Mathematical Sciences: Some Probabilistic Aspects of Harmonic Analysis

几个数学问题位于谐波分析和概率论之间的接口将在这个项目中解决。概率论在数学分析问题中的应用在重新聚焦观点和提供新的、强大的方法方面有着丰富的成功历史。这项工作的目标之一是检查Khintchine的不等式为空三角级数。这个不等式通过系数的平方和限定函数的Rademacher展开式的p范数。如果使用空白展开，则得到相同类型的不等式。我们将努力表明，下界的常数可以足够精确地估计，以测量它对p次幂的依赖性。第二项研究涉及调和函数的迭代对数的尾律。这个定律是由Kolmogorov在1950年发现的，它测量了三角级数的部分和和系数的平方和。尾定律与之相似，只是它只比较级数的尾和系数的尾。最近也得到了类似的结果，将调和函数与其梯度的迭代对数进行了比较。目前的工作旨在给出相应尾部比较的估计。其他工作试图将指数鞅的发展与经典的Littlewood-Paley理论联系起来。

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.