分形的傅里叶分析

The applicant plans to research two Fourier-analytic problems associated with fractals. The long term research objective is to identify extremal fractals that satisfy certain Fourier-analytic properties in an optimal way. Probabilistic methods are often employed to produce near-optimal results in such problems. However, obtaining truly optimal results tends to require substantial new ideas exploiting hidden correlation in the problem. The applicant proposes the following main projects: ..(1) Construction of fractals that satisfy the best possible Fourier restriction property under the dimension constraint. In joint work with Andreas Seeger, the applicant has obtained a partial result in the case when the dimension of the fractal divides the ambient Euclidean dimension. In the general case, the applicant proposes to approach the problem by extending Bourgain's result on Lambda(p) sets to the fractal setting. ..(2) Construction of Ahlfors regular Salem fractals. In 2004, Mattila posed the question whether there exists a fractal on the real line that satisfy both the Ahlfors regularity and the endpoint Fourier decay property. The applicant proposes to investigate this question by reducing it to related problems in the discrete setting, where in particular one needs to understand an intriguing uncertainty-type principle.

申请人计划研究两个与分形相关的傅里叶分析问题.主要研究目标是对给定的分形维数寻找最优的分形集,以满足特定的傅里叶性质.虽然这类问题往往可以通过随机构造的方法得到近似最优的结果,然而要找到真正最优的分形集还需深入挖掘问题本身的内在结构.申请人计划研究以下具体问题:..(一)对给定的分形维数,构造满足最佳傅里叶限制性估计的分形.申请人与Seeger在2017年对分形维数整除欧氏空间维数的情形解决了该问题.对于一般情形,申请人计划通过把Bourgain关于Lambda(p)集的工作推广到分形集来解决这个问题...(二)构造具有Ahlfors正则性的Salem分形.Mattila在2004年提出一个问题,即实轴上是否存在同时满足Ahlfors正则性和最佳傅里叶衰减率的分形.申请人计划将此问题归结为离散情形的相关问题来研究,特别地归结为理解离散傅里叶变换的某类不确定原理.

本项目研究与分形相关的若干傅里叶分析问题,研究目标在于寻求分形集的最优化构造,使之满足特定的傅里叶性质.研究内容包括傅里叶级数的发散点集、缺项三角级数的最大模估计及Λ(p)性质、平面Kakeya集的最小化问题等.研究成果包括与合作者在球面上推广Kolmogorov的几乎处处发散傅里叶级数例子、证明球面上线性薛定谔方程解的最佳光滑化估计、建立半周期线性薛定谔方程解的局部光滑化估计等.上述成果分别发表在《Rev.Mat.Iberoam.》(2020)、《J.Math.Pures.Appl.》(2022)及投稿至《J.Funct.Anal.》.

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