Harmonic Analysis and Linear Elliptic Equations

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ABSTRACTThis proposal is concerned with research on singular integrals, the theory ofweights, with applications to linear elliptic partial differential equations of second and of higher order. There are three main areas ofcurrent focus. First, we study the elliptic measure associated toa second order elliptic operator - both divergence and nondivergence form. Second, we are studying the boundary values of solutions tocertain higher order elliptic operators in non-smooth domains. The thirdpart of this proposal concerns the maximal functions and singularintegrals associated to certain product domains - such operators involvenonisotropic dilations and results require a study of the geometry offamilies of rectangles arising from this dilation structure.In the middle of this century, revolutionary ideas in analysis werebeing forged on two fronts: the theory of singular integrals in harmonic analysis (the Calderon-Zygmund school of analysis) and thetheory of elliptic equations (Di Giorgi, J. Nash). Each of thesesubjects has provided fascinating applications - the first in suchareas as applied harmonic analysis/wavelets/Fourier analysis, and thesecond in the theory of nonlinear partial differential equations, oftremendous value in applications. But at the beginning of these studies, andfor a period of twenty or so years, the development of each of thesesubjects proceeded independently, and with little interaction. However,since the early 1980's, the connections between these areas havebeen increasingly exploited and understood, to the great benefit ofthe fields of harmonic analysis and partial differential equations.This proposer's research is in this field of harmonic analysis, withan emphasis on its connection to and inspiration from partial differential equations.