{"paper":{"title":"On Fourier transforms of fractal measures on the parabola","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Aleksi Py\\\"or\\\"al\\\"a, Carmelo Puliatti, Tuomas Orponen","submitted_at":"2024-01-31T14:26:13Z","abstract_excerpt":"Let $s \\in [0,1]$ and $t \\in [0,\\min\\{3s,s + 1\\})$. Let $\\sigma$ be a Borel measure supported on the parabola $\\mathbb{P} = \\{(x,x^{2}) : x \\in [-1,1]\\}$ satisfying the $s$-dimensional Frostman condition $\\sigma(B(x,r)) \\leq r^{s}$. Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \\geq 1$ such that $$\\|\\hat{\\sigma}\\|_{L^{p}(B(R))} \\leq C_{s,t}R^{(2 - t)/p}, \\qquad R \\geq 1.$$ Moreover, when $s \\geq 2/3$ and $t \\in [0,s + 1)$, the previous inequality is true for $p \\geq 6$.\n  We also obtain the following fractal geometric counterpart of the previous re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.17867","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.17867/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}