{"title":"Local cone multipliers and Cauchy–Szegö projections in bounded symmetric domains","authors":"Fernando Ballesta Yagüe, Gustavo Garrigós","doi":"10.1112/jlms.12986","DOIUrl":null,"url":null,"abstract":"<p>We show that the cone multiplier satisfies local <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mi>p</mi>\n </msup>\n <annotation>$L^p$</annotation>\n </semantics></math>-<span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mi>q</mi>\n </msup>\n <annotation>$L^q$</annotation>\n </semantics></math> bounds only in the trivial range <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>⩽</mo>\n <mi>q</mi>\n <mo>⩽</mo>\n <mn>2</mn>\n <mo>⩽</mo>\n <mi>p</mi>\n <mo>⩽</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$1\\leqslant q\\leqslant 2\\leqslant p\\leqslant \\infty$</annotation>\n </semantics></math>. To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we answer negatively a question by Békollé and Bonami, regarding the continuity from <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>L</mi>\n <mi>p</mi>\n </msup>\n <mo>→</mo>\n <msup>\n <mi>L</mi>\n <mi>q</mi>\n </msup>\n </mrow>\n <annotation>$L^p\\rightarrow L^q$</annotation>\n </semantics></math> of the Cauchy–Szegö projections associated with a class of bounded symmetric domains in <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>n</mi>\n </msup>\n <annotation>${\\mathbb {C}}^n$</annotation>\n </semantics></math> with rank <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$r\\geqslant 2$</annotation>\n </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 4","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12986","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12986","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We show that the cone multiplier satisfies local - bounds only in the trivial range . To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we answer negatively a question by Békollé and Bonami, regarding the continuity from of the Cauchy–Szegö projections associated with a class of bounded symmetric domains in with rank .
我们证明，锥乘法器仅在微不足道的范围 1 ⩽ q ⩽ 2 ⩽ p ⩽ ∞ 1\leqslant q\leqslant 2\leqslant p\leqslant \infty$ 中满足局部 L p $L^p$ - L q $L^q$ 约束。为此，我们把费弗曼对球乘法器的证明适当地调整到这个环境中。因此，我们否定地回答了贝科雷和博纳米提出的一个问题，即从 L p → L q $L^p\rightarrow L^q$ 与 C n 中一类秩为 r ⩾ 2 $r\geqslant 2$ 的有界对称域相关的考奇-塞戈投影的连续性问题。
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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