Harmonic Analysis and the Theory of Elliptic Measure

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).There are two parts to the research program in this proposal. Part one concerns multi-parameter Fourier analysis, the higher dimensional (or product) theory with origins in the theory of several complex variables, convergence of multiple Fourier series and analysis on symmetric spaces. Some of the modern questions proposed here are related to singular integrals, Hardy spaces, averaging, bilinear operators, and reach into probability theory through multi-parameter martingales, and higher dimensional Brownian sheets. Part two of this proposal concerns the theory of second order elliptic partial differential equations, in both divergence and non-divergence form. The questions are mainly centered on the fine properties of the elliptic measure associated with the solvability of boundary value problems and are posed in two settings: when the operator itself has non-smooth coefficients, or for smooth operators in domains whose boundary is less regular yet satisfies some geometric conditions (Lipschitz, chord-arc, Reifenberg flat, for example).In 1811, Joseph Fourier proposed a mathematical theory of heat which took the point of view that heat was a ``flow" and should be described by differential equations. This was the starting point of the theory now called Fourier analysis, a field which has an impact on every area of mathematics and the physical sciences. In this proposal, we consider some basic open problems related to such differential equations, and try to understand how their solutions depend on boundary conditions. Solving these problems may lead to a better understanding of how to obtain approximations to solutions (using computational tools) when it is impossible to write down a formula which describes them exactly. This grant will partially support graduate students and undergraduates in this field by providing them the opportunity to do research in this area (individually and in teams) and to travel to conferences where they can present their own results.