Hessian Estimates for Nondivergence Parabolic and Elliptic Equations with Partially BMO Coefficients

Abstract

In this paper, we mainly prove a local Calderon–Zygmund estimate in the Lorentz spaces with a variable power $$p(\cdot )$$ to the Hessian of nondivergence parabolic equations $$u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t)=f(x,t)$$, under assumptions that the variable exponent $$p(\cdot )$$ is $$\log $$-Holder continuous, and the coefficient is a small partially BMO matrix which means that $$a_{ij}(x,t)$$ is merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables. In addition, we also derive a similar result for nondivergence elliptic equations $$a_{ij}(x)D_{ij}u(x)=f(x)$$ with small partially BMO coefficients.