{"paper":{"title":"Boundedness of Pseudodifferential Operators on Banach Function Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Alexei Yu. Karlovich","submitted_at":"2013-09-02T09:11:00Z","abstract_excerpt":"We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\\mathbb{R}^n)$ and on its associate space $X'(\\mathbb{R}^n)$, then a pseudodifferential operator $\\operatorname{Op}(a)$ is bounded on $X(\\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\\\"ormander class $S_{\\rho,\\delta}^{n(\\rho-1)}$ with $0<\\rho\\le 1$, $0\\le\\delta<1$ or to the the Miyachi class $S_{\\rho,\\delta}^{n(\\rho-1)}(\\varkappa,n)$ with $0\\le\\delta\\le\\rho\\le 1$, $0\\le\\delta<1$, and $\\varkappa>0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\\cdot)}(\\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0328","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"}