{"paper":{"title":"On the maximum principle for the Riesz transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Fedor Nazarov, Vladimir Eiderman","submitted_at":"2017-01-17T01:00:18Z","abstract_excerpt":"Let $\\mu$ be a measure in $\\mathbb R^d$ with compact support and continuous density, and let $$ R^s\\mu(x)=\\int\\frac{y-x}{|y-x|^{s+1}}\\,d\\mu(y),\\ \\ x,y\\in\\mathbb R^d,\\ \\ 0<s<d. $$ We consider the following conjecture: $$ \\sup_{x\\in\\mathbb R^d}|R^s\\mu(x)|\\le C\\sup_{x\\in\\text{supp}\\,\\mu}|R^s\\mu(x)|,\\quad C=C(d,s). $$ This relation was known for $d-1\\le s<d$, and is still an open problem in the general case. We prove the maximum principle for $0< s<1$, and also for $0<s<d$ in the case of radial measure. Moreover, we show that this conjecture is incorrect for non-positive measures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04500","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"}