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

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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.