Poincaré inequalities and $A_p$ weights on bow-ties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909219926573056 |
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| author | Björn, Anders Björn, Jana Christensen, Andreas |
| author_facet | Björn, Anders Björn, Jana Christensen, Andreas |
| contents | 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 $μ$ on $X$ supports a $(q,p)$--Poincaré inequality on $X$ if and only if $X$ satisfies a quasiconvexity-type condition, $μ$ supports a $(q,p)$-Poincaré 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-Poincaré 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_07491 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Poincaré inequalities and $A_p$ weights on bow-ties Björn, Anders Björn, Jana Christensen, Andreas Metric Geometry Functional Analysis 26D10 (Primary) 30L15, 31C15, 31C45, 31E05, 46E36 (Secondary) 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 $μ$ on $X$ supports a $(q,p)$--Poincaré inequality on $X$ if and only if $X$ satisfies a quasiconvexity-type condition, $μ$ supports a $(q,p)$-Poincaré 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-Poincaré 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. |
| title | Poincaré inequalities and $A_p$ weights on bow-ties |
| topic | Metric Geometry Functional Analysis 26D10 (Primary) 30L15, 31C15, 31C45, 31E05, 46E36 (Secondary) |
| url | https://arxiv.org/abs/2202.07491 |