Product Formulas for Positive Measures and Applications
We will discuss a simple product formula for general positive measures on Euclidean space. This formula was introduced (as far as I know) by R. Fefferman, C. Kenig, and J. Pipher) in their work on harmonic measure for certain elliptic PDE's. I will discuss applications to various problems. The first of these (joint work with D. Bassu, L. Ness, V. Rokhlin) is to analysis of volume-like signals that arise in communication networks. We will show how one various coefficients that come from the product formula can be used for classification. We then discuss how an arbitrary measure on the unit circle can be related to a (unique) curve in the plane. Certain classes of random curves arising in SLE processes are intimately related to this procedure (joint work with K. Astala, A. Kupiainen, E. Saksman). Finally we will present a few of the ideas that arise in recent work with M. Csörnyei on a seemingly unrelated problem. An old theorem of Rademacher states that a Lipschitz mapping from one Euclidean space to another is differentiable almost everywhere. We show that, in any dimension, given a set E of Lebesgue measure zero, there is a mapping from R^n to itself that is nowhere differentiable on E. This was previously known when n = 1 (where it is very simple) and n = 2 (which is a difficult recent result due to others). Along the way we need to introduce some new types of objects in the classical Calderon-Zygmund machinery.