{"paper":{"title":"On weighted norm inequalities for the Carleson and Walsh-Carleson operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Andrei K. Lerner, Francesco Di Plinio","submitted_at":"2013-12-03T14:19:31Z","abstract_excerpt":"We prove $L^p(w)$ bounds for the Carleson operator ${\\mathcal C}$, its lacunary version $\\mathcal C_{lac}$, and its analogue for the Walsh series $\\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\\le q\\le p$. In particular, we show that, exactly as for the Hilbert transform, $\\|{\\mathcal C}\\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\\le q<p$.\n  We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \\cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$.\n  Our approach works in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0833","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"}