Solving second order elliptic boundary value problems with rough (i.e. merely bounded and measurable) coefficients has been of interest for some time, both in the analysis of PDEs and in harmonic analysis. A 'first-order approach' to these problems was introduced by Auscher, Axelsson, and McIntosh in 2010, and since then this approach has been elaborated upon, extending its range of applicability.
When one considers such elliptic boundary value problems with data in L^p smoothness spaces (for example, Hardy-Sobolev and Besov spaces), this first-order approach involves abstract function spaces adapted to differential operators. These abstract spaces are built using tent spaces and holomorphic functional calculus.
In this talk, I will explain the above paragraphs in considerably more detail than in this abstract. An longer and more technical version of this talk will be given in Friday's harmonic analysis workshop.