A remark on a paper by F. Chiarenza and M. Frasca

Pub Date : 2024-04-19 DOI:10.1090/proc/16885

N. Krylov

{"title":"A remark on a paper by F. Chiarenza and M. Frasca","authors":"N. Krylov","doi":"10.1090/proc/16885","DOIUrl":null,"url":null,"abstract":"<p>In 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue b u EndAbsoluteValue Superscript p\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>b</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|bu|^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> through the integral of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper D u EndAbsoluteValue Superscript p\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|Du|^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Formally their proof is valid only for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We present here further generalization with a different proof in which <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced with the fractional power of the Laplacian for any dimension <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d\\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16885","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

In 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of | b u | p |bu|^{p} through the integral of | D u | p |Du|^{p} for p > 1 p>1 . Formally their proof is valid only for d ≥ 3 d\geq 3 . We present here further generalization with a different proof in which D D is replaced with the fractional power of the Laplacian for any dimension d ≥ 2 d\geq 2 .

查看原文

对 F. Chiarenza 和 M. Frasca 论文的评论

1990 年，F. Chiarenza 和 M. Frasca 发表论文，将 C. Fefferman 关于 p > 1 p>1 时通过 | D u | p |Du|^{p} 的积分来估计 | b u | p |bu|^{p} 的积分的结果加以推广。形式上，他们的证明仅对 d ≥ 3 d\geq 3 有效。我们在这里用一个不同的证明来进一步概括，在这个证明中，对于任何维度 d ≥ 2 d\geq 2，D D 被替换为拉普拉奇的分数幂。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文