Mathematical Sciences: Harmonic Analysis, Sobolev Inequalities and Geometry

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Three main themes highlight this mathematical research. The first combines ideas of harmonic analysis and probability theory. They involve infinite series of Haar functions - orthogonal step functions defined on an interval. The power norm of any alternating sum is always dominated by a fixed multiple of the original norm. Although the sharpest multiplier is known, the alternating sums which maximize or minimize the norm are not easily obtained. In this project, work will be done in determining optimal choices of sign change based on the coefficients of the original series. A second line of investigation will focus on exponential multiples of the Laplace operator on the surface of a sphere. The logarithm of the determinant of such an operator contains a nonnegative energy term which vanishes only when the exponential is the potential in the basic conformality equation. This result has implications in one lower dimension in that it becomes the classical Lebedev-Milin inequality which, in turn, derives from Szego's theorem. Efforts will be made in reversing the argument in higher dimensions; from a version of the Szego theorem to the energy inequality. Underlying this research are basic questions concerning the spectrum of the Laplacian. Work of a more geometric flavor will continue on the question of which functions defined on a sphere can be the curvature functions of conformally related metrics. Sufficient topological conditions are known depending on the critical points of the function. Necessary and sufficient conditions based on analytic properties of the function will be sought.