Characterisations of Sobolev spaces and constant functions over metric spaces

In a doubling metric measure space $(X,\rho ,\mu )$ supporting a Poincaré inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of the finiteness of certain integrals through a new approach. As a key tool of independent potential, we introduce a novel ``macroscopic’’ Poincaré 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 $[f,T]$ of pointwise multipliers and singular integrals. 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.