Variational Estimates for Nonnegative Harmonic Functions

Abstract

Consider a nonnegative harmonic function $u$ on a domain $D\subset {\mathbb{R}}^{d-1}$. We establish boundedness of Jones' variational integral on an ultradense, $d-1$ dimensional subset of the boundary of $C^{1,\mathrm{Dini}}$ domains. This result was previously only known for the unit ball.