Young Integration with respect to Hölder Charges

Abstract

We present a multidimensional Young integral that enables to integrate Hölder continuous functions with respect to a Hölder charge. It encompasses the integration of Hölder differential forms introduced by R. Züst: if $f$, $g_1,\dots ,g_d$ are merely Hölder continuous functions on the cube $[0,1{]}^d$ whose Hölder exponents satisfy a certain condition, it is possible to interpret $\mathrm{d}{g}_1\wedge \cdots \wedge \mathrm{d}{g}_d$ as a Hölder charge and thus to make sense of the integral \[ \int_B f \mathrm{d} g_1 \wedge \cdots \wedge \mathrm{d}g_d \] over a set $B\subset [0,1{]}^d$ of finite perimeter.