Calderon-Zygmund (CZ) theory and Homogenization of Random Elliptic and Parabolic PDE

This talk will be concerned with some classical results (CZ and Aronson theorems) in harmonic analysis and their application to homogenization of linear uniformly elliptic and parabolic PDE with random coefficients. The CZ theorem (1952) concerns the boundedness of singular integral operators on p integrable functions. The Aronson theorem (1967) gives bounds on the Green’s function for uniformly elliptic and parabolic PDE which depend only on the ellipticity constant. We will show how these theorems can be used to obtain rate of convergence results in homogenization of PDE with random coefficients, and in Euclidean field theories with uniformly convex Lagrangians.