Function Spaces and Norm Inequalities

1.傅里叶变换是数学中最重要的算符，在理论和实际应用中都处于领先地位。尽管经过了几个世纪的审查，但关于它的一些根本性问题仍然悬而未决。 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称为有界的，前提是该运算不会将此大小增加超过固定的系数。 我建议研究以下问题：刻画这些权重u和v，使得傅里叶变换作为从指标p和权重u的勒贝格空间到指标q和权重v的勒贝格空间的映射有界。 2.拟凹函数是满足两个单调性条件的函数--当与一个固定函数相乘时，它们是递增的，当与另一个固定函数相乘时，它们是递减的。了解这类函数如何与算子和范数相关，对于勒贝格和相关的洛伦兹空间的工作是很重要的。以前对递减函数类的理解在过去的工作中有很大的好处，我们希望从这项研究中获得类似的好处。 3.角度等价性是一种新的联系方式，即不同的物体大小度量(通常是函数)对应于这些相同对象之间的不同角度度量。我们已经证明了角等价的基本性质。现在是时候扩大我们对这一新概念的理解，并将其与现有的数学联系起来。

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.