{"paper":{"title":"Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.FA","authors_text":"Dachun Yang, Pingxu Hu, Wen Yuan, Yangyang Zhang, Yinqin Li","submitted_at":"2025-05-22T01:30:42Z","abstract_excerpt":"Let $X$ be a ball Banach function space on $\\mathbb{R}^n$, $k\\in\\mathbb{N}$, $h\\in\\mathbb{R}^n$, and $\\Delta^k_h$ denote the $k${\\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that, for both parameters $q$ and $\\gamma$ in \\emph{sharp} ranges which are related to $X$ and for any locally integrable function $f$ on ${\\mathbb{R}^n}$ satisfying $|\\nabla^k f|\\in X$, $$ \\sup_{\\lambda\\in(0,\\infty)}\\lambda \\left\\|\\left[\\int_{\\{h\\in\\mathbb{R}^n:\\ |\\Delta_h^k f(\\cdot)|>\\lambda|h|^{k+\\frac{\\gamma}{q}}\\}} \\left|h\\right|^{\\gamma-n}\\,dh\\right]^\\frac{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.16110","kind":"arxiv","version":1},"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/2505.16110/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"}