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.
See here for the abstract.
In this talk we investigate the behavior of the "typical" Laplacian eigenfunction of a compact smooth Riemannian manifold. In particular, motivated by both Yau's conjecture on nodal sets and Berry's ansatz on planar random waves, we consider Gaussian eigenfunctions on the sphere and study the distribution of the length of their nodal lines in the high energy limit. This result raises several questions regarding both the distribution of other geometric functionals and the behavior of nodal statistics of random eigenfunctions on a "generic" manifold. (This talk is mainly based on a joint work with D. Marinucci and I. Wigman)
This talk surveys several developments in the study of directional singular integrals and maximal directional averages. These questions are motivated in part by differentiation along directions and by the maximal Kakeya and Nikodym conjectures, while their singular integral counterparts have a deep connection to Carleson’s theorem on pointwise convergence of Fourier series and, in higher dimensions, to the problem of spherical summation. Part of the talk will give a high-level description of some strategies of proof leading to the recent resolution of the sharp L²-bounds for maximal directional averages in higher dimensions. This talk reports on joint work with Francesco Di Plinio (Università di Napoli Federico II).
The Schrödinger map equation, sometimes referred to as the Landau-Lifshitz equation, is a continuum model describing the dynamics of the spin in ferromagnetic materials. The main objective of this talk is to present our recent advances in understanding the dynamical behaviour of solutions to this model. We will see how a geometric approach, that has been fruitful in the study of self-similar solutions to 1D-Schrödinger maps and other related equations, can shed light on the study of equivariant Schrödinger maps in two spatial dimensions.
In this lecture, we present a modern perspective on function smoothness and regularity by employing sharp tools from Harmonic Analysis. The first part of the talk moves beyond classical pointwise analysis to explore functional smoothness through the lens of local averages. This approach provides a more flexible and robust framework for understanding analytical behavior, particularly in settings where traditional derivatives are not or cannot be considered.
In the second part, we apply these harmonic analysis techniques to derive improved local Poincaré-Sobolev estimates, which are instrumental in proving the celebrated De Giorgi regularity theorem. This framework also provides a proof of the well-known John–Nirenberg theorem for BMO functions. The central theme of the discussion will be the ’self-improving’ property—a remarkable phenomenon where modest local control over oscillations leads to significantly stronger global integrability. As another application, we present an extension of the celebrated Nash inequality (which yields another proof of De Giorgi’s theorem), as well as an improved generalization of the Gagliardo–Nirenberg–Sobolev theorem using Campanato spaces.
The aim of this talk is to give an overview of the program of extending the classical Birkhoff ergodic theorem to the case of polynomial orbits. In particular, we will discuss recent progress toward resolving the Furstenberg–Bergelson–Leibman conjecture, which concerns multilinear averages involving a family of measure-preserving transformations that generate a nilpotent group. We will also highlight the role of various tools from harmonic analysis in this context. The presentation is based on joint work with Mariusz Mirek, Sarah Peluse, Renhui Wan, and James Wright.
Let (X, d) be a geodesic Gromov hyperbolic space and let μ be a positive measure on X. Assume that μ is locally doubling and that the measure of metric balls satisfies ca^r ≤ μ(B_r(x)) ≤ Cb^r for all r > 1, x ∈ X, for some 1 < a ≤ b < a^2. We prove that the centred Hardy–Littlewood maximal operator is of weak type (log_a b, log_a b), and that this exponent is optimal. The proof is based on a geometric approximation of hyperbolic spaces by suitable graphs, called spiderwebs, which are 1–quasi-isometric to the original space. We also discuss the behaviour of the uncentred maximal operator, showing that its boundedness properties differ drastically from the centred case. Moreover, we show that, by replacing balls with suitable subsets in the definition of the uncentred maximal operator, one can recover strong-type L^p bounds for all p > 1. Based on joint works with N. Chalmoukis, S. Meda, and E. Papageorgiou.