Mathematical Sciences: Multi-Parameter Harmonic Analysis andLittlewood-Paley Inequalities

本项目的工作将集中在现代谐波分析理论及其与偏微分方程的关系中出现的问题。当微分方程用各种方法转换成积分表达式或运算符时，这种关系自然产生。1976年，得到了Calderon-Zygmund运算界与加权函数范数的基本不等式。随后进行了大量的估算工作，但大多数工作仅限于单参数操作。在这个项目中，提出了一个多参数理论进行研究。在这种情况下，由于高维并矢区间的复杂性，基本不等式更难实现。然而，对于两个参数的情况已经建立了不等式。应用于椭圆算子，如双参数薛定谔算子和相应测不准原理的发展。

Work to be done on this project will focus on problems arising in the modern theory of harmonic analysis and its relationship with partial differential equations. This relationship arises naturally when differential equations are transformed by various means into integral expressions or operators. In 1976, the fundamental inequalities relating bounds of Calderon-Zygmund operate to weighted function norms were obtained. Extensive work on these estimates followed, but most work was confined to one-parameter operators. In this project, a multiparameter theory is laid out for investigation. The basic inequalities are much harder to achieve in this context because of the complexities of higher dimentional analogues to dyadic intervals. Nevertheless, inequalities have been established for the two parameter case. Applications are to be made to elliptic operators such as two-parameter Schrodinger operators and the development of a correspondinguncertainty principle.