Diego Maldonado

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.

A note on the engulfing property and the ¿^1^+^a^l^p^h^a-regularity of convex functions in Carnot groups

We study the engulfing property for convex functions in Carnot groups. As an application we show that the horizontal gradient of functions with this property is Holder continuous.

A visual formalism for weights satisfying reverse inequalities

In this expository article we introduce a diagrammatic scheme to represent reverse classes of weights and some of their properties.

Harnack inequality for the fractional nonlocal linearized Monge–Ampère equation

The fractional nonlocal linearized Monge–Ampère equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge–Ampère sections is proved.

On the Preservation of Eccentricities of Monge–Ampère Sections

A study on the preservation of eccentricities of Monge–Ampere sections under an integral Dini-type condition on the Monge–Ampere measure is presented. The approach is based solely on C2, α-estimates for solutions to the Monge–Ampere equation. The main results are then related to the local quasi-conformal Jacobian problem and to a priori estimates for solutions to the linearized Monge–Ampere equation.

Bilinear Calderón–Zygmund Operators

This chapter is based on the presentation “Generalized bilinear Calderon –Zygmund operators and applications” delivered by the author during the 2008 February Fourier Talks at the Norbert Wiener Center for Harmonic Analysis and Applications, Department of Mathematics, University of Maryland, College Park, on February 21st. In turn, that presentation was based on material from the article “Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity,” J. Fourier Anal. Appl. 15 (2), (2009), 218–261, by Virginia Naibo and the author. This chapter also surveys some more recent results concerning the symbolic calculus and mapping properties of bilinear pseudo-differential operators.