Characterisations of Sobolev spaces and constant functions over metric spaces
Tuomas Hytönen, Riikka Korte
TL;DR
This work extends derivative-free and constant-recognition approaches for Sobolev-type spaces from Euclidean settings to general complete doubling metric measure spaces $(X,\rho,\mu)$ that support a $(1,p)$-Poincaré inequality. Central to the analysis is a novel macroscopic Poincaré inequality, which controls mean oscillations at a macroscopic scale by oscillations at a smaller scale, enabling a robust link between the Hajłasz–Sobolev space $\dot M^{1,p}(\mu)$ and a Besov-type oscillation functional $m_f$ measured in the weak-$L^p$ scale $L^{p,\infty}(\nu_p)$. The authors prove that $f\in \dot M^{1,p}(\mu)$ iff $m_f\in L^{p,\infty}(\nu_p)$, with equivalence of norms, and obtain both a lower bound (via a direct adaptation of Frank) and an upper bound (via Lipschitz approximation and the macroscopic inequality); they also show a sharp limit relation for the $s\to1$ transition in this general setting. Consequences include a local characterisation of constants through Besov-type norms and a framework for analyzing commutator mappings in general domains, with a companion paper extending these ideas to quantitative commutator compactness on the same class of 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 naïve 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 $(X,ρ,μ)$ supporting a $(1,p)$-Poincaré inequality. As a key tool of independent potential, we introduce a new ``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 $[T,f]$ 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 $(X,ρ,μ)$, is obtained in a companion paper.