The talk gives first an overview of the method called Compensated Compactness. We work out the special role of the so called div -curl lemma and of Fourier-Multipliers estimates. The survey is based on Work of Morrey, Murat, Tartar, J. Ball, and S. Mueller.
The second part presents a solution to a problem of Tartar concerning weak lower semicontinuity of separately convex Lagrangians. The proof exploits Riesz Transforms, Calderon Zygmund operators and interpolatory estimates for directional Haar projections.