Muckenhoupt Weights are ubiquitous in the field of harmonic analysis. In particular, they are appropriate weights for weighted $L^p$ bounds for many classical harmonic analysis objects (maximal functions, Riesz transforms, etc.) These weights also play a role in the solvability of the $L^p$ Dirichlet problem for the Laplacian in `rough’ sets. In particular, if the harmonic measure is a Muckenhoupt weight (in some sense) then the $L^p$ Dirichlet problem is solvable for some p. Moreover, if the harmonic measure is a Muckenhoupt weight then one can gain geometric information about the domain or its boundary (this is a free boundary problem).

This talk will briefly introduce Muckenhoupt weights and harmonic measure, then we will discuss some recent developments in the study of free boundary problems for harmonic measure. In particular, I will discuss (two-phase) free boundary problems where both the harmonic measure for a domain and its exterior are `asymptotically optimal' Muckenhoupt weights.

This is joint work with M. Engelstein, M. Goering, S. Hofmann, T. Toro and Z. Zhao.