Interior Hölder regularity of the linearized Monge-Ampère equation

In this paper, we investigate the interior Hölder regularity of solutions to the linearized Monge-Ampère equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic equation and singular Abreu equations. In the two-dimensional case, we give a new proof of the Caffarelli-Gutiérrez Hölder estimate ($\mathit{\text{Amer.}}\mathit{\text{J.}}\mathit{\text{Math.}}$ $\mathbf{\text{119}}$ (1997), no.,2, 423-465) and the result of Le ($\mathit{\text{Comm.}}\mathit{\text{Math.}}\mathit{\text{Phys.}}$ $\mathbf{\text{360}}$ (2018), no.,1, 271-305) for the linearized Monge-Ampère equation with singular right-hand side term in divergence form. The main new ingredient in the proof contains the application of the partial Legendre transform to the linearized Monge-Ampère equation. Building on this idea, we also establish a new Moser-Trudinger type inequality in dimension two. In higher dimensions, we derive the interior Hölder estimate under certain integrability assumptions on the coefficients using De Giorgi’s iteration.