Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces

Abstract

Given a complete doubling metric measure space $(X,\rho ,\mu )$ supporting a Poincaré inequality, we prove weak-type characterizations of the Sobolev space ${\dot{W}}^{1,p}(\mu )$ and the space of functions of bounded variation, achieving a full analogy in general Poincaré spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in (1,\infty )$ and $\gamma \in \mathbb{R}\setminus \{0\}$, then, for any $f\in L_{\mathrm{loc}}^1(\mu )$, \begin{equation*}\tag{$\ast$} \|f\|_{\dot W^{1,p}(μ)} \sim \|ρ^{-1}ϕ^{-γ}F\|_{L^{p,\infty}(ϕ^{γp}V^{-1})}, \qquad F\in\{Δf,m_f\},\quad ϕ\in\{ρ,V\}, \end{equation*} where the homogeneous Sobolev space ${\dot{W}}^{1,p}(\mu )$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu (B(x,\rho (x,y)))$ and $\Delta f(x,y):=|f(x)-f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho (x,y))$. For $p=1$, the equivalence $(\ast )$ holds after replacing $\Vert f{\Vert }_{{\dot{W}}^{1,1}(\mu )}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma \in (-\infty ,-1)\cup (0,\infty )$ for $\varphi =\rho$ or $\gamma \in (-\infty ,-\frac{1}{d})\cup (0,\infty )$ for $\varphi =V$, where $d\in (0,\infty )$ is the lower dimension of $X$.

arXiv comment47 pages