{"title":"Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups","authors":"Ji Li , Xiao Xiong , Fulin Yang","doi":"10.1016/j.matpur.2024.05.013","DOIUrl":null,"url":null,"abstract":"<div><p>We provide full characterization of the Schatten properties of <span><math><mo>[</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>,</mo><mi>T</mi><mo>]</mo></math></span>, the commutator of Calderón–Zygmund singular integral <em>T</em> with symbol <em>b</em> <span><math><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>b</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> on stratified Lie groups <span><math><mi>G</mi></math></span>. We show that, when <em>p</em> is larger than the homogeneous dimension <span><math><mi>Q</mi></math></span> of <span><math><mi>G</mi></math></span>, the Schatten <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> norm of the commutator is equivalent to the Besov semi-norm <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>Q</mi><mo>/</mo><mi>p</mi></mrow></msubsup></math></span> of the function <em>b</em>; but when <span><math><mi>p</mi><mo>≤</mo><mi>Q</mi></math></span>, the commutator belongs to <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> if and only if <em>b</em> is a constant. For the endpoint case at the critical index <span><math><mi>p</mi><mo>=</mo><mi>Q</mi></math></span>, we further show that the Schatten <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>Q</mi><mo>,</mo><mo>∞</mo></mrow></msub></math></span> norm of the commutator is equivalent to the Sobolev norm <span><math><msup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mn>1</mn><mo>,</mo><mi>Q</mi></mrow></msup></math></span> of <em>b</em>. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"188 ","pages":"Pages 73-113"},"PeriodicalIF":2.1000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782424000606","RegionNum":1,"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 provide full characterization of the Schatten properties of , the commutator of Calderón–Zygmund singular integral T with symbol b on stratified Lie groups . We show that, when p is larger than the homogeneous dimension of , the Schatten norm of the commutator is equivalent to the Besov semi-norm of the function b; but when , the commutator belongs to if and only if b is a constant. For the endpoint case at the critical index , we further show that the Schatten norm of the commutator is equivalent to the Sobolev norm of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.