{"title":"On the integrability of multi-dimensional rare maximal functions","authors":"I. Japaridze, G. Oniani","doi":"10.1007/s10474-023-01367-x","DOIUrl":null,"url":null,"abstract":"<div><p>We characterize the translation invariant monotone collections of multi-dimensional intervals for which the analogue of Stein's criterion for the integrability of the Hardy--Littlewood maximal function is true. Namely, we characterize the collections <span>\\(B\\)</span> of the mentioned type for which the conditions <span>\\(\\int_{[0,1]^d}M_B(f)<\\infty\\)</span> and\n <span>\\(\\int_{[0,1]^d}\\vert f\\vert \\log^+\\vert f\\vert <\\infty\\)</span> are equivalent for functions <span>\\(f\\)</span>\nsupported on the unit cube <span>\\([0,1]^d\\)</span>. Here <span>\\(M_B\\)</span> denotes the maximal operator associated to a collection <span>\\(B\\)</span>.\n</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-023-01367-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We characterize the translation invariant monotone collections of multi-dimensional intervals for which the analogue of Stein's criterion for the integrability of the Hardy--Littlewood maximal function is true. Namely, we characterize the collections \(B\) of the mentioned type for which the conditions \(\int_{[0,1]^d}M_B(f)<\infty\) and
\(\int_{[0,1]^d}\vert f\vert \log^+\vert f\vert <\infty\) are equivalent for functions \(f\)
supported on the unit cube \([0,1]^d\). Here \(M_B\) denotes the maximal operator associated to a collection \(B\).
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.