The Dirichlet problem with BMO boundary data and almost-real coefficients

Abstract

It is known that a function, harmonic in a Lipschitz domain, is the Poisson extension of a BMO function if and only if its gradient satisfies a Carleson-measure condition. We show that the same is true of functions that satisfy elliptic equations in two-dimensional Lipschitz domains, provided the coefficients are independent of one coordinate and have small imaginary part.