{"paper":{"title":"Estimates in Beurling--Helson type theorems. Multidimensional case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Vladimir Lebedev","submitted_at":"2012-01-02T00:34:00Z","abstract_excerpt":"We consider the spaces $A_p(\\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\\mathbb T^m$ such that the sequence of the Fourier coefficients $\\hat{f}=\\{\\hat{f}(k), ~k \\in \\mathbb Z^m\\}$ belongs to $l^p(\\mathbb Z^m), ~1\\leq p<2$. The norm on $A_p(\\mathbb T^m)$ is defined by $\\|f\\|_{A_p(\\mathbb T^m)}=\\|\\hat{f}\\|_{l^p(\\mathbb Z^m)}$. We study the rate of growth of the norms $\\|e^{i\\lambda\\varphi}\\|_{A_p(\\mathbb T^m)}$ as $|\\lambda|\\rightarrow \\infty, ~\\lambda\\in\\mathbb R,$ for $C^1$ -smooth real functions $\\varphi$ on $\\mathbb T^m$ (the one-dimensional case was investigated by the au"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0403","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"}