与拉盖尔多项式展开相关的谐波分析算子的终点估计值

Pub Date : 2024-03-27 DOI:10.1002/mana.202300088

Jorge J. Betancor, Estefanía Dalmasso, Pablo Quijano, Roberto Scotto

{"title":"与拉盖尔多项式展开相关的谐波分析算子的终点估计值","authors":"Jorge J. Betancor, Estefanía Dalmasso, Pablo Quijano, Roberto Scotto","doi":"10.1002/mana.202300088","DOIUrl":null,"url":null,"abstract":"In this paper, we give a criterion to prove boundedness results for several operators from the Hardy-type space <math>\n <semantics>\n <mrow>\n <msup>\n <mi>H</mi>\n <mn>1</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <mi>d</mi>\n </msup>\n <mo>,</mo>\n <msub>\n <mi>γ</mi>\n <mi>α</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$H^1((0,\\infty)^d,\\gamma _\\alpha)$</annotation>\n </semantics></math> to <math>\n <semantics>\n <mrow>\n <msup>\n <mi>L</mi>\n <mn>1</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <mi>d</mi>\n </msup>\n <mo>,</mo>\n <msub>\n <mi>γ</mi>\n <mi>α</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$L^1((0,\\infty)^d,\\gamma _\\alpha)$</annotation>\n </semantics></math> and also from <math>\n <semantics>\n <mrow>\n <msup>\n <mi>L</mi>\n <mi>∞</mi>\n </msup>\n <mrow>\n <mo>(</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <mi>d</mi>\n </msup>\n <mo>,</mo>\n <msub>\n <mi>γ</mi>\n <mi>α</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$L^\\infty ((0,\\infty)^d,\\gamma _\\alpha)$</annotation>\n </semantics></math> to the space of functions of bounded mean oscillation <math>\n <semantics>\n <mrow>\n <mi>B</mi>\n <mi>M</mi>\n <mi>O</mi>\n <mo>(</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <mi>d</mi>\n </msup>\n <mo>,</mo>\n <msub>\n <mi>γ</mi>\n <mi>α</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$\\textup {BMO}((0,\\infty)^d,\\gamma _\\alpha)$</annotation>\n </semantics></math>, with respect to the probability measure <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <msub>\n <mi>γ</mi>\n <mi>α</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <msubsup>\n <mo>∏</mo>\n <mrow>\n <mi>j</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <mi>d</mi>\n </msubsup>\n <mfrac>\n <mn>2</mn>\n <mrow>\n <mi>Γ</mi>\n <mo>(</mo>\n <msub>\n <mi>α</mi>\n <mi>j</mi>\n </msub>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n </mfrac>\n <msubsup>\n <mi>x</mi>\n <mi>j</mi>\n <mrow>\n <mn>2</mn>\n <msub>\n <mi>α</mi>\n <mi>j</mi>\n </msub>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msubsup>\n <msup>\n <mi>e</mi>\n <mrow>\n <mo>−</mo>\n <msubsup>\n <mi>x</mi>\n <mi>j</mi>\n <mn>2</mn>\n </msubsup>\n </mrow>\n </msup>\n <mi>d</mi>\n <msub>\n <mi>x</mi>\n <mi>j</mi>\n </msub>\n </mrow>\n <annotation>$d\\gamma _\\alpha (x)=\\prod _{j=1}^d\\frac{2}{\\Gamma (\\alpha _j+1)} x_j^{2\\alpha _j+1} \\text{e}^{-x_j^2} dx_j$</annotation>\n </semantics></math> on <math>\n <semantics>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <mi>d</mi>\n </msup>\n <annotation>$(0,\\infty)^d$</annotation>\n </semantics></math> when <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>=</mo>\n <mo>(</mo>\n <msub>\n <mi>α</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <msub>\n <mi>α</mi>\n <mi>d</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$\\alpha =(\\alpha _1, \\dots,\\alpha _d)$</annotation>\n </semantics></math> is a multi-index in <math>\n <semantics>\n <msup>\n <mfenced>\n <mo>−</mo>\n <mfrac>\n <mn>1</mn>\n <mn>2</mn>\n </mfrac>\n <mo>,</mo>\n <mi>∞</mi>\n </mfenced>\n <mi>d</mi>\n </msup>\n <annotation>$\\left(-\\frac{1}{2},\\infty \\right)^d$</annotation>\n </semantics></math>. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions\",\"authors\":\"Jorge J. 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We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-03-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300088\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300088","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

在本文中，我们给出了从哈代型空间 H1((0,∞)d,γα)$H^1((0、\到 L1((0,∞)d,γα)$L^1((0,\infty)^d,\gamma _\alpha)$ 以及从 L∞((0,∞)d,γα)$L^\infty((0,\infty)^d、(0,\infty)^d,\gamma_\alpha)$到有界均值振荡函数空间 BMO((0,∞)d,γα)$textup {BMO}((0,\infty)^d,\gamma _\alpha)$、关于概率度量 dγα(x)=∏j=1d2Γ(αj+1)xj2αj+1e-xj2dxj$d\gamma _\alpha (x)=\prod _{j=1}^d\frac{2}{Gamma (\alpha _j+1)} x_j^{2\alpha _j+1}\当 α=(α1,⋯,αd)$\alpha =(\alpha _1, \dots,\alpha _d)$是(-12,∞)d$left(-\frac{1}{2},\infty \right)^d$中的多指数时，在(0,∞)d$(0,\infty)^d$上的text{e}^{-x_j^2} dx_j$。我们将应用它来建立里兹变换、最大算子、利特尔伍德-帕利函数、拉普拉斯变换类型的乘法器、分数积分以及拉盖尔设置中的变算子的端点估计。

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Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

In this paper, we give a criterion to prove boundedness results for several operators from the Hardy-type space $H^{1} ({(0, \infty)}^{d}, γ_{α})$ to $L^{1} ({(0, \infty)}^{d}, γ_{α})$ and also from $L^{\infty} ({(0, \infty)}^{d}, γ_{α})$ to the space of functions of bounded mean oscillation $B M O ({(0, \infty)}^{d}, γ_{α})$ , with respect to the probability measure $d γ_{α} (x) = \prod_{j = 1}^{d} \frac{2}{Γ (α_{j} + 1)} x_{j}^{2 α_{j} + 1} e^{- x_{j}^{2}} d x_{j}$ on ${(0, \infty)}^{d}$ when $α = (α_{1}, \dots, α_{d})$ is a multi-index in ${(- \frac{1}{2}, \infty)}^{d}$ . We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.

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