Topics in Harmonic Analysis

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Methods from mathematical analysis have found wide applications in understanding physical phenomena in the natural sciences and engineering. This project is concerned with topics in harmonic analysis that are designed to provide effective mathematical tools for these disciplines and that could well contribute to the unification of seemingly unrelated areas. The main goal of the project is to expand the mathematical toolbox in harmonic analysis along these lines. The project involves mentoring of graduate students and postdoctoral researchers.The principal investigator will to work on several projects in harmonic analysis. The first project is concerned with the functional calculus for selfadjoint elliptic and subelliptic partial differential operators. The research will focus on a model example in the subelliptic case, the Laplacian on the Heisenberg group. The goal is to derive new multiplier results on Lebesgue spaces, in particular for the model case of Bochner-Riesz means. In the Euclidean case such multiplier results can be based on Fourier restriction theory, which is not available on the Heisenberg group. A new approach is proposed based on the fine structure of the wave kernel on the Heisenberg group. A second project is motivated by a problem on the cost of mixing for incompressible flows that are generated by time-dependent vector fields. It turns out that this problem is related to boundedness questions on certain bilinear singular integral operators. A general theory will be developed that incorporates multilinear generalizations and is of independent interest. A third project is concerned with problems in approximation theory that involve function spaces of low order of smoothness. In many cases there are gaps between the known results for the categories of Besov and Sobolev spaces. As an example, the range for unconditional convergence of Haar wavelet expansions is well understood for Besov spaces but remains open for Sobolev spaces. One goal for this project is to determine the correct range for Sobolev spaces.