New characterizations of Muckenhoupt $A_p$ distance weights for $p>1$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918147386245120 |
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| author | Vargas, Ignacio Gómez |
| author_facet | Vargas, Ignacio Gómez |
| contents | We characterize the collection of sets $E \subset \mathbb{R}^n$ for which there exists $θ\in \mathbb{R}\setminus\{0\}$ such that the distance weight $w(x) = \operatorname{dist}(x, E)^θ$ belongs to the Muckenhoupt class $A_p$, where $p > 1$. These sets exhibit a certain balance between the small-scale and large-scale pores that constitute their complement$-$a property we show to be more general than the so-called weak porosity condition, which in turn, and according to recent results, characterizes the sets with associated distance weights in the $A_1$ case. Furthermore, we verify the agreement between this new characterization and the properties of known examples of distance weights, that are either $A_p$ weights or merely doubling weights, by means of a probabilistic approach that may be of interest by itself. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_18805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New characterizations of Muckenhoupt $A_p$ distance weights for $p>1$ Vargas, Ignacio Gómez Classical Analysis and ODEs 42B37, 42B25, 28A75, 42B25 We characterize the collection of sets $E \subset \mathbb{R}^n$ for which there exists $θ\in \mathbb{R}\setminus\{0\}$ such that the distance weight $w(x) = \operatorname{dist}(x, E)^θ$ belongs to the Muckenhoupt class $A_p$, where $p > 1$. These sets exhibit a certain balance between the small-scale and large-scale pores that constitute their complement$-$a property we show to be more general than the so-called weak porosity condition, which in turn, and according to recent results, characterizes the sets with associated distance weights in the $A_1$ case. Furthermore, we verify the agreement between this new characterization and the properties of known examples of distance weights, that are either $A_p$ weights or merely doubling weights, by means of a probabilistic approach that may be of interest by itself. |
| title | New characterizations of Muckenhoupt $A_p$ distance weights for $p>1$ |
| topic | Classical Analysis and ODEs 42B37, 42B25, 28A75, 42B25 |
| url | https://arxiv.org/abs/2507.18805 |