Study of Multilinear singular Integral and Littlewood-Paley Operator

1.多线性奇异积分与Littlewood-Paley算子我们给出了关于多线性奇异积分和分数次积分的一个已知Lp估计和Lipschitz估计的简单证明，并利用这个新思想得到了已知事实的一个推广版本(文献2)。我们得到了Campanato空间中带有粗糙核的参量化Marcinkiewicz积分的存在性和有界性的几乎最终结果(文献1)。引入一些新的思想，我们得到了Marcinkiewicz积分沿变核曲面的新的L2估计(文献3，12，13)。我们还得到了Littlewood-Paley算子的加权估计(第14条)。奇异积分关于Lp和Lp之间的合并空间上的奇异积分，我们得到了与经典情形相同的估计，并给出了对偏微分方程解的一个应用(文献15)。我们还得到了一个关于沿可变曲面的奇异积分的结果(文章第11号)。关于傅立叶乘子和伪微分算子，我们也得到了三个相关的结果(文章9，17，18)。本文建立了雅可比多项式正交级数在Hardy空间中的一个移植定理，并在实直线区间上的加权Hardy空间中证明了Burkhold-Gundy-Siverstein定理的类比(文献4、5、16)。给出了紧支撑函数原子分解的一种构造性方法(文章8)。相关函数空间我们证明了可以用广义原子来定义广义Morrey-Campanato空间的准对偶，推广了著名的Coifman-Weiss结果(文献19)。我们还得到了关于Campanato空间、Morrey空间和Orlitz-Morrey空间的新结果(文献6，10)。

1. Multilinear singular integrals and Littlewood-Paley operatorsWe gave a simple proof of a known Lp and Lipschitz estimates about multilinear singular and fractional integrals, and by this new idea we got an extended version of known facts (Article No.2). We got almost final results about existence and boundedness of parametrized Marcinkiewicz integral with rough kernel in Campanato spaces (Article No.1). Introducing some new ideas, we obtained new L2 estimates for Marcinkiewicz integrals along surfaces with variable kernels (Article No.3, 12, 13). We also got weighted estimates for Littlewood-Paley operator (Article No. 14).2. Singular integralsAbout singular integrals on amalgum spaces between Lp and lp, we obtained the same estimates as in the classical case, and gave an application to PDE (Article No. 15). We also got an result about singular integrals along variable surfaces (Article No. 11).3. About Fourier multipliers and Pseudo differential operators, we also got three results as related theme (Article No.9, 17, 18).4. Hardy space theory as related themeWe established a transplantation theorem in Hardy spaces for Jacobi polynomials orthogonal series, and we showed an analogy of Burkholder-Gundy-Siverstein's theorem in the weighted Hardy spaces on intervals of the real line (Article No.4, 5, 16). We gave an constructive method of an atomic decomposition of functions with compact support (article No.8).5. Related function spacesWe showed that one can define predual of generalized Morrey-Campanato spaces in terms of generalized atoms, which extends the known famous Coifman-Weiss result (Article No.19). We also obtain new results on Campanato spaces, Morrey spaces, and Orlitz-Morrey spaces (Article No.6, 10).