{"paper":{"title":"An Hardy estimate for commutators of pseudo-differential operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ha Duy Hung, Luong Dang Ky","submitted_at":"2014-06-28T12:42:29Z","abstract_excerpt":"Let $T$ be a pseudo-differential operator whose symbol belongs to the H\\\"ormander class $S^m_{\\rho,\\delta}$ with $0\\leq \\delta<1, 0< \\rho\\leq 1, \\delta \\leq \\rho$ and $-(n+1)< m \\leq - (n+1)(1-\\rho)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\\sup_{{\\rm balls}\\; B\\subset \\mathbb R^n} \\frac{\\log(e+ 1/|B|)}{(1+ |B|)^\\theta} \\frac{1}{|B|}\\int_{B} \\Big|f(x)- \\frac{1}{|B|}\\int_{B} f(y) dy\\Big|dx <\\infty$$ for some $\\theta\\in [0,\\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\\mathbb R^n)$ introduced by Goldberg \\cite{Go}.\n  A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7393","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"}