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: 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

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.傅里叶变换是数学中最重要的运算符，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审查，仍然有一些关于它的基本问题仍然悬而未决。 加权勒贝格空间是函数的集合，其“大小”由某个积分公式给出。将函数映射到函数的运算符被称为有界的，只要该运算不会使该大小增加超过一个固定因子。 我建议研究以下问题：刻画这些权重u和v，使得傅立叶变换有界为从索引为p且权重为u的勒贝格空间到索引为q且权重为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，并且在乘以另一个固定函数时减少。理解这类函数与算子和范数的关系对于勒贝格空间和相关洛伦兹空间的工作很重要。之前对递减函数类所取得的理解在过去的工作中取得了巨大的好处，我们预计这项研究也会带来类似的好处。 3.角度等价是一种新的方法，它将物体（通常是函数）大小的不同度量与这些相同物体之间角度的不同度量相对应。我们已经证明了角等价的基本性质。现在是时候扩展我们对这个新概念的理解，并将其与已有的数学联系起来了。