Mathematical Sciences: Fourier Analysis and Partial Differential Equations

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Mathematical research on problems involving the application of harmonic analysis to partial differential equations will be carried out under this award. The work falls into three groups, the first of which concerns the Zygmund conjecture on multiple parameter families of rectangles. It was long believed that the k-parameter maximal function should have boundedness properties depending only on the parameter k and not on the dimension of the underlying space. That this was not necessarily the case was shown by F. Soria in 1986. Since this maximal function is central to any multi-dimensional differentiation theory, it is still of interest to find what the correct formulation of the bound on the maximal function should be. A second line of investigation concerns product singular integrals, a theory which developed from the one-variable theory only after the investment of considerable effort (the natural one-dimensional generalizations failed to carry over). Work is now progressing on the goal of showing that singular integrals can be expressed as pointwise limits of multiply truncated singular integrals. Finally, work in partial differential equations will concentrate on elliptic equations defined in a ball in Euclidean space. If the coefficients of the elliptic part of the operator are bounded and measurable, and the boundary values are required to be in one of the Lebesgue spaces, there are (somewhat surprisingly) cases where the equation has no solution. The question of existence has attracted the attention of many mathematicians over the past decade. What is now evolving are methods of determining existence in terms of the proximity of the operator to other operators for which the Dirichlet problem can be solved. This work will continue along the same line except that the domain under consideration will be unbounded. Significant differences arise in this setting, even for elliptic operators with smooth coefficients.