{"title":"二维矩形算子的加权Hardy不等式——E. Sawyer定理的推广","authors":"V. Stepanov, E. Ushakova","doi":"10.7153/mia-2021-24-3","DOIUrl":null,"url":null,"abstract":"A characterization is obtained for those pairs of weights $v$ and $w$ on $\\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\\mathbb{R}^2_+)$ to $L^q_w(\\mathbb{R}^2_+)$ for $1<p\\not= q<\\infty$, which is an essential complement to E. Sawyer's result \\cite{Saw1} given for $1<p\\leq q<\\infty$. Besides, we declare that the E. Sawyer theorem is actual if $p=q$ only, for $p<q$ the criterion is less complicated. The case $q<p$ is new.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On weighted Hardy inequality with two-dimensional rectangular operator -- extension of the E. Sawyer theorem\",\"authors\":\"V. Stepanov, E. Ushakova\",\"doi\":\"10.7153/mia-2021-24-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A characterization is obtained for those pairs of weights $v$ and $w$ on $\\\\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\\\\mathbb{R}^2_+)$ to $L^q_w(\\\\mathbb{R}^2_+)$ for $1<p\\\\not= q<\\\\infty$, which is an essential complement to E. Sawyer's result \\\\cite{Saw1} given for $1<p\\\\leq q<\\\\infty$. Besides, we declare that the E. Sawyer theorem is actual if $p=q$ only, for $p<q$ the criterion is less complicated. The case $q<p$ is new.\",\"PeriodicalId\":8426,\"journal\":{\"name\":\"arXiv: Functional Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Functional Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7153/mia-2021-24-3\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7153/mia-2021-24-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On weighted Hardy inequality with two-dimensional rectangular operator -- extension of the E. Sawyer theorem
A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to $L^q_w(\mathbb{R}^2_+)$ for $1
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