Young Integration with respect to H\"older Charges

Abstract

We present a multidimensional Young integral that enables to integrate H\"older continuous functions with respect to a H\"older charge. It encompasses the integration of H\"older differential forms introduced by R. Z\"ust: if $f$, $g_1, \dots, g_d$ are merely H\"older continuous functions on the cube $[0, 1]^d$ whose H\"older exponents satisfy a certain condition, it is possible to interpret $\mathrm{d}g_1 \wedge \cdots \wedge \mathrm{d}g_d$ as a H\"older 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.