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

IF 0.8 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":"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":0.8000,"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://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

Abstract

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.