Decomposition into Martingales: An Invariance Principle

Abstract

Our goal in this chapter is to show that arbitrary harmonic functions in the upper half space can be approximated by dyadic martingales so that both the error terms in this approximation and the area function of the martingale are controlled by the area function of the harmonic function. This technique is very similar to the invariance principles discussed in Chapter 6, hence the title for this chapter. Much of what we do here was done in Bañuelos, Klemes, and Moore [BKM1], and in Bañuelos and Moore [BM2]. However, it is to be emphasized that the techniques of these two references are essentially a refinement of those found in Chang, Wilson, and Wolff [CWW]. In order to understand the motivation for the results and techniques here, as well as the historical precedents, we first discuss the Chang-Wilson-Wolff result.