Doubling measures and Poincaré inequalities for sphericalizations of metric spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913988549279744 |
|---|---|
| author | Björn, Anders Björn, Jana Li, Xining |
| author_facet | Björn, Anders Björn, Jana Li, Xining |
| contents | The identification between the complex plane and the Riemann sphere preserves holomorphic and harmonic functions and is a classical tool. In this paper we consider a similar mapping from an unbounded metric space $X$ to a bounded space and show how it preserves $p$-harmonic functions and Poincaré inequalities. When $X$ is Ahlfors regular, this was shown in our earlier paper (J. Math. Anal. Appl. 474 (2019), 852-875). Here we only require the much weaker (and more natural) doubling property of the measure. Furthermore, we consider a broader class of transformed measures. The sphericalization is then applied to obtain new results for the Dirichlet boundary value problem in unbounded sets and for boundary regularity at infinity for $p$-harmonic functions. Some of these results are new also for unweighted $\mathbf{R}^n$, $n \ge 2$ and $p\ne2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09795 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Doubling measures and Poincaré inequalities for sphericalizations of metric spaces Björn, Anders Björn, Jana Li, Xining Functional Analysis Analysis of PDEs Primary: 31E05, Secondary: 26D10, 30L10, 30L15, 31C45, 35J66, 35J92, 49Q20 The identification between the complex plane and the Riemann sphere preserves holomorphic and harmonic functions and is a classical tool. In this paper we consider a similar mapping from an unbounded metric space $X$ to a bounded space and show how it preserves $p$-harmonic functions and Poincaré inequalities. When $X$ is Ahlfors regular, this was shown in our earlier paper (J. Math. Anal. Appl. 474 (2019), 852-875). Here we only require the much weaker (and more natural) doubling property of the measure. Furthermore, we consider a broader class of transformed measures. The sphericalization is then applied to obtain new results for the Dirichlet boundary value problem in unbounded sets and for boundary regularity at infinity for $p$-harmonic functions. Some of these results are new also for unweighted $\mathbf{R}^n$, $n \ge 2$ and $p\ne2$. |
| title | Doubling measures and Poincaré inequalities for sphericalizations of metric spaces |
| topic | Functional Analysis Analysis of PDEs Primary: 31E05, Secondary: 26D10, 30L10, 30L15, 31C45, 35J66, 35J92, 49Q20 |
| url | https://arxiv.org/abs/2508.09795 |