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. 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\":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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助