Title: Understanding BMO and VMO using elliptic systems in the upper-half space

Abstract: Harmonic Analysis plays a fundamental role in the study of boundary value problems for elliptic operators. In the simplest case of the Laplacian in the upper half-space, the Dirichlet boundary value problem with data in BMO (i.e., having bounded mean oscillation) is solved using the harmonic Poisson kernel, and the corresponding solutions satisfy a Carleson measure estimate. This was established by Fabes, Johnson, and Neri who also obtained a Fatou type result allowing them to recover the space BMO from the non-tangential traces of harmonic functions in the upper half-space satisfying a Carleson measure condition.

The goal of this talk is to extend the previous ideas to the case of second-order, homogeneous, elliptic systems, with constant complex coefficients, such as the Lam\’e system of elasticity. Furthermore, we also consider the Dirichlet problem with data in VMO (i.e., having vanishing mean oscillation) and find the corresponding Fatou type result which allows us to identify VMO with the traces of null solutions satisfying a vanishing Carleson measure condition.  As a consequence, we obtain that the set of smooth functions in BMO are dense in VMO, improving  Sarason’s classical result describing VMO as the closure of uniformly continuous functions in BMO. In hindsight, we are able to obtain a harmonic analysis result using PDE techniques.