Function Spaces and Norm Inequalities

函数空间和范数不等式

基本信息

批准号：
RGPIN-2015-06750
负责人：
Sinnamon, Gord
金额：
$ 1.02万
依托单位：
University of Western Ontario
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2016
资助国家：
加拿大
起止时间：
2016-01-01 至 2017-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614338
关键词：
Function Spaces Norm Inequalities

项目摘要

1. The Fourier transform is the single most important operator in Mathematics, leading the pack in both theoretical and practical applications. Despite centuries of scrutiny, there are still fundamental questions about it that remain open. Weighted Lebesgue spaces are collections of functions whose "size" is given by a certain integral formula. An operator that maps functions to functions is said to be bounded provided the operation does not increase this size by more than a fixed factor. I propose to work on the following problem: Characterize those weights u and v such that the Fourier Transform is bounded as a map from the Lebesgue space with index p and weight u to the Lebesgue space with index q and weight v. 2. Quasiconcave functions are functions satisfying two monotonicity conditions – they are increasing when multiplied by one fixed function and are decreasing when multiplied by another. Understanding how this class of functions relates to operators and norms is important for work in Lebesgue and the related Lorentz spaces. The understanding previously achieved for the class of decreasing functions has been of substantial benefit in past work and we expect similar benefits from this study. 3. Angular equivalence is a new means of relating the way different measures of the size of objects (usually functions) correspond to different measures of the angles between these same objects. We have already proved basic properties of angular equivalence. Now it is time to expand our understanding of this new concept and relate it to established mathematics.

1. 傅立叶变换是数学中最重要的算子，在理论和实际应用中都处于领先地位，尽管经过几个世纪的研究，仍然存在一些基本问题。加权勒贝格空间是其“大小”由某个积分公式给出的函数的集合，如果该操作不会使该大小增加超过一个固定因子，则将函数映射到函数的运算符被认为是有界的。我建议解决以下问题：表征这些权重 u 和 v，使得傅里叶变换被限制为从具有索引 p 和权重 u 的勒贝格空间到具有索引 q 和权重 v 的勒贝格空间的映射。 2. 拟凹函数是满足两个单调性条件的函数——它们在乘以一个固定函数时增加，在乘以另一个固定函数时减少。了解此类函数与算子和范数的关系对于勒贝格和相关洛伦兹空间的工作很重要。先前对递减函数类别的理解在过去的工作中具有很大的好处，我们期望从这项研究中获得类似的好处。 3. 角度等效是一种将物体（通常是函数）的不同尺寸度量与这些相同物体之间的角度的不同度量相关联的新方法。我们已经证明了角度等效的基本性质。扩展我们对这个新概念的理解，并将其与已建立的数学联系起来。