Interior H\"older regularity of the linearized Monge-Amp\`ere equation

In this paper, we investigate the interior H\"older regularity of solutions to the linearized Monge-Amp\`ere 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-Guti\'errez H\"older estimate (Amer. J. Math. 119 (1997), no.\,2, 423-465) and the result of Le (Comm. Math. Phys. 360 (2018), no.\,1, 271-305) for the linearized Monge-Amp\`ere 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-Amp\`ere equation. Building on this idea, we also establish a new Moser-Trudinger type inequality in dimension two. In higher dimensions, we derive the interior H\"older estimate under certain integrability assumptions on the coefficients using De Giorgi's iteration.