Area integrals and the exponential square theorem for elliptic operators with coefficients supported in Whitney type cubes

Abstract

We provide a direct proof of a result comparing the area functions of solutions of two second order linear elliptic operators, when the discrepancy between their main coefficients is supported on Whitney type cubes of the unit ball of n dimensional Euclidean space. Our arguments are specialized to this type of operators, and the vanishing Carleson condition that we adopt is inspired by work of C. Sweezy. The comparison between area functions implies the preservation of the so called exponential square theorem assuming the aforementioned discrepancy of the coefficients. Mathematics subject classification (2010): 42B25, 42B35, 31A20.