本项目研究奇异拉东变换、球面极大算子、多线性极大算子的离散化算子及其变差算子的若干调和分析问题. 主要研究内容为：离散奇异拉东变换和挠曲离散奇异拉东变换的(p, p)有界性与端点估计；离散球面极大算子的加幂权(p, p)有界性和加幂权端点估计，并探讨其加幂权端点外估计；离散抛物平均算子族、离散抛物Hilbert变换算子族、离散球面平均算子族和一般离散奇异拉东变换算子族对应的变差算子的(p, p)有界性及加权估计和端点估计；离散多线性极大算子的有界性和端点正则性估计；多线性分数次极大算子的正则性及其离散算子的有界性和端点正则性. 这些问题来源于调和分析，并与遍历理论、周期函数的偏微分方程理论、Fourier级数的收敛性和数论及鞅论等研究领域有着密切联系，其结果将丰富和完善奇异积分理论和极大函数理论，在上述相关研究领域有着重要应用. 这些问题的处理涉及调和分析方法和数论中的一些技巧的巧妙结合.

This project is to study the discrete analogues of several classes of operators, including singular Radon transforms, maximal operators of spherical means, maximal functions of multilinear operators and multilinear fractional operators, in harmonic analysis. First, the Lp-boundedness and the endpoint estimates on the discrete singular Radon transforms and the twisted discrete singular Radon transforms with non-smooth kernels are considered. Second, the weighted Lp-boundedness with power weights as well as the corresponding estimates at or beyond the endpoints for the discrete maximal functions of spherical means are studied. Third, the variation inequalities on the families of the discrete singukar averaging operators along parabolic curves, the discrete parabolic Hilbert transforms and the discrete spherical means as well as the discrete singular Radon transforms are considered. Finally, the regularities on the Sobolev spaces for the multilinear maximal operators and the multilinear fractional maximal operators as well as their discrete analogues are investgated. These problems involve with the ergodic theory, the partial differential equations for periodic functions, the convergence of Fourier series, the number theory and probability. The methods employed include the classical analytic methods as well as techniques from number theory, including the circle method, theta functions, exponential sums and Diophantine (rational) approximations.