Function Spaces and Norm Inequalities

函数空间和范数不等式

• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675402
• 关键词: Function; Spaces; Norm; Inequalities

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运算符，在理论和实际应用中均领先。尽管进行了几个世纪的审查，但仍然有基本问题仍然开放。******加权的Lebesgue空间是功能集合的集合，其“大小”由某种整体公式给出。据说，映射功能映射到功能的操作员只要操作不超过固定因素而增加此尺寸。 Quasiconcave函数是满足两个单调性条件的函数 - 当乘以一个固定函数并乘以另一个函数时，它们正在增加。了解这类与运营商和规范相关的功能对于Lebesgue和相关Lorentz空间的工作很重要。以前对降低功能类别的理解在过去的工作中具有很大的好处，我们希望这项研究相似。****** 3。角度等效性是一种新的手段，即将对象大小的不同度量（通常功能）与这些相同对象之间的角度的不同度量相对应。我们已经证明了角度等效的基本特性。现在是时候扩大我们对这一新概念的理解并将其与已建立的数学联系起来的时候了。 **