Regularity of harmonic functions in Cheeger-type Sobolev spaces (Q1771565)

The Cheeger-type Sobolev spaces under discussion are spaces of functions on metric spaces satisfying the doubling condition and some weak Poincaré inequality of type \((1,2)\), equipped with an energy form defined by \textit{J. Cheeger} [Geom. Funct. Anal. 9, 428--517 (1999; Zbl 0942.58018)]. The studies are based on geometric considerations of finite-dimensional Banach spaces with a kind of upper curvature bound on the unit spheres. Assuming that the cotangent space of the source space allows such a bound, Hölder continuity of harmonic (i.e., energy minimizing) functions with respect to the energy form is proven.