{"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-2021-24-43","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":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2021-01-01","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-2021-24-43\",\"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\":49868,\"journal\":{\"name\":\"Mathematical Inequalities & Applications\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Inequalities & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7153/mia-2021-24-43\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Inequalities & Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7153/mia-2021-24-43","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","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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''Mathematical Inequalities & Applications'' (''MIA'') brings together original research papers in all areas of mathematics, provided they are concerned with inequalities or their role. From time to time ''MIA'' will publish invited survey articles. Short notes with interesting results or open problems will also be accepted. ''MIA'' is published quarterly, in January, April, July, and October.
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