几类变指标函数空间上的算子有界性及应用

The function spaces with variable exponents are put forward in the studies of the boundary value problems of nonlinear Dirichlet with nonstandard growth conditions. The study of the function spaces with variable exponents and related operators is one of the most lively problems in the area of harmonic analysis, since it essentially generalized the classic function spaces and is closely relevant to many branches of mathematics and mechanics, such as PDEs, probability theory and fluid dynamics. Now the theory of the function spaces with variable exponents is widely used in the related studies of finance and image restoration. Therefore, its further study has great theoretical significance and broad application prospect. The project aims at intensively studying boundedness of operators on several function spaces with variable exponents and their applications. The main contents of the project are: the boundedness of strongly singular operators, multilinear operators, pseudo-differential operators and their commutators on Morrey type spaces with variable exponents; new Herz type spaces with variable exponents and the boundedness of some operators and their commutators on above spaces. As applications, we will study the related problems on PDEs. For example, we will establish the estimate of operator semigroups on Lebesgue spaces with variable exponent and Besov spaces with variable exponents, and give the well-posedness of the Cauchy problem for the Navier-Stokes equations on the corresponding function spaces.

变指标函数空间是在研究具有非标准增长条件的非线性Dirichlet边值问题时提出的。由于它从本质上推广了经典函数空间且与偏微分方程、概率论以及流体力学等数学、力学中的多个分支有非常紧密的联系，变指标函数空间及相关算子为核心的研究已成为近年来调和分析领域最活跃的课题之一。目前变指标函数空间理论已被广泛应用到金融学、图像复原等相关问题的研究中，其进一步研究具有重要的理论意义和广阔的应用前景。本项目旨在对几类变指标函数空间上的算子有界性及应用进行深入研究。主要内容包括：变指标Morrey型空间中强奇异算子、多线性算子、拟微分算子及其交换子的有界性；新的变指标Herz型空间及若干积分算子在其中的有界性；将上述内容应用到偏微分方程相关问题的研究中，如建立算子半群在变指标Lebesgue空间及变指标Besov空间中的估计并给出Navier-Stokes方程的Cauchy问题在相应函数空间中的适定性。

变指标函数空间及相关算子为核心的研究已成为近年来调和分析领域最活跃的课题之一，其研究具有重要的理论意义和广阔的应用前景。本项目研究了几类变指标函数空间上的算子有界性及应用。主要成果有：证明了一类多次线性奇异积分算子在乘积变指标中心Morrey空间上的有界性，并在此基础上得到了多线性Calderón-Zygmund奇异积分算子和两类多线性奇异积分交换子在变指标中心Morrey空间上的有界性；研究了带粗糙核的Marcinkiewicz积分交换子在变指标Herz型空间上的连续性；定义了弱变指标Herz空间和弱变指标Herz型Hardy空间，并建立了一些算子在上述空间中的有界性估计和在端点处的估计，以及一些交换子在这类空间上的弱型BMO估计和弱型Lipschitz估计；作为应用，研究了变指标Herz型Hardy空间和变指标Herz-Morrey空间上带VMO系数的非散度型椭圆方程解的正则性。这些工作丰富了变指标函数空间及算子理论的研究，并对其在偏微分方程中的应用进行了有益的探索。