Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

Abstract

We prove distributional inequalities that imply the comparability of the $L^p$ norms of the multiplicative square function of $u$ and the nontangential maximal function of $\log u$, where $u$ is a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case when $u$ is a harmonic function is considered.