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

Suppose $X$ is an $\mathrm{R}\mathrm{C}\mathrm{D}(K,N)$ space with $K\in \mathbb{R}$ and $N\in (1,\mathrm{\infty })$. 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.