Nonlinear Aspects of Calderón-Zygmund Theory

Abstract

Calderon-Zygmund theory is classically a linear fact and amounts to get sharp integrability and differentiability properties of solutions of linear equations depending on those of the given data. A typical question is for instance: Given the Poisson equation −∆u=μ, in which Lebesgue space do Du or D 2 u lie if we assume that μ∈L γ for some γ≥1? Questions of this type have been traditionally answered using the theory of singular integrals and using Harmonic Analysis methods, which perfectly fit in the case of linear equations. The related results lie at the core of nowadays analysis of partial differential equations (pde) as they often provide the first regularity information after which further qualitative properties of solutions can be established. In the last years there has anyway been an ever growing number of results concerning nonlinear equations: put together, they start shaping what we may call a nonlinear Calderon-Zygmund theory. This means a theory which reproduces for non-linear equations the results and phenomena known for linear ones, without necessarily appealing to linear techniques and tools. The approaches are in this case suited to the special equations under consideration. Yet, although bypassing general Harmonic Analysis tools, in some way they preserve the general spirit of some the basic Harmonic Analysis ideas, applying them directly at a pde level. This is a report on some of the main results available in this context.