Topics in harmonic analysis

点击翻译按钮获取中文摘要

Harmonic analysis has classical roots and global reach. In this proposal, four distinct problems will be studied: 1. Analysis and geometry of sparse sets, 2. Resolution of singularities with applications to analysis, 3. Convex geometry and several complex variables, 4. Kakeya-type problems and directional maximal operators. 1. I aim to explore arithmetic and geometric patterns in thin Euclidean sets, which have zero Lebesgue measure but nontrivial Hausdorff or Fourier dimension. I will also examine whether such sets support measures with rich enough structure to admit operator-theoretic extensions of geometric phenomena. 2. Resolution of singularities is an area of algebraic geometry concerned with structures of zero sets of polynomials. I plan to develop explicit and effective algorithms for resolving polynomial singularities in any dimension with a view towards application in analytical problems where such zero sets play an important role. 3. This aspect of the proposal is concerned with function-theoretic problems in several complex variables, such as estimation of Bergman and Szego kernels on weakly pseudo-convex domains, which can be approached using tools from convex geometry, such as structure of polytopes and duality. 4. Another proposed avenue of research is to establish Lebesgue bounds for Euclidean directional maximal operators, a problem with Kakeya-type features.