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.
We introduce the Schrödinger equation with a fractional Laplacian on real hyperbolic spaces and their discrete analogues, homogeneous trees. While on real hyperbolic spaces, the Strichartz estimates for the fractional Laplacian exhibit a loss of derivatives (due to the Knapp phenomenon), in the setting of homogeneous trees, this loss vanishes due to the triviality of the estimates for small times. This is a joint work with Jean-Philippe Anker and Yannick Sire.
The purpose of this talk is to present some new insights on a classical model of visual neurons in brain's primary visual cortex V1 from the point of view of harmonic analysis.
Most of the linear behaviour of a single V1 neuron can be described as a wavelet coefficient of the visual stimulus, associated with the quasiregular representation of the SE(2) group on a gaussian mother wavelet. However, the set of available coefficients (as observed with neurophysiological measurements on V1) has a puzzling distribution, which does not immediately resemble known sampling sets. We want to present - with analytical and numerical arguments - an approach that can allow us to obtain frame conditions, and quantify the accuracy of the representation, for a relevant function space of images. We will also show an elementary reconstruction algorithm that does not require the explicit computation of a dual frame.