{"title":"端点$\\ well ^r$改善素数平均值的估计","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":"{\"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}","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}
Endpoint $\ell^r$ improving estimates for prime averages
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.
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