{"title":"On Unique Minimal $$\\boldsymbol{L}^{\\boldsymbol{p}}$$ -Norm Harmonic or Holomorphic Function Which Takes Given Value in a Fixed Point","authors":"T. Ł. Żynda","doi":"10.3103/s1068362324700134","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>First, it will be shown that Banach spaces <span>\\(V\\)</span> of harmonic or holomorphic functions with <span>\\(L^{p}\\)</span> norm satisfy minimal norm property, i.e., in any set</p><span>$$V_{z,c}:=\\{f\\in V\\>|\\>f(z)=c\\},$$</span><p>if nonempty, there is exactly one element with minimal norm. Later, it will be proved that this element depends continuously on a deformation of a norm and on an increasing sequence of domains in a precisely defined sense.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3103/s1068362324700134","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
First, it will be shown that Banach spaces \(V\) of harmonic or holomorphic functions with \(L^{p}\) norm satisfy minimal norm property, i.e., in any set
$$V_{z,c}:=\{f\in V\>|\>f(z)=c\},$$
if nonempty, there is exactly one element with minimal norm. Later, it will be proved that this element depends continuously on a deformation of a norm and on an increasing sequence of domains in a precisely defined sense.
期刊介绍：
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) is an outlet for research stemming from the widely acclaimed Armenian school of theory of functions, this journal today continues the traditions of that school in the area of general analysis. A very prolific group of mathematicians in Yerevan contribute to this leading mathematics journal in the following fields: real and complex analysis; approximations; boundary value problems; integral and stochastic geometry; differential equations; probability; integral equations; algebra.
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