与分数次相关的问题一直是调和分析的核心研究内容之一.分数次积分、分数阶傅里叶变换和分数次容量在偏微分方程、位势分析以及信号处理等领域都有广泛的应用.申请人与合作者在相关于分数次积分和分数次容量方面进行了系列研究.本课题拟进一步研究与分数次积分和分数阶傅里叶变换相关问题及其在分数次容量和分数阶Bochner-Riesz平均等问题中的应用,其中包括利用分数次积分交换子的紧性刻画函数空间;完善一维分数阶傅里叶变换理论;发展高维分数阶傅里叶变换的理论并引进和研究分数阶Bochner-Riesz平均;结合分数次积分在各类函数空间中的有界性和紧性等性质系统研究分数次Sobolev型容量的基本性质、等价刻画、位势理论及其在分数阶偏微分方程初边值等问题中的应用.

The problems of fractional correlation are always one of the core contents of harmonic analysis.Fractional integral,fractional Fourier transform and fractional capacity have a wide application in the field of partial differential equations,potential analysis and signal processing. The applicant and his collaborators have studied several fractional integrals and fractional capacity.This project aims to further study some problems related to fractional integral,fractional Fourier transform and their applications in fractional capacity and fractional Bochner-Riesz means,including the characterization of function spaces by the compactness of commutators for fractional integral operators;perfect one dimensional fractional Fourier transform theory;establish the high dimensional theory of fractional Fourier transform and then define and study fractional Bochner-Riesz means;With the boundedness and compactness of fractional integral on function spaces,we study the basic properties,equivalent characterization,potential theory of fractional Sobolev type capacity and its applications to the initial boundary value problems of fractional partial differential equations.