Functions of bounded fractional variation and fractal currents

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 $α$ if there is a finite constant $C \geq 0$ such that the estimate $\biggl|\int u(x) \det D(f,g_1,\dots,g_{n-1})_x \, dx\biggr| \leq C\operatorname{Lip}^α(f) \operatorname{Lip}(g_1) \cdots \operatorname{Lip}(g_{n-1})$ 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.

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