| Autor: |
Björn, Anders, Björn, Jana, Christensen, Andreas |
| Rok vydání: |
2022 |
| Předmět: |
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| Zdroj: |
J. Math. Anal. Appl. 539 (2024), Paper No. 128483, 28 pp. (Open choice) |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1016/j.jmaa.2024.128483 |
| Popis: |
A metric space $X$ is called a \emph{bow-tie} if it can be written as $X=X_{+} \cup X_{-}$, where $X_{+} \cap X_{-}=\{x_0\}$ and $X_{\pm} \ne \{x_0\}$ are closed subsets of $X$. We show that a doubling measure $\mu$ on $X$ supports a $(q,p)$--Poincar\'e inequality on $X$ if and only if $X$ satisfies a quasiconvexity-type condition, $\mu$ supports a $(q,p)$-Poincar\'e inequality on both $X_{+}$ and $X_{-}$, and a variational \p-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at $x_0$. In particular, we study the bow-tie $X_{\mathbf{R}^n}$ consisting of the positive and negative hyperquadrants in $\mathbf{R}^n$ equipped with a radial doubling weight and characterize the validity of the \p-Poincar\'e inequality on $X_{\mathbf{R}^n}$ in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin. |
| Databáze: |
arXiv |
| Externí odkaz: |
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