非交换鞅变换的混合L p$ L^p$估计

IF 1 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-11-11 DOI:10.1112/blms.13184

Adam Osękowski

{"title":"非交换鞅变换的混合L p$ L^p$估计","authors":"Adam Osękowski","doi":"10.1112/blms.13184","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo><</mo>\n <mi>p</mi>\n <mo><</mo>\n <mspace></mspace>\n <mi>q</mi>\n <mo><</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$1<p<\\, q<\\infty $</annotation>\n </semantics></math>. The paper is devoted to the study of <math>\n <semantics>\n <mrow>\n <msup>\n <mi>L</mi>\n <mi>q</mi>\n </msup>\n <mo>→</mo>\n <msup>\n <mi>L</mi>\n <mi>p</mi>\n </msup>\n </mrow>\n <annotation>$L^q\\rightarrow L^p$</annotation>\n </semantics></math> estimates for transforms of noncommutative martingales, under the assumption that the transforming sequence takes values in <math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mi>r</mi>\n </msup>\n <annotation>$L^r$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>/</mo>\n <mi>r</mi>\n <mo>=</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mi>p</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mi>q</mi>\n </mrow>\n <annotation>$1/r=1/p-1/q$</annotation>\n </semantics></math>. This goes beyond the usual context of <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>=</mo>\n <mi>q</mi>\n </mrow>\n <annotation>$p=q$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>r</mi>\n <mo>=</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$r=\\infty$</annotation>\n </semantics></math> studied so far in the literature. The obtained constants are of optimal order at the endpoints, in addition the approach allows to obtain sharp values in the range <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>⩽</mo>\n <mn>2</mn>\n <mo>⩽</mo>\n <mi>q</mi>\n </mrow>\n <annotation>$p\\leqslant 2\\leqslant q$</annotation>\n </semantics></math>. The proof rests on real interpolation-type arguments for martingale transforms, which are of independent interest.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 1","pages":"96-114"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Mixed \\n \\n \\n L\\n p\\n \\n $L^p$\\n estimates for transforms of noncommutative martingales\",\"authors\":\"Adam Osękowski\",\"doi\":\"10.1112/blms.13184\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo><</mo>\\n <mi>p</mi>\\n <mo><</mo>\\n <mspace></mspace>\\n <mi>q</mi>\\n <mo><</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$1<p<\\\\, q<\\\\infty $</annotation>\\n </semantics></math>. The paper is devoted to the study of <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>L</mi>\\n <mi>q</mi>\\n </msup>\\n <mo>→</mo>\\n <msup>\\n <mi>L</mi>\\n <mi>p</mi>\\n </msup>\\n </mrow>\\n <annotation>$L^q\\\\rightarrow L^p$</annotation>\\n </semantics></math> estimates for transforms of noncommutative martingales, under the assumption that the transforming sequence takes values in <math>\\n <semantics>\\n <msup>\\n <mi>L</mi>\\n <mi>r</mi>\\n </msup>\\n <annotation>$L^r$</annotation>\\n </semantics></math>, <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mi>r</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mi>p</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mi>q</mi>\\n </mrow>\\n <annotation>$1/r=1/p-1/q$</annotation>\\n </semantics></math>. This goes beyond the usual context of <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>=</mo>\\n <mi>q</mi>\\n </mrow>\\n <annotation>$p=q$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n <mo>=</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$r=\\\\infty$</annotation>\\n </semantics></math> studied so far in the literature. The obtained constants are of optimal order at the endpoints, in addition the approach allows to obtain sharp values in the range <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>⩽</mo>\\n <mn>2</mn>\\n <mo>⩽</mo>\\n <mi>q</mi>\\n </mrow>\\n <annotation>$p\\\\leqslant 2\\\\leqslant q$</annotation>\\n </semantics></math>. The proof rests on real interpolation-type arguments for martingale transforms, which are of independent interest.\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 1\",\"pages\":\"96-114\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.13184\",\"RegionNum\":3,\"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":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.13184","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

设1 ＆lt；P ＆lt；Q ＆lt；∞$1<p<\, q<\infty $。本文研究了非交换鞅变换的lq→lp $L^q\rightarrow L^p$估计。假设变换序列取L r $L^r$, 1 / r = 1 / p−1 / q $1/r=1/p-1/q$。这超出了迄今为止在文献中研究的p = q $p=q$和r =∞$r=\infty$的通常背景。得到的常数在端点处是最优的顺序，此外，该方法允许在p≤2≤q $p\leqslant 2\leqslant q$范围内获得尖锐的值。证明依赖于鞅变换的实插值型参数，这是独立的兴趣。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

Mixed L p $L^p$ estimates for transforms of noncommutative martingales

Let $1 < p < q < \infty$ . The paper is devoted to the study of $L^{q} \to L^{p}$ estimates for transforms of noncommutative martingales, under the assumption that the transforming sequence takes values in $L^{r}$ , $1 / r = 1 / p - 1 / q$ . This goes beyond the usual context of $p = q$ and $r = \infty$ studied so far in the literature. The obtained constants are of optimal order at the endpoints, in addition the approach allows to obtain sharp values in the range $p ⩽ 2 ⩽ q$ . The proof rests on real interpolation-type arguments for martingale transforms, which are of independent interest.