{"paper":{"title":"Norm estimates of the partial derivatives for harmonic and harmonic elliptic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"SH. Chen, S. Ponnusamy, X. Wang","submitted_at":"2020-08-26T13:34:47Z","abstract_excerpt":"Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\\mathbb{T}$ and $\\dot{F}\\in L_p(0, 2\\pi)$, where $\\dot{F}(e^{it})=\\frac{d}{dt} F(e^{it})$ and $p\\geq 1$. Recently, the author in \\cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\\leq p< 2$, then $f_{z}$ and $\\overline{f_{\\overline{z}}}\\in \\mathcal{B}^{p}(\\mathbb{D}),$ the classical Bergman spaces of $\\mathbb{D}$ \\cite[Theorem 1.2]{Zhu}; $(2)$ if $f$ is a harmonic quasiregular mapping and $1\\leq p\\leq \\infty$, then $f_{z},$ $\\overline{f_{\\overli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.11553","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.11553/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}