Mathematical Sciences: Fourier Analysis and Partial Differential Equations

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Fefferman 9311692 This project will focus on several ongoing research themes in harmonic analysis and partial differential equations. In particular work will continue on the extension of the theory of singular integrals on product spaces. Here the objective is to analyze the properties of operators which commute with various groups of dilations of Euclidean space. So far, a complete theory is available only for the classical dilations and product dilations. The theory evolving is quite different from the Calderon-Zygmund theory. Work will also continue on problems from elliptic equations with non-smooth coefficients. In particular, the solvability of the Dirichlet problem with Lp coefficients will be treated. An important goal in this research is to determine the extent that one can perturb coefficients of the equation and still guarantee that the solutions still remain in the same Lebesgue space. In the same spirit, elliptic operators in nondivergence form with coefficients which are only assumed to be measurable will be studied. The approach used will be that of approximating solutions by solutions to equations with smooth coefficients and determine whether or not the process leads to unique limits regardless of the smoothing of the original operator. *** Partial differential operators form a basis for studying the generic forms of partial differential equations. The analysis of these operators through deep abstract methods leads to surprisingly complete information about related classes of equations. Present research concentrates on operators in which smoothness assumptions are limited to the bare essentials. In this way, results reflect a more realistic picture of the physical world which the differential equations seek to represent.