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

Abstract

In the setting of a complete, doubling metric measure space $(X,d,\mu )$ supporting a $(1,1)$-Poincaré inequality, we show that for all $0<\theta <1$, the following fractional Poincaré 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 $(1,1)$-Poincaré 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 Poincaré inequality. We also show that $(X,d,\mu )$ supports a $(1,1)$-Poincaré inequality if and only if the above fractional Poincaré 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.

54 pages, 1 figure