{"title":"Bochner–Riesz means at the critical index: weighted and sparse bounds","authors":"David Beltran, Joris Roos, Andreas Seeger","doi":"10.1007/s00208-024-02962-1","DOIUrl":null,"url":null,"abstract":"<p>We consider Bochner–Riesz means on weighted <span>\\(L^p\\)</span> spaces, at the critical index <span>\\(\\lambda (p)=d(\\frac{1}{p}-\\frac{1}{2})-\\frac{1}{2}\\)</span>. For every <span>\\(A_1\\)</span>-weight we obtain an extension of Vargas’ weak type (1, 1) inequality in some range of <span>\\(p>1\\)</span>. To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension <span>\\(d= 2\\)</span>; partial results as well as conditional results are proved in higher dimensions. For the means of index <span>\\(\\lambda _*= \\frac{d-1}{2d+2}\\)</span> we prove fully optimal sparse bounds.\n</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-024-02962-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider Bochner–Riesz means on weighted \(L^p\) spaces, at the critical index \(\lambda (p)=d(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}\). For every \(A_1\)-weight we obtain an extension of Vargas’ weak type (1, 1) inequality in some range of \(p>1\). To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension \(d= 2\); partial results as well as conditional results are proved in higher dimensions. For the means of index \(\lambda _*= \frac{d-1}{2d+2}\) we prove fully optimal sparse bounds.
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.