Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

Abstract

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

Asymptotically Mean Value Harmonic Functions in Subriemannian and RCD Settings

“…In this paper, we continue the investigations of and complement results from [2] by considering functions with the amv property in subriemannian and RCD settings. As in [2] we consider two notions of asymptotically mean value harmonic (amv-harmonic) functions, arising from different ways to interpret the limit in (1). More precisely, we define strong and weak amv-harmonicity as follows.…”

“…In the claim above H W 1,2 loc ( ) denotes the horizontal Sobolev space and G is the subelliptic laplacian (13). Theorem 1.2 characterizes amv-harmonicity (in both senses) for a large class of pseudonorms, namely those satisfying (2), and in particular for the Koranyi gauge (12). See [26] for similar results for p-harmonic functions on Carnot groups with the Koranyi gauge.…”

“…Such a result would need very good control over the rate of change of the H N -measure of balls centered at nearby points: the presence of a weight (e.g. the setting of weakly non-collapsed RCD-spaces, see [24]) rules out the possibility of having even the weak amv property for harmonic functions, see [2,Sect. 6].…”

“…Proposition 3.2 Suppose X is a metric measure space where the Korevaar-Schoen energy E 2 K S exists in the sense of Definition 2.2. Then the following are equivalent for a function u ∈ K S 1,2 loc ( ) in a domain ⊂ X:…”

“…For recent work on amv functions on general metric measure spaces we refer the reader to [42,43] and in particular to [2] where we focus on functions with the amv property in doubling metric measure spaces and show their local Hölder regularity and the mv property for their blow-ups. As an important special case, in [2] we consider weighted Euclidean setting and find an elliptic PDE satisfied by amv-harmonic functions.…”