Characterisations of Sobolev spaces and constant functions over metric spaces

In a doubling metric measure space $(X,ρ,μ)$ 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 $(X,ρ,μ)$, is obtained in a companion paper.

💡 Research Summary

The paper investigates first‑order Sobolev spaces and constant functions on very general metric measure spaces (X,ρ,μ) that are doubling and support a (1,p)‑Poincaré inequality. The authors introduce a “macroscopic” Poincaré inequality, which relates oscillations of a function on a large scale to oscillations on a smaller scale uniformly over all pairs of scales. This tool is pivotal for the two main results.

The first main theorem (Theorem 1.1) provides a derivative‑free characterisation of the homogeneous Hajłasz–Sobolev space (\dot M_{1,p}(μ)). For a locally integrable function f, define the mean‑oscillation function
(m_f(x,t)=\frac{1}{\mu(B(x,t))}\int_{B(x,t)}|f(y)-\langle f\rangle_{B(x,t)}|,d\mu(y))
and the product measure (\nu_p) on (X\times(0,\infty)) by (d\nu_p(x,t)=\frac{d\mu(x),dt}{t^{p+1}}). The theorem states that
\