{"title":"(Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n","authors":"Tianyi Ren","doi":"10.1142/S1664360719500036","DOIUrl":null,"url":null,"abstract":"We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2019-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/S1664360719500036","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 1
Abstract
We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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