度量测度空间上Sobolev空间与有界变差空间的弱型刻画
Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces
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中文总结 AI 辅助
本文在满足Poincaré不等式的完备加倍度量测度空间上，证明了无需预先假设光滑性的Sobolev空间与有界变差空间的弱型刻画，补充了相关范数等价性的适用范围。
中文摘要 AI 辅助
给定满足Poincaré不等式的完备加倍度量测度空间$(X,\rho,\boldsymbol{\u03bc})$，我们证明了Sobolev空间$\boldsymbol{\u1e8f}^{1,p}(\boldsymbol{\u03bc})$与有界变差空间的弱型刻画，在一般Poincaré空间中实现了与Brezis等人[Anal. PDE 17 (2024), 943-979]的欧氏结果的完全类比。本文的主要创新点在于：弱型范数的有限性仅涉及函数$f$的差或平均振荡，无需预先假设任何光滑性，即可保证$f$属于对应的Sobolev或BV空间。这一成果区别于Dai等人[Adv. Math. 502 (2026), Paper No. 111153]的近期工作，后者在预先假设$f$为Lipschitz函数的条件下得到了相关范数等价性。本文方法的关键中间步骤是一种新的局部化Bourgain-Brezis-Mironescu型刻画。更准确地说，我们证明：若$p\in(1,\infty)$且$\gamma\in\mathbb{R}\setminus\{0\}$，则对任意$f\in L^1_{\mathrm{loc}}(\boldsymbol{\u03bc})$，有$\\|f\\|_{\boldsymbol{\u1e8f}^{1,p}(\boldsymbol{\u03bc})}\sim\\|\rho^{-1}\phi^{-\gamma}F\\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}$，其中$F\in\{\Delta f,m_f\}$，$\phi\in\{\rho,V\}$；此处齐次Sobolev空间$\boldsymbol{\u1e8f}^{1,p}(\boldsymbol{\u03bc})$由最小$p$-弱上梯度定义，对任意$x,y\in X$，记$V(x,y):=\boldsymbol{\u03bc}(B(x,\rho(x,y)))$，$\Delta f(x,y):=|f(x)-f(y)|$，$m_f(x,y)$为$f$在球$B(x,\rho(x,y))$上的平均振荡。当$p=1$时，等价式$(*)$成立，只需将$\\|f\\|_{\boldsymbol{\u1e8f}^{1,1}(\boldsymbol{\u03bc})}$替换为有界变差范数，并将参数限制在最优范围：当$\phi=\rho$时，$\gamma\in(-\infty,-1)\cup(0,\infty)$；当$\phi=V$时，$\gamma\in(-\infty,-\frac{1}{d})\cup(0,\infty)$，其中$d\in(0,\infty)$为$X$的下维数。
英文摘要
Given a complete doubling metric measure space $(X,ρ,μ)$ supporting a Poincaré inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(μ)$ 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 $γ\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(μ)$, \begin{equation*}\tag{$*$} \|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 $\dot{W}^{1,p}(μ)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=μ(B(x,ρ(x,y)))$ and $Δf(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,ρ(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(μ)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $γ\in(-\infty,-1)\cup(0,\infty)$ for $ϕ=ρ$ or $γ\in (-\infty,-\frac1d)\cup(0,\infty)$ for $ϕ=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.
