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

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.