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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1701.04500","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-17T01:00:18Z","cross_cats_sorted":[],"title_canon_sha256":"adcca9f4b1d671a696eb1a16078803b20029b13cd371caa7269053696adf4c88","abstract_canon_sha256":"71aec53dcc03d659168c6295347cf48da2c3d0a8e0422d4916e1bad133a659de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:43.524749Z","signature_b64":"eSoPJ8q1YtAHklON+YoJVSrEfQH4EAreIyPyCR13wt6vqsKZeona4pSoYsImFsjWEWLlWqOsfRJFE/IJ5mXHAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b399096a0b920d3a26a6061638c7b0d698e4d6b16d91e272fab1b6237c1bd02","last_reissued_at":"2026-05-18T00:52:43.524209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:43.524209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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