Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

TL;DR

This work extends the Maz'ya–Shaposhnikova endpoint representation to ball quasi-Banach function spaces on spaces of homogeneous type by introducing the weak reverse doubling (WRD) and weak measure density (WMD) conditions. The authors derive upper and lower bounds for the Maz'ya–Shaposhnikova limit via extrapolation from weighted Lebesgue spaces and Fatou arguments, and they show the necessity of WRD/WMD through counterexamples. They further adapt the representation to domains and demonstrate its applicability across ten broad classes of function spaces, including weighted, variable, Lorentz, Orlicz, Morrey, and their generalized block variants. The results unify and extend endpoint Sobolev-type characterizations in non-Euclidean settings and provide a versatile framework for analyzing quasi-norms in diverse function spaces. Notably, several specialized instances recover and generalize known Euclidean results, while many of the applications are new in spaces of homogeneous type. The paper also highlights the sharpness of the WRD/WMD conditions through explicit counterexamples, emphasizing their essential role in the representation theory.

Abstract

Let $Y(\mathcal{X})$ be a ball quasi-Banach function space on the space of homogeneous type $(\mathcal{X},ρ,μ)$ satisfying some mild additional assumptions, $q\in(0,\infty)$, and $\dot{W}^{s,q}_Y(\mathcal{X})$ with $s\in(0,1)$ be the homogeneous fractional Sobolev space associated with $Y(\mathcal{X})$. In this article, we show that, for any $f\in Y(\mathcal{X})\cap\bigcup_{s\in(0,1)} \dot{W}^{s,q}_Y(\mathcal{X})$, \begin{align*} \|f\|_{Y(\mathcal{X})} &\lesssim\varliminf_{s \to 0^+} s^{\frac{1}{q}}\left\| \left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}}\\ &\leq \varlimsup_{s \to 0^+} s^{\frac{1}{q}}\left\|\left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}} \lesssim\|f\|_{Y(\mathcal{X})}, \end{align*} where $U(x,y):=\min\{μ(B(x,ρ(x,y))),\,μ(B(y,ρ(x,y)))\}$ for any $x,y\in\mathcal{X}$ and the implicit positive constants are independent of $f$, which is applied to ten specific ball quasi-Banach function spaces and hence is of wide generality. In particular, when $Y(\mathcal{X})=L^q(\mathbb{R}^n)$ with $q\in[1,\infty)$, the above formula is closely related to the celebrated result of Maz'ya and Shaposhnikova in 2002. We also establish the above representation formula on domains of $\mathcal{X}$. The main novelty lies in proposing two new concepts, namely the weak reverse doubling condition (for $\mathcal{X}$) and the weak measure density condition (for domains of $\mathcal{X}$), which are proved to be necessary in some sense. In addition, we find an interesting fact that, when the underlying space under consideration is bounded, the above Maz'ya--Shaposhnikova-type limit always tends to zero.

Theorems & Definitions (138)

• Theorem 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Definition 1.9
• Remark 1.10