“…Futhermore, if f is close to a constant, then W 2,p loc estimates are proved in [4] for the classical Monge-Ampère equation (i.e., when A ≡ 0), and in [20] under some stronger conditions on A (namely, the inequality in (1.6) should be strict when ξ, η = 0). distance function on the sphere and its perturbations [8,9,14,15,22], it looks plausible to us that, at least in some particular regimes, this result may have applications in the study of generalized semi-geostrophic system on the sphere [7].…”