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 (Am J Math 119(2):423–465, 1997) and the result of Le (Commun Math Phys 360(1):271–305, 2018) 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.