Harmonic measure in higher co-dimension

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Speaker: 
S. Mayboroda
Institution: 
U. Minnesota
Time: 
Thu, 04/13/2017 - 4:00pm
Location: 
RH 306

 

 

Over the past century an effort to understand dimension and structure of the harmonic measure spanned many spectacular developments in Analysis and in Geometric Measure Theory. Uniform rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of the harmonic Riesz transform in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions. While many of geometric and analytic problems remain relevant in sets of higher co-dimension (e.g., a curve in $\RR^3$), the concept of the harmonic measure is notoriously missing. In this talk, we introduce a new notion of a "harmonic" measure, associated to a linear PDE, which serves the higher co-dimensional sets. We discuss its basic properties and give large strokes of the argument to prove that our measure is absolutely continuous with respect to the Hausdorff measure on Lipschitz graphs with small Lipschitz constant.