Cristian Rios

The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds

We prove the Kato conjecture for degenerate elliptic operators on R n . More precisely, we consider the divergence form operator Lw = −w 1 divA∇, where w is a Muckenhoupt A2 weight and A is a complex-valued n × n matrix such that w 1 A is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup e tLw satisfies Gaussian bounds, then the weighted Kato square root estimate, kL 1/2 w f k L2(w) ≈ k∇ f k L2(w), holds.

From Sobolev inequality to doubling

In various analytical contexts, it is proved that a weak Sobolev inequality implies a doubling property for the underlying measure.

THE KATO PROBLEM FOR OPERATORS WITH WEIGHTED ELLIPTICITY

We consider second order operators Lw = w 1 divAwr with ellipticity controlled by a Muckemphout A2 weight w. We prove that the Kato square root estimate L 1=2 w f L2(w) kr fkL2(w) holds in the weighted space L 2 (w).

Smoothness of radial solutions to Monge-Ampère equations

We prove that generalized convex radial solutions to the generalized Monge-Ampere equation det D 2 u = f(|x| 2 /2, u, |∇u| 2 /2) with f smooth are always smooth away from the origin. Moreover, we characterize the global smoothness of these solutions in terms of the order of vanishing of f at the origin.