Speakers

In this talk we will discuss some results related to the sampling of functions in Paley Wiener space. We will start by analyzing whether it is possible to sample near the theoretical critical density given by Landau's theorem. Then we will focus on the frame bounds, since they are very important to guarantee a stable reconstruction of the signal from the samples. In this direction, we will analyze whether it is possible to obtain "good" frame bounds with a density near to the critical one.

Many problems in analysis of the continuum have discrete counterparts, in which often their essential features are more visible and easier to deal with. Sometimes, the problem in the continuum and its discrete ghost are equivalent. We illustrate this principle with some examples from holomorphic space theory and potential theory.

In this talk we will introduce new results concerning greedy-like bases. Concretely, we will talk about the characterization of greediness using polynomials of constant coefficients and, moreover, we will show some relations between approximation spaces and some greedy-like classes of approximation. The results presented in the talk are in some joint preprints with David González, Hung V. Chu, Miguel Berasategui and Eugenio Hernández.

I will talk about a joint result with Gianmarco Giovannardi, Andrea Pinamonti and Simone Verzellesi, concerning the asymptotic behavior of solutions to the subelliptic $p-$Poisson equation as $p\to \infty$ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the infinity Laplacian and the Eikonal equation.

We study the large-time behaviour of solutions to inhomogeneous heat equations in the whole space, with a diffusion operator which may be local or nonlocal both in space and time. We find that the asymptotic profiles depend strongly on the space-time scale and on the time behaviour of the spatial $L^1$ norm of the forcing term. Some of our results are surprising even for the classical heat equation in the somewhat studied case in which the right-hand side is globally integrable in space and time. On the other hand, our assumptions on the source term allow for the space integral to grow to infinity as time goes to infinity. This is joint work with Noemí Wolanski (IMAS-UBA-CONICET, Argentina) and Carmen Cortázar (PUC, Chile), done mostly within the framework of GHAIA.

I will describe recent works related to the heat flow and heat kernel for the Rumin complex of differential forms on Heisenberg groups. After introducing the complex and few preliminary results, I will explain how to derive basic properties on the heat kernel associated to the Rumin laplacian. Having in mind applications to div-curl lemmas, I will provide an application of such results to a Calderon reproducing formula.