{"paper":{"title":"On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Zhiwen Zhao","submitted_at":"2023-10-02T17:21:34Z","abstract_excerpt":"In this paper, a class of anisotropic weights having the form of $|x'|^{\\theta_{1}}|x|^{\\theta_{2}}|x_{n}|^{\\theta_{3}}$ in dimensions $n\\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal range of $(\\theta_{1},\\theta_{2},\\theta_{3})$ such that this type of weights belongs to the Muckenhoupt class $A_{p}$. Then we further study its doubling property, which shows that it provides an example of a doubling measure but is not in $A_{p}$. As a consequence, we obtain anisotropic weighted Poincar\\'{e} and Sobolev inequalities, which are used to study the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.01359","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.01359/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}