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仔细研究，但关于它的基本问题仍然是悬而未决的。*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算子称为有界的，前提是该操作不增加超过固定因子的大小。*我建议研究以下问题：刻画那些权重u和v，使得傅立叶变换有界于从指数p和权重u的勒贝格空间到指数为q和权重v的勒贝格空间的映射。拟凹函数是满足两个单调性条件的函数--当乘以一个固定函数时，它们是递增的，而当乘以另一个固定函数时，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，我们期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方法，可以将物体(通常是函数)大小的不同度量对应于这些相同对象之间的角度的不同度量。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，并将其与已有的数学联系起来。

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. **