Topics in harmonic analysis

调和分析有着经典的根源和全球性的影响。本提案将研究四个不同的问题： 1.稀疏集的分析和几何， 2.奇异性的解析及其在分析中的应用， 3.凸几何和几个复变量， 4. Kakeya型问题与方向极大算子。 1.我的目标是探索薄欧几里德集的算术和几何模式，它具有零勒贝格测度，但非平凡的Hausdorff或傅立叶维数。我还将研究是否支持这些集的措施，足够丰富的结构，承认运营商理论的几何现象的扩展。 2.奇点的分解是代数几何中的一个领域，涉及多项式零集的结构。我计划制定明确的和有效的算法解决多项式奇点在任何层面上的应用，在分析问题，这样的零集发挥了重要作用。 3.这方面的建议是关于函数理论的问题，在几个复杂的变量，如估计伯格曼和Szego核弱伪凸域，这可以接近使用工具，从凸几何，如结构的多面体和对偶。 4.另一个建议的研究途径是建立欧几里德方向极大算子的勒贝格界，这是一个具有Kakeya型特征的问题。

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.