# Product Formulas for Measures and Applications to Analysis

# Product Formulas for Measures and Applications to Analysis

We will discuss elementary product formalisms for positive measures. These appeared in analysis for purposes of examining "harmonic measures" related to elliptic equations (work of R. Fefferman, J. Pipher, C. Kenig). We will discuss three topics where product formulas appear: applied projects related to signal processing; SLE; and Geometric measure theory. For the first topic we will explain some work arising in analysis of network failures. For the second topic (SLE) we will show the relations between some models of random measures, and relations to SLE. The third topic (geometric measure theory) will be a discussion of joint work with Marianna Csörnyei. The main point here is how product formulas can detect directionality in sets. The new result concerns Lebesgue measurable sets $E$ of finite measure in the unit cube (in any dimension). The set $E$ can be decomposed into a bounded number of sets with the property that each (sub)set has a nice "tangent cone". This yields strong results on points of non-differentiability for Lipschitz functions. The main technical result needed is a $d$-dimensional, measure theoretic version of (a geometric form of) the Erdös-Szekeres theorem, which holds when $d=2$. In what is perhaps a small surprise, certain ideas from random measures can be used effectively in the deterministic setting.