{"title":"Endpoint $\\ell^r$ improving estimates for prime averages","authors":"M. Lacey, H. Mousavi, Yaghoub Rahimi","doi":"10.4310/mrl.2022.v29.n6.a6","DOIUrl":null,"url":null,"abstract":"Let $ \\Lambda $ denote von Mangoldt's function, and consider the averages \\begin{align*} A_N f (x)&=\\frac{1}{N}\\sum_{1\\leq n \\leq N}f(x-n)\\Lambda(n) . \\end{align*} We prove sharp $ \\ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\\subset [0,N]$ there holds \\begin{equation*} N ^{-1} \\langle A_N \\mathbf 1_{F} , \\mathbf 1_{G} \\rangle \\ll \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigl( \\operatorname {Log} \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigr) ^{t}, \\end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ \\sup_N A_N \\mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Research Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/mrl.2022.v29.n6.a6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
Let $ \Lambda $ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x)&=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)\Lambda(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\subset [0,N]$ there holds \begin{equation*} N ^{-1} \langle A_N \mathbf 1_{F} , \mathbf 1_{G} \rangle \ll \frac{\lvert F\rvert \cdot \lvert G\rvert} { N ^2 } \Bigl( \operatorname {Log} \frac{\lvert F\rvert \cdot \lvert G\rvert} { N ^2 } \Bigr) ^{t}, \end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ \sup_N A_N \mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.
期刊介绍:
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