Functions of bounded fractional variation and fractal currents

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 $C\ge 0$ such that the estimate \[ \biggl|\int u(x) \det D(f,g_1,\dots,g_{n-1})_x \, dx\biggr| \leq C\operatorname{Lip}^α(f) \operatorname{Lip}(g_1) \cdots \operatorname{Lip}(g_{n-1}) \] 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 Hölder 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 Hölder 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 Hölder maps defined on domains with fractal boundaries.

53 pages, 1 figure