{"title":"Construction of a curved Kakeya set","authors":"Tongou Yang, Yue Zhong","doi":"arxiv-2408.01917","DOIUrl":null,"url":null,"abstract":"We construct a compact set in R2 of measure 0 containing a piece of a\nparabola of every aperture between 1 and 2. As a consequence, we improve lower\nbounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a\nrange of $p$, $q$. Moreover, our construction can be generalised from parabolas\nto a family of curves with cinematic curvature.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.01917","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We construct a compact set in R2 of measure 0 containing a piece of a
parabola of every aperture between 1 and 2. As a consequence, we improve lower
bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a
range of $p$, $q$. Moreover, our construction can be generalised from parabolas
to a family of curves with cinematic curvature.
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