Alexandrov estimates for polynomial operators by determinant majorization

Abstract

We obtain estimates on the supremum, infimum and oscillation of solutions for a wide class of inhomogeneous fully nonlinear elliptic equations on Euclidean domains where the differential operator is an I-central Garding-Dirichlet operator in the sense of Harvey-Lawson (2024). The argument combines two recent results: an Alexandrov estimate of Payne-Redaelli (2025) for locally semiconvex functions based on the area formula and a determinant majorization estimate of Harvey-Lawson (2024). The determinant majorization estimate has as a special case the arithmetic - geometric mean inequality, so the result includes the classical Alexandrov-Bakelman-Pucci estimate for linear operators. A potential theoretic approach is used involving subequation subharmonics and their dual subharmonics. Semiconvex approximation plays a crucial role.