Kirchheim currents. Consequently we discuss extensions to H\"older differential forms, higher integrability, an isoperimetric inequality, a Lusin type property and change of variables. As an application we obtain sharp integrability results for Brouwer degree functions with respect to H\"older maps defined on domains with fractal boundaries.