Applications of the Calderon-Zygmund theory to PDE's

Question

I am planning to build a PDE topics course focussing on the Calderon-Zygmund theory. I know some important applications of the Calderon-Zygmund theory to elliptic PDEs, but I don't know enough to get the whole picture (I have an impression that it is used in hyperbolic problems also). My question is, what are my choices on the "applications" part of the course? One application per answer, and please include references if possible.

To get started, I have the following examples (please expand these if you want)

• Standard elliptic estimates for the Laplacian in the $L^p$ setting can be established by using the Calderon-Zygmund theory applied to the Riesz potential.

• Elliptic estimates for more general constant coefficient operators can be proved by using the Littlewood-Paley theory, which in turn is established by using the Calderon-Zygmund type estimates.

Answer 1

starting with the laplacian is a good idea. I suggest Stein old book about Singular integrals. Then, constant coefficient operators in principal part is just change of choordinates. Then you could move to the variable coefficients operators. I recommend you to talk about Meyer counterexample (see Gianquinta book on Multiple integrals in the Calculus of Variations). Then the theory has different branches accordind to the smoothness of the leading coefficients. If you want to consider the continuous coefficients case I suggest to give a look to Campanato papers in the 60's If you want to consider some discontinuity of the coefficients you may look at the paper by Campanato about the Cordes condition i.e. the eigenvalues of the elliptic matrix are close to those of the laplacian. Otherwise you may consider the case of the VMO discontinuity. In that case, the theory is more recent. See the papers by Chiarenza Frasca and Longo (two papers) 1991-1993 They consider the non variational operators and the strong solutions Otherwise, the divergence form operators are considered in Di Fazio 1996. If you need any other suggestion let me know.