Probabilistic Littlewood -Paley theory

Abstract

In this dissertation, we study the solution to a Dirichlet problem on the upper-half space [special characters omitted]. We define the harmonic extension of an L^P[special characters omitted] function with respect to a product process of a d-dimensional symmetric stable process and 1-dimensional Brownian motion. Using this harmonic extension, we introduce Littlewood-Paley functions and prove some results on norm comparability of a function f and its corresponding Littlewood-Paley function. As a result of the operators obtained, we prove a multiplier theorem.