{"title":"关于平移不变凸微分基的比较","authors":"Irakli Japaridze","doi":"10.1515/gmj-2023-2070","DOIUrl":null,"url":null,"abstract":"It is known that if <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{\\prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{\\prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{\\prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in view of their positive equivalence.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the comparison of translation invariant convex differentiation bases\",\"authors\":\"Irakli Japaridze\",\"doi\":\"10.1515/gmj-2023-2070\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is known that if <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2070_eq_0070.png\\\" /> <jats:tex-math>{B^{\\\\prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2070_eq_0070.png\\\" /> <jats:tex-math>{B^{\\\\prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2070_eq_0070.png\\\" /> <jats:tex-math>{B^{\\\\prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in view of their positive equivalence.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/gmj-2023-2070\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2023-2070","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
众所周知，如果 B 和 B ′ {B^{/prime}} 是平移不变的凸密度微分基，并且与它们相关的最大算子在局部上相互大化，那么 B 和 B ′ {B^{/prime} 就微分同一类非负函数的积分。我们证明，在同样的条件下，鉴于 B 和 B ′ {B^{prime} 的正等价性，不可能断言它们的微分性质有更多的相似性。
On the comparison of translation invariant convex differentiation bases
It is known that if B and B′{B^{\prime}} are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then B and B′{B^{\prime}} differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of B and B′{B^{\prime}} in view of their positive equivalence.
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