{"paper":{"title":"Gradient estimates for heat kernels and harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA","math.DG"],"primary_cat":"math.MG","authors_text":"Adam Sikora, Pekka Koskela, Renjin Jiang, Thierry Coulhon","submitted_at":"2017-03-06T23:53:43Z","abstract_excerpt":"Let $(X,d,\\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\\E$ deriving from a \"carr\\'e du champ\". Assume that $(X,d,\\mu,\\E)$ supports a scale-invariant $L^2$-Poincar\\'e inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\\in (2,\\infty]$:\n  (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup;\n  (ii) $(RH_p)$: $L^p$-reverse H\\\"older inequality for the gradients of harmonic functions;\n  (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\\infty$);\n  (iv) $(GBE)$: a g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02152","kind":"arxiv","version":2},"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"}