Function Spaces and Norm Inequalities

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。尽管经过了几个世纪的仔细研究，关于它的一些基本问题仍然没有解决。*******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操作符被称为有界操作符，前提是该操作不会使该操作符的大小增加超过一个固定因子。*******我建议解决以下问题：描述这些权值u和v的特征使得傅里叶变换被限定为从指标p和权值u的勒贝格空间到指标q和权值v的勒贝格空间的映射*******2。拟凹洞函数是满足两个单调性条件的函数，它们与一个固定函数相乘时递增，与另一个固定函数相乘时递减。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中的工作是很重要的。以前对递减函数类的理解已经在过去的工作中取得了实质性的好处，我们期望从这项研究中获得类似的好处。*******3。角等效是一种新的方法，它将物体（通常是函数）的不同尺寸对应于这些相同物体之间不同角度的方式联系起来。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，并把它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. **