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

摘要

We prove distributional inequalities that imply the comparability of theLpnorms of the multiplicativesquare function ofuand the nontangential maximal function oflogu, whereuis a positive solution of a nondivergenceelliptic equation. We also give criteria for singularity and mutual absolute continuity with respect toharmonic 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 isintroduced and where the case whenuis a harmonic function is considered.