{"title":"L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus","authors":"Ciprian Demeter, Pierre Germain","doi":"10.1017/s0013091524000099","DOIUrl":null,"url":null,"abstract":"We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their <jats:italic>L</jats:italic><jats:sup>2</jats:sup> to <jats:italic>L<jats:sup>p</jats:sup></jats:italic> operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000099_inline1.png\" /> <jats:tex-math>$\\ell^2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> decoupling, small cap decoupling and estimates of exponential sums.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0013091524000099","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.
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