{"title":"Injectivity of Spherical Means on H-Type Groups","authors":"E. K. Narayanan, P. K. Sanjay, K. T. Yasser","doi":"10.1007/s00041-024-10089-9","DOIUrl":null,"url":null,"abstract":"<p>We establish injectivity results for three different spherical means on an <i>H</i>-type group, <i>G</i>. The first one is the standard spherical mean operator, which is defined as the average of a function over the spheres in the complement of the center, the second one is the average over the product of spheres in the center and its complement, and the third one is the average over the spheres defined by a homogeneous norm on <i>G</i>. If <i>m</i> is the dimension of the center of <i>G</i>, injectivity of these spherical means is proved for the range <span>\\(1 \\le p \\le \\frac{2m}{m-1}\\)</span>. Examples are provided to show the sharpness of our results in the first two cases.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00041-024-10089-9","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We establish injectivity results for three different spherical means on an H-type group, G. The first one is the standard spherical mean operator, which is defined as the average of a function over the spheres in the complement of the center, the second one is the average over the product of spheres in the center and its complement, and the third one is the average over the spheres defined by a homogeneous norm on G. If m is the dimension of the center of G, injectivity of these spherical means is proved for the range \(1 \le p \le \frac{2m}{m-1}\). Examples are provided to show the sharpness of our results in the first two cases.
我们建立了 H 型群 G 上三种不同球均值的注入性结果。第一种是标准球均值算子，它被定义为函数在中心补集球面上的平均值；第二种是在中心球面与其补集球面乘积上的平均值；第三种是在 G 上由同质规范定义的球面上的平均值。如果 m 是 G 的中心维数，那么在 \(1 \le p \le \frac{2m}{m-1}\) 的范围内，这些球面均值的注入性将得到证明。举例说明了我们在前两种情况下的结果的尖锐性。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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