{"title":"与反射群有关的Dunkl微分算子的Hilbert变换","authors":"I. A. López P","doi":"10.1007/s10476-023-0189-3","DOIUrl":null,"url":null,"abstract":"<div><p>The aim of this paper is to introduce the Dunkl—Hilbert transform <i>H</i><sup><i>k</i></sup>, with <i>k</i> ≥ 0, induced by the Dunkl differential operator and associated with the reflection group ℤ<sub>2</sub>. For this end, we establish that the Dunkl—Poisson kernel and the conjugate Dunkl—Poisson kernel satisfy the Cauchy—Riemann equations in the Dunkl context. We prove the continuity of <i>H</i><sup><i>k</i></sup> on <i>L</i><sup>p</sup>(<i>w</i><sub><i>k</i></sub>) for 1 < <i>p</i> < ∞, where <i>w</i><sub><i>k</i></sub>(<i>x</i>) = ∣<i>x</i>∣<sup>2<i>k</i></sup>. Finally, we introduce the maximal Hilbert operator <i>H</i><span>\n <sup><i>k</i></sup><sub>*</sub>\n \n </span> and establish an analogue of Cotlar’s theorem.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Hilbert Transform for Dunkl Differential Operators Associated to the Reflection Group ℤ2\",\"authors\":\"I. A. López P\",\"doi\":\"10.1007/s10476-023-0189-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The aim of this paper is to introduce the Dunkl—Hilbert transform <i>H</i><sup><i>k</i></sup>, with <i>k</i> ≥ 0, induced by the Dunkl differential operator and associated with the reflection group ℤ<sub>2</sub>. For this end, we establish that the Dunkl—Poisson kernel and the conjugate Dunkl—Poisson kernel satisfy the Cauchy—Riemann equations in the Dunkl context. We prove the continuity of <i>H</i><sup><i>k</i></sup> on <i>L</i><sup>p</sup>(<i>w</i><sub><i>k</i></sub>) for 1 < <i>p</i> < ∞, where <i>w</i><sub><i>k</i></sub>(<i>x</i>) = ∣<i>x</i>∣<sup>2<i>k</i></sup>. Finally, we introduce the maximal Hilbert operator <i>H</i><span>\\n <sup><i>k</i></sup><sub>*</sub>\\n \\n </span> and establish an analogue of Cotlar’s theorem.</p></div>\",\"PeriodicalId\":55518,\"journal\":{\"name\":\"Analysis Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-01-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10476-023-0189-3\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis Mathematica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10476-023-0189-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Hilbert Transform for Dunkl Differential Operators Associated to the Reflection Group ℤ2
The aim of this paper is to introduce the Dunkl—Hilbert transform Hk, with k ≥ 0, induced by the Dunkl differential operator and associated with the reflection group ℤ2. For this end, we establish that the Dunkl—Poisson kernel and the conjugate Dunkl—Poisson kernel satisfy the Cauchy—Riemann equations in the Dunkl context. We prove the continuity of Hk on Lp(wk) for 1 < p < ∞, where wk(x) = ∣x∣2k. Finally, we introduce the maximal Hilbert operator Hk* and establish an analogue of Cotlar’s theorem.
期刊介绍：
Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx).
The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx).
The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
