Grant: Discovery Grants Program - Individual

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 an certain integral formula. An operator that maps functions to functions is 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. Recent advances in the theory of positive operators will drive this research. The Fourier transform is not a positive operator but it interacts closely with a class of operators called convolution operators. Convolution operators can in turn be simply estimated by positive operators. I have had success proving Fourier inequalities in other spaces and, recently, developed a technique for generating new Fourier inequalities in weighted Lebesgue spaces. There is still a long way to go before a complete characterization can be given.