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 $(X,d,μ)$ 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.