Functions of bounded fractional variation and fractal currents

Extending the notion of bounded variation, a function u is an element of L-c(1) (R-n) is of bounded fractional variation with respect to some exponent a if there is a finite constant C >= 0 such that the estimate vertical bar integral u(x) det D(f, g(1),...,g(n-1))(x) dx vertical bar <= C Lip(alpha)(f) Lip(g(1)) . . . Lip(g(n-1)) holds for all Lipschitz functions f, g(1), . . . , g(n-1) on R-n. Among such functions are characteristic functions of domains with fractal boundaries and Holder 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 Holder 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 Holder maps defined on domains with fractal boundaries.