Quantitative harmonic approximations and Dorronsoro's Theorem in metric measure spaces

TL;DR

This paper characterizes Newtonian-Sobolev spaces on RCD spaces through harmonic approximation measures, extending Dorronsoro's Theorem to metric measure spaces and providing new insights even in Euclidean settings.

Contribution

It introduces novel characterizations of Sobolev spaces via harmonic approximation in RCD spaces, including a new proof of Dorronsoro's Theorem in Euclidean spaces.

Findings

New harmonic approximation characterization of Sobolev spaces.

Extension of Dorronsoro's Theorem to RCD metric measure spaces.

Provides a new proof of a special case of Dorronsoro's Theorem in Euclidean space.

Abstract

Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally (across all scales and locations) well-approximated by harmonic functions. A similar characterisation is obtained which further takes into account the local oscillations of the approximating harmonic functions. The first characterisation is new even when $X = \mathbb{R}^n$; the second characterisation is a version of Dorronsoro's Theorem in RCD spaces and gives a new proof of (a special case) of this theorem in Euclidean space.