Bochner–Riesz means at the critical index: weighted and sparse bounds

IF 1.3 2区 数学 Q1 MATHEMATICS Mathematische Annalen Pub Date : 2024-09-02 DOI:10.1007/s00208-024-02962-1
David Beltran, Joris Roos, Andreas Seeger
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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.

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临界指数上的 Bochner-Riesz 均值:加权和稀疏边界
我们考虑加权(L^p\ )空间上的博赫纳-里兹手段,在临界指数 \(\lambda (p)=d(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}\).对于每一个(A_1)-权重,我们都会得到瓦尔加斯弱型(1,1)不等式在某个范围内的(p>1\)扩展。为了证明这个结果,我们为稀疏支配建立了新的端点结果。这些结果在维度(d= 2)上几乎是最优的;部分结果以及条件结果在更高维度上也得到了证明。对于索引 \(λ_*= \frac{d-1}{2d+2}\) 的手段,我们证明了完全最优的稀疏边界。
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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
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.
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