Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces (Q1403843)

Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces is established by means of properties of the heat kernel. The metric spaces under consideration are those endowed with a doubling measure supporting a \((1,2)\)-Poincaré inequality and the corresponding Sobolev-Poincaré type inequality for the modification of the measure obtained via the heat kernel. Sharp examples are also provided showing that in the general settings the best possible regularity for the Cheeger-harmonic functions is the Lipschitz one.