{"paper":{"title":"Jump and variational inequalities for rough operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guixiang Hong, Honghai Liu, Yong Ding","submitted_at":"2015-08-16T21:57:06Z","abstract_excerpt":"In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones {\\it et al}. More precisely, we show some jump and variational inequalities for the families $\\mathcal T:=\\{T_\\varepsilon\\}_{\\varepsilon>0}$ of truncated singular integrals and $\\mathcal M:=\\{M_t\\}_{t>0}$ of averaging operators with rough kernels, which are defined respectively by $$ T_\\varepsilon f(x)=\\int_{|y|>\\varepsilon}\\frac{\\Omega(y')}{|y|^n}f(x-y)dy $$ and $$M_t f(x)=\\frac1{t^n}\\int_{|y|<t}\\Omega(y')f(x-y)dy, $$ where the kernel $\\Omega$ belongs to $L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03872","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":""},"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"}