Pub Date : 2023-10-28DOI: 10.1007/s10476-023-0243-1
H. Rafeiro, S. Samko, S. Umarkhadzhiev
The approach to “locally” aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of “aggrandizer”, is combined with the usual “global” aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces.
{"title":"Grand Lebesgue Spaces with Mixed Local and Global Aggrandization and the Maximal and Singular Operators","authors":"H. Rafeiro, S. Samko, S. Umarkhadzhiev","doi":"10.1007/s10476-023-0243-1","DOIUrl":"10.1007/s10476-023-0243-1","url":null,"abstract":"<div><p>The approach to “locally” aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of “aggrandizer”, is combined with the usual “global” aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10476-023-0243-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134797686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-09DOI: 10.1007/s10476-023-0238-y
D. D. Haroske, H.-G. Leopold, S. D. Moura, L. Skrzypczak
We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain Ω ⊂ ℝd. This covers, in particular, the well-known situation for spaces of Besov and Triebel–Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in ∓r spaces, which we could recently extend to the vector-valued setting needed here.
{"title":"Nuclear and Compact Embeddings in Function Spaces of Generalised Smoothness","authors":"D. D. Haroske, H.-G. Leopold, S. D. Moura, L. Skrzypczak","doi":"10.1007/s10476-023-0238-y","DOIUrl":"10.1007/s10476-023-0238-y","url":null,"abstract":"<div><p>We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain Ω ⊂ ℝ<sup><i>d</i></sup>. This covers, in particular, the well-known situation for spaces of Besov and Triebel–Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in <i>∓</i><sub><i>r</i></sub> spaces, which we could recently extend to the vector-valued setting needed here.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10476-023-0238-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134795697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}