{"paper":{"title":"On an estimate of Calder\\'on-Zygmund operators by dyadic positive operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Andrei K. Lerner","submitted_at":"2012-02-09T00:29:58Z","abstract_excerpt":"Given a general dyadic grid ${\\mathscr{D}}$ and a sparse family of cubes ${\\mathcal S}=\\{Q_j^k\\}\\in {\\mathscr{D}}$, define a dyadic positive operator ${\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}$ by $${\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}f(x)=\\sum_{j,k}f_{Q_j^k}\\chi_{Q_j^k}(x).$$ Given a Banach function space $X({\\mathbb R}^n)$ and the maximal Calder\\'on-Zygmund operator $T_{\\natural}$, we show that $$\\|T_{\\natural}f\\|_X\\le c(n,T)\\sup_{{\\mathscr{D}},{\\mathcal S}}\\|{\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}f\\|_{X}.$$\n  This result is applied to weighted inequalities. In particular, it implies:"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1860","kind":"arxiv","version":3},"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"}