Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912690438406144 |
|---|---|
| author | Kline, Josh Lahti, Panu Li, Jiang Zhou, Xiaodan |
| author_facet | Kline, Josh Lahti, Panu Li, Jiang Zhou, Xiaodan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces Kline, Josh Lahti, Panu Li, Jiang Zhou, Xiaodan Functional Analysis Metric Geometry 30L15 46E36 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. |
| title | Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces |
| topic | Functional Analysis Metric Geometry 30L15 46E36 |
| url | https://arxiv.org/abs/2511.04187 |