  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.

Functional Analysis;Metric Geometry