Geometric inequalities related to fractional perimeter: fractional Poincar\'e, isoperimetric, and boxing inequalities in metric measure spaces

TL;DR

This paper establishes fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces, revealing their equivalence and dependence on the measure's regularity, with implications for Sobolev functions and geometric analysis.

Contribution

It introduces fractional inequalities with a scaling constant (1-θ) in metric measure spaces and proves their equivalence to classical Poincaré inequalities and measure regularity conditions.

Findings

Fractional Poincaré inequality with (1-θ) scaling constant.

Equivalence between fractional inequalities and measure regularity.

Global fractional isoperimetric and Sobolev inequalities under Ahlfors regularity.

Abstract

In the setting of a complete, doubling metric measure space $(X,d,\mu)$ supporting a $(1,1)$-Poincar\'e inequality, we show that for all $0<\theta<1$, the following fractional Poincar\'e inequality holds for all balls $B$ and locally integrable functions $u$, $$ \int_{B}|u-u_B|d\mu\le C(1-\theta)\,\text{rad}(B)^\theta\int_{\tau B}\int_{\tau B}\frac{|u(x)-u(y)|}{d(x,y)^\theta\mu(B(x,d(x,y)))}d\mu(y)d\mu(x), $$ where $C\ge 1$ and $\tau\ge 1$ are constants depending only on the doubling and $(1,1)$-Poincar\'e inequality constants. Notably, this inequality features the scaling constant $(1-\theta)$ 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 $(1-\theta)$ and defined in terms of the fractional $\theta$-perimeter, and prove equivalences with the above fractional Poincar\'e inequality. We also show that $(X,d,\mu)$ supports a $(1,1)$-Poincar\'e inequality if and only if the above fractional Poincar\'e inequality holds for all $\theta$ sufficiently close to $1$. Under the additional assumption of lower Ahlfors $Q$-regularity of the measure $\mu$, 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 $(1-\theta)$. Moreover, we prove that such inequalities are equivalent with the lower Ahlfors $Q$-regularity condition on the measure.