We will discuss a recent improvement on the principle of sparse domination which preserves the cancellative structure of the domain. One of the novelties of the proof is that it avoids the typical weak-type (1,1) approach (which is not strong enough for our purposes). Some applications to Hardy spaces will be explained.
This is a joint work with José M. Conde-Alonso and Emiel Lorist.
The Schrödinger equation on the Heisenberg group is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available.
We obtain refined Strichartz estimates for the sub-Riemannian Schrödinger equation on Heisenberg and H-type Carnot groups reinterpreting Strichartz estimates as Fourier restriction theorems for noncommutative nilpotent groups. The same argument permits to obtain refined Strichartz estimates for the wave equation.