{"paper":{"title":"Hilbert Transformation and Representation of ax+b Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Hua Liu, Pei Dang, Tao Qian","submitted_at":"2017-11-13T10:51:49Z","abstract_excerpt":"In this paper we study the Hilbert transformations over $L^2(\\mathbb{R})$ and $L^2(\\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\\mathbb{R})$ commutative with the ax+b group we show that the operator is of the form $\\lambda I+\\eta H, $ where $I$ and $H$ are the identity operator and Hilbert transformation respectively, and $\\lambda,\\eta$ are complex numbers. In the related literature this result was proved through first invoking the boundedness result of the operator, proved though a big machinery. In our setting the boundedness is a consequence of the boundednes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04514","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"}