{"paper":{"title":"On the norm of the operator $aI+bH$ on $L^p(\\mathbb R)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Kai Zhu, Loukas Grafakos, Yong Ding","submitted_at":"2017-02-16T03:37:33Z","abstract_excerpt":"We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky \\cite{HKV}: let $H$ be the Hilbert transform and let $a,b$ be real constants. Then for $1<p<\\infty$ the norm of the operator $aI+bH$ from $L^p(\\mathbb R)$ to $L^p(\\mathbb R)$ is equal to $$ \\bigg(\\max_{x\\in \\Bbb R}\\frac{|ax-b+(bx+a)\\tan \\frac{\\pi}{2p}|^p+|ax-b-(bx+a)\\tan \\frac{\\pi}{2p}|^p}{|x+\\tan \\frac{\\pi}{2p}|^p+|x-\\tan \\frac{\\pi}{2p}|^p} \\bigg)^{\\frac 1p}. $$ Our proof avoids passing through the analogous result for the conjugate function on the circle, as in \\cite{HKV}, and is given directly on the lin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.04848","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"}