沿非凸曲线和旋转曲面的极大奇异积分的$L^p$有界性

Singular integrals associated to surfaces(or curves) are intimately connected with many important mathematical problems, such as the boundary-value problem for strongly pseudo-convex domains, the estimates for solutions of constant coefficient parabolic differential equation, the Cauchy-Riemann equations for domains in several complex variables and the characterization of interface in a two-layer fluid system, their boundedness are hot and difficult problems in international mathematical study. Singular integrals associated to surfaces(or curves) are defined by principal value integrals, the boundedness for maximal singular integrals associated to surfaces(or curves) gives the pointwise existence of corresponding principal value singular integrals. Based on applicant’s early works about the boundedness for singular integrals associated to surfaces on Triebel-Lizorkin spaces, applicants will consider the $L^p$ boundedness for maximal singular integrals along nonconvex curves, singular integrals and maximal singular integrals associated to surfaces of revolution by using Fourier transform estimates, Littlewood-Paley theory and interpolation theorem. This project will complement singular integrals theory further and lay a foundation for applicants studying the properties of singular integrals associated to surfaces(or curves) in depth.

沿曲面(或曲线)的奇异积分与许多重大数学问题有密切联系，如强伪凸域上的边值问题、常系数抛物方程解的正则性估计、多复变域上的Cauchy-Riemann方程、双层流体的界面方程刻画，它的有界性问题是国际数学研究的热点和难点。沿曲面(或曲线)的奇异积分是由主值积分定义的，要考虑该主值奇异积分的点态存在性就需探讨相应极大奇异积分的有界性。基于申请人前期对沿曲面的奇异积分在Triebel-Lizorkin空间上的有界性的研究，本项目将运用Fourier变换估计、Littlewood-Paley理论和实插值方法，探讨沿非凸曲线的极大奇异积分、沿旋转曲面的粗糙奇异积分和极大粗糙奇异积分在$L^p$空间上的有界性。本课题将丰富奇异积分理论，为申请人进一步探讨沿曲面(或曲线)的奇异积分的性质奠定基础。

该项目比既定计划提前圆满完成，预期成果顺利取得。项目组证明了沿非凸曲线的极大Hilbert变换的L^p有界性，沿非凸旋转曲面的粗糙奇异积分的L^p有界性及沿非凸旋转曲面的极大奇异积分的L^p有界性。作为后续研究，项目组还探讨了沿凸曲线的Hilbert变换在L^p(l^q)上的有界性和带变量核的Carleson极大算子的L^p有界性。相关研究成果已以学术论文的形式展现，项目组共撰写相关学术论文4篇，其中三篇分别在SCI收录期刊"Nonlinear Analysis Forum"、"Journal of Osaka Mathematics" 和"Chinese Annals of Mathematics " 上发表，另一篇在"ISRN Mathematical Analysis"上发表。