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

$$\begin{aligned} \biggl |\int u(x) \det D(f,g_1,\ldots ,g_{n-1})_x \, dx\biggr | \le C\text{ Lip }^\alpha (f) \text{ Lip }(g_1) \cdots \text{ Lip }(g_{n-1}) \end{aligned}$$

holds for all Lipschitz functions $f,g_1,\ldots ,g_{n-1}$ 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.