Mathematical Sciences: Fourier Analysis and Partial Differential Equations

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Work to be done during the term of this project will cover several problem areas of mathematical analysis. They include studies of maximal functions and their control of the differentiation of integrals, multiple parameter singular integrals and the theory of elliptic partial differential equations. Much of the present work continues earlier efforts to extend fundamental one (complex) variable results to higher dimensions. Basic concepts such as maximal functions and parametrized families of rectangles for differentiation do not extend directly. Special classes have been substituted and gradually a full Hardy-space environment is being developed. The goals are the same: to obtain bounds on the fundamental operators of analysis such as the Riesz transform and the multidimensional Hilbert transform. Connecting maximal operators with classical singular integrals much be accomplished before these bounds and the resulting applications can be made. The study of Dirichlet problems on half-space relates directly to the questions of harmonic analysis related above. The boundary function and its nontangential maximal function are to be in some Lebesque class. This work is distinguished from earlier investigations in that it treats equations defined on unbounded domains. A general theory of existence of solutions has yet to be established. This research will concentrate on developing criteria in terms of the coefficients of the matrix determining the equation. Additional work will be done in extending classical restriction theorems for the Fourier transform to the setting of compact Riemannian manifolds without boundary and to study smoothness of Radon transforms over lower dimensional submanifolds.