Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces

TL;DR

This work extends BBM-type fractional analysis to complete, doubling metric measure spaces that support a (1,1)-Poincaré inequality, proving a fractional Poincaré inequality with the sharp (1−θ) scaling for all 0<θ<1. From this, the authors derive fractional relative isoperimetric and boxing inequalities, establish their mutual equivalences, and connect these nonlocal inequalities to lower Ahlfors regularity, yielding global fractional Sobolev and isoperimetric inequalities under suitable regularity. A key nonlocal improvement of the relative isoperimetric inequality (Lemma 3.1) underpins the fractional Poincaré inequality and its equivalences with the relative isoperimetric framework. The paper also shows that fractional Poincaré inequalities for θ near 1 imply, and are implied by, first-order Poincaré inequalities, illustrating a tight bridge between nonlocal and local analyses. Under lower Ahlfors Q-regularity, global BBM-type inequalities are obtained and shown to be equivalent to the regularity condition, unifying nonlocal geometric-analytic inequalities in broad metric-measure spaces, including Euclidean, Riemannian, and sub-Riemannian contexts.

Abstract

In the setting of a complete, doubling metric measure space $(X,d,μ)$ supporting a $(1,1)$-Poincaré inequality, we show that for all $0<θ<1$, the following fractional Poincaré inequality holds for all balls $B$ and locally integrable functions $u$, $$ \int_{B}|u-u_B|dμ\le C(1-θ)\,\text{rad}(B)^θ\int_{τB}\int_{τB}\frac{|u(x)-u(y)|}{d(x,y)^θμ(B(x,d(x,y)))}dμ(y)dμ(x), $$ where $C\ge 1$ and $τ\ge 1$ are constants depending only on the doubling and $(1,1)$-Poincaré inequality constants. Notably, this inequality features the scaling constant $(1-θ)$ present in the Bourgain-Brezis-Mironescu theory characterizing Sobolev functions via nonlocal functionals. From this inequality, we obtain a fractional relative isoperimetric inequality as well as global and local versions of a fractional boxing inequality, each featuring the same scaling constant $(1-θ)$ and defined in terms of the fractional $θ$-perimeter, and prove equivalences with the above fractional Poincaré inequality. We also show that $(X,d,μ)$ supports a $(1,1)$-Poincaré inequality if and only if the above fractional Poincaré inequality holds for all $θ$ sufficiently close to $1$. Under the additional assumption of lower Ahlfors $Q$-regularity of the measure $μ$, we additionally use the aforementioned results to establish global inequalities, in the form of fractional isoperimetric and fractional Sobolev inequalities, which also feature the scaling constant $(1-θ)$. Moreover, we prove that such inequalities are equivalent with the lower Ahlfors $Q$-regularity condition on the measure.

Figures (1)

• Figure 1: Inequality implications

Theorems & Definitions (46)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Theorem 1.8
• Lemma 2.1
• proof