{"title":"Absence of local unconditional structure in spaces of smooth functions on the torus of arbitrary dimension","authors":"A. Tselishchev","doi":"10.4064/sm200629-21-12","DOIUrl":null,"url":null,"abstract":"Consider a finite collection {T1, . . . , TJ} of differential operators with constant coefficients on Tn (n ≥ 2) and the space of smooth functions generated by this collection, namely, the space of functions f such that Tjf ∈ C(T n), 1 ≤ j ≤ J . We prove that if there are at least two linearly independent operators among their senior parts (relative to some mixed pattern of homogeneity), then this space does not have local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of C(S).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/sm200629-21-12","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Consider a finite collection {T1, . . . , TJ} of differential operators with constant coefficients on Tn (n ≥ 2) and the space of smooth functions generated by this collection, namely, the space of functions f such that Tjf ∈ C(T n), 1 ≤ j ≤ J . We prove that if there are at least two linearly independent operators among their senior parts (relative to some mixed pattern of homogeneity), then this space does not have local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of C(S).
期刊介绍:
The journal publishes original papers in English, French, German and Russian, mainly in functional analysis, abstract methods of mathematical analysis and probability theory.