Interior $C^2$ Estimates and a Liouville Theorem for the Complex Monge--Amp\`ere Equation

Abstract

We establish an interior $C^2$ estimate for locally bounded plurisubharmonic Bedford-Taylor solutions of the complex Monge--Amp\`ere equation $\det(u_{i\bar j})=f$ under weak quadratic growth bounds on sufficiently large balls, where $f$ is positive and Dini-continuous. We assume a distributional lower bound for the Laplacian of $f^{1/n}$. The main ingredient is a pseudo-$C^2$ estimate using the Green-function averaging error, valid also when $f$ is nonnegative. As an application, we prove that every entire Bedford-Taylor solution of $\det(u_{i\bar j})=1$ with weak quadratic growth is a real quadratic polynomial, resolving a conjecture of G\'abor Sz\'ekelyhidi.