Harmonic functions on metric measure spaces

In the talk we introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. We discuss some of the properties of such functions including Harnack estimates on balls and compact sets, maximum and comparison principles, the H\"older and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. Furthermore, we present the Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method. We employ the Perron method to construct harmonic functions with continuous boundary data.

The properties of the underlying measure play important role in our investigations and, in particular, we discuss results for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition.

If time permits, we will discuss the mean value-harmonic functions in the setting of the Carnot--Carath\'eodory groups, focusing on regularity and relations of such functions to the sub-Laplace equation.