{"paper":{"title":"Maximally Modulated Singular Integral Operators and their Applications to Pseudodifferential Operators on Banach Function Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Alexei Yu. Karlovich","submitted_at":"2014-08-19T17:22:44Z","abstract_excerpt":"We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\\mathbb{R}^n)$ and on its associate space $X'(\\mathbb{R}^n)$ and a maximally modulated Calder\\'on-Zygmund singular integral operator $T^\\Phi$ is of weak type $(r,r)$ for all $r\\in(1,\\infty)$, then $T^\\Phi$ extends to a bounded operator on $X(\\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\\cdot)}(\\mathbb{R})$ under natural assumptions on the variable exponent $p:\\mathbb{R}\\to(1,\\infty)$. Applications of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4400","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"}