{"title":"Some properties of the extremal function for the Fuglede p-modulus","authors":"Małgorzata Ciska-Niedziałomska","doi":"10.1007/s43034-023-00303-y","DOIUrl":null,"url":null,"abstract":"<div><p>We deal with the Fuglede <i>p</i>-modulus of a system of measures, focusing on three aspects. First, we combine results concerning Badger’s criterion for the extremal function, i.e., the function which realizes the <i>p</i>-modulus, and plans with barycenter in <span>\\(L^q\\)</span>, which give an alternative—in a sense, probabilistic—approach to <i>p</i>-modulus. It seems that the correlation of these results has not yet been established. Second, we deal with families of measures for which the integral of the extremal function is one. On such a family, the <i>p</i>-modulus as well as the optimal plans are concentrated. We consider closures of these families and relate them with generic families of measures for which the extremal function exists. Finally, we compute the <i>p</i>-modulus and extremal function for finite families of measures.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43034-023-00303-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-023-00303-y","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
We deal with the Fuglede p-modulus of a system of measures, focusing on three aspects. First, we combine results concerning Badger’s criterion for the extremal function, i.e., the function which realizes the p-modulus, and plans with barycenter in \(L^q\), which give an alternative—in a sense, probabilistic—approach to p-modulus. It seems that the correlation of these results has not yet been established. Second, we deal with families of measures for which the integral of the extremal function is one. On such a family, the p-modulus as well as the optimal plans are concentrated. We consider closures of these families and relate them with generic families of measures for which the extremal function exists. Finally, we compute the p-modulus and extremal function for finite families of measures.
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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