Gradient potential estimates for linearized Monge-Ampère equations

This paper establishes a pointwise gradient potential estimate for solutions to linearized Monge-Amp\`ere equations and derives a modulus of continuity estimate for the gradient in terms of the associated section adapted potential. As applications, we obtain a sufficient criterion for the continuity of the gradient, local BMO and VMO estimates for the gradient with respect to Monge-Amp\`ere sections, and a local $C^{1,\gamma}$ regularity criterion.