Mathematical Sciences: Some Probabilistic Aspects of Harmonic Analysis

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Several mathematical problems situated at the interface between harmonic analysis and probability theory will be addressed in this project. The application of probability theory to questions of mathematical analysis has a rich history of success in refocusing the point of view and providing new, powerful methods. Among the targets of this work is an examination of Khintchine's inequality for lacunary trigonometric series. This inequality bounds the p-norm of the Rademacher expansion of a function by sums of (squares of) the coefficient. The same type of inequality obtains if a lacunary expansion is used. Efforts will be made to show that the constant in the lower bound can be estimated accurately enough to measure its dependence on the p-th power. A second line of investigation concerns tail laws of the iterated logarithm for harmonic functions. This law, discovered by Kolmogorov in 1950, measures partial sums of trigonometric series against the sum of squares of its coefficients. The tail law is similar except that it compares only the tail of the series and tail of the coefficients. Recently analogous results have been obtained in which harmonic functions are compared with iterated logarithms of their gradients. Current work seeks to give estimates of corresponding tail comparison. Other work seeks to relate developments of exponential martingales with classical Littlewood-Paley theory.