{"paper":{"title":"A family of singular integral operators which control the Cauchy transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Joan Mateu, Petr Chunaev, Xavier Tolsa","submitted_at":"2018-03-07T19:41:14Z","abstract_excerpt":"We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\\frac{(\\Re z)^{3}}{|z|^{4}}+t\\cdot \\frac{\\Re z}{|z|^{2}}, \\quad t\\in \\mathbb{R},\\qquad k_\\infty(z):=\\frac{\\Re z}{|z|^{2}}\\equiv \\Re \\frac{1}{z},\\quad z\\in \\mathbb{C}\\setminus\\{0\\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\\mathcal{H}^1\\lfloor E)$-bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02854","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"}