{"paper":{"title":"Harmonic measure and Riesz transform in uniform and general domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mihalis Mourgoglou, Xavier Tolsa","submitted_at":"2015-09-28T16:48:25Z","abstract_excerpt":"Let $\\Omega\\subsetneq\\mathbb R^{n+1}$ be open and let $\\mu$ be some measure supported on $\\partial\\Omega$ such that $\\mu(B(x,r))\\leq C\\,r^n$ for all $x\\in\\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $\\Omega$ satisfies some scale invariant $A_\\infty$ type conditions with respect to $\\mu$, then the $n$-dimensional Riesz transform $$R_\\mu f(x) = \\int \\frac{x-y}{|x-y|^{n+1}}\\,f(y)\\,d\\mu(y)$$ is bounded in $L^2(\\mu)$. We do not assume any doubling condition on $\\mu$. We also consider the particular case when $\\Omega$ is a bounded uniform domain. To this end, we need first to obt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.08386","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"}