{"paper":{"title":"On singularity of $p$-energy measures on metric measure spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.FA","authors_text":"Meng Yang","submitted_at":"2025-05-18T15:46:23Z","abstract_excerpt":"For $p>1$, we prove that, for a $p$-energy on a volume doubling metric measure space, the Poincar\\'e inequality and the cutoff Sobolev inequality, both with $p$-walk dimension strictly larger than $p$, imply that the associated $p$-energy measure is singular with respect to the underlying measure. Under the slow volume regularity condition, we further prove that these two inequalities are equivalent to the resistance estimate; in particular, as part of the proof, we give a simple and direct derivation of the cutoff Sobolev inequality from the Poincar\\'e inequality and the capacity upper bound."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.12468","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/2505.12468/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"}