Title:Product Formulas for Positive Measures and Applications
Abstract: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 volumelike 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 CalderonZygmund machinery.
