{"title":"A compact extension of Journé’s 𝑇1 theorem on product spaces","authors":"Mingming Cao, Kôzô Yabuta, Dachun Yang","doi":"10.1090/tran/9206","DOIUrl":null,"url":null,"abstract":"<p>We prove a compact version of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Baseline 1\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding=\"application/x-tex\">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C upper M upper O\"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>M</mml:mi> <mml:mi>O</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">CMO</mml:annotation> </mml:semantics> </mml:math> </inline-formula> condition, and the product <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C upper M upper O\"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>M</mml:mi> <mml:mi>O</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">CMO</mml:annotation> </mml:semantics> </mml:math> </inline-formula> condition, then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding=\"application/x-tex\">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be extended to a compact operator on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p Baseline left-parenthesis w right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^p(w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 greater-than p greater-than normal infinity\"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">1>p>\\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"w element-of upper A Subscript p Baseline left-parenthesis double-struck upper R Superscript n 1 Baseline times double-struck upper R Superscript n 2 Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">w \\in A_p(\\mathbb {R}^{n_1} \\times \\mathbb {R}^{n_2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Even in the unweighted setting, it is the first time to give a compact extension of Journé’s <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Baseline 1\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> theorem on product spaces.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9206","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 prove a compact version of the T1T1 theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator TT admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal CMOCMO condition, and the product CMOCMO condition, then TT can be extended to a compact operator on Lp(w)L^p(w) for all 1>p>∞1>p>\infty and w∈Ap(Rn1×Rn2)w \in A_p(\mathbb {R}^{n_1} \times \mathbb {R}^{n_2}). Even in the unweighted setting, it is the first time to give a compact extension of Journé’s T1T1 theorem on product spaces.
我们证明了双参数奇异积分 T 1 T1 定理的紧凑版本。也就是说，如果双参数奇异积分算子 T T 承认紧凑的全核和偏核表示，并且满足弱紧凑性、对角线 C M O CMO 条件和积 C M O CMO 条件，那么 T T 可以扩展为 L p ( w ) L^p(w) 上的紧凑算子，对于所有 1 >；p > ∞ 1>p>\infty 且 w∈ A p ( R n 1 × R n 2 ) w \in A_p(\mathbb {R}^{n_1} \times\mathbb {R}^{n_2}) .即使在无权设置中，这也是第一次给出儒尔内 T 1 T1 定理在积空间上的紧凑扩展。
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