Characterisations of Sobolev spaces and constant functions over metric spaces

Abstract: The defect'', that the fractional Sobolev norm $\|f\|_{\dot W^{s,p}}$, for $s\in(0,1)$, fails to converge to the first-order Sobolev norm $\|f\|_{\dot W^{1,p}}$ as $s\to 1$, is a source of two types of substitute results: (1) characterisations of constant functions by integral conditions obtained by na\"ive substitution of $s=1$ into the formula of $\|f\|_{\dot W^{s,p}}$, and (2) derivative-free characterisations of $\dot W^{1,p}$ by suitable modifications of the said formula. In this paper, we obtain extensions of both types of results for the versions of these function spaces over any complete doubling metric measure space $(X,\rho,\mu)$ supporting a $(1,p)$-Poincar\'e inequality. As a key tool of independent potential, we introduce a newmacroscopic'' Poincar\'e inequality, whose right-hand side has oscillations of the same form as the left-hand side, but at a smaller macroscopic scale $r\in(0,R)$. Besides intrinsic interest, these results are motivated by applications to quantitative compactness properties of commutators $[T,f]$ of singular integrals and pointwise multipliers. With pivotal use of the present results, a characterisation of commutator mapping properties, over the same class of general domains $(X,\rho,\mu)$, is obtained in a companion paper.