A central property of harmonic functions in Euclidean domains is the mean value property. Although it is very specific to the Euclidean setting - and need not hold on more general Riemannian manifolds - the mv property can be stated using only the underlying metric and measure. In this talk I will introduce an extension of this, asymptotic mean value (amv) harmonicity, which can be defined on any metric measure space and which characterizes harmonicity in Riemannian manifolds. I will describe connections between harmonicity and amv harmonicity beyond the smooth setting, in Heisenberg groups and RCD spaces, and also present some regularity results for amv harmonic functions on general doubling metric measure spaces. This is joint work with T. Adamowicz and A. Kijowski.

Meeting ID: 844 1062 0269 Passcode: 282091