{"title":"曲线上平均值的平滑特性","authors":"Hyerim Ko, Sanghyuk Lee, Sewook Oh","doi":"10.1017/fmp.2023.2","DOIUrl":null,"url":null,"abstract":"Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve \n$\\gamma $\n in \n$\\mathbb R^d$\n , \n$d\\ge 3$\n . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal \n$L^p$\n Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every \n$d\\ge 3$\n . As a result, we establish, for the first time, nontrivial \n$L^p$\n boundedness of the maximal average over dilations of \n$\\gamma $\n for \n$d\\ge 4$\n .","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"Sharp smoothing properties of averages over curves\",\"authors\":\"Hyerim Ko, Sanghyuk Lee, Sewook Oh\",\"doi\":\"10.1017/fmp.2023.2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve \\n$\\\\gamma $\\n in \\n$\\\\mathbb R^d$\\n , \\n$d\\\\ge 3$\\n . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal \\n$L^p$\\n Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every \\n$d\\\\ge 3$\\n . As a result, we establish, for the first time, nontrivial \\n$L^p$\\n boundedness of the maximal average over dilations of \\n$\\\\gamma $\\n for \\n$d\\\\ge 4$\\n .\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2021-05-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/fmp.2023.2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fmp.2023.2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
Sharp smoothing properties of averages over curves
Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve
$\gamma $
in
$\mathbb R^d$
,
$d\ge 3$
. Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal
$L^p$
Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every
$d\ge 3$
. As a result, we establish, for the first time, nontrivial
$L^p$
boundedness of the maximal average over dilations of
$\gamma $
for
$d\ge 4$
.