2023/2024
Organizers: V. Casarino, B. Gariboldi, A. Martini and A. Monguzzi
Organizers: V. Casarino, B. Gariboldi, A. Martini and A. Monguzzi
11/10/23 - Gustavo Garrigós (University of Murcia) - Pointwise convergence of Poisson integrals associated with the Hermite operator
25/10/23 - Cyril Letrouit (CNRS & University of Paris-Saclay) - Propagation of singularities in subelliptic PDEs
In this talk we consider the wave equation where the Laplacian is replaced by a sub-Laplacian (also called “Hormander sum of square”), which is an hypoelliptic operator. We handle the problem of describing the propagation of singularities in such equations: the main new phenomenon that we describe is that singularities can propagate along abnormal curves at any speed between 0 and 1. This general result extends an idea due to R. Melrose, and we then illustrate it on an example, the Martinet case, following a joint work with Y. Colin de Verdiere. Our statements are part of a classical/quantum correspondence between sub-Riemannian geometry (on the classical side) and the hypoelliptic operator (on the quantum side).
08/11/23 - Luca Brandolini (University of Bergamo) - Irregularities of distribution for bounded sets and half-spaces
Let P_N be a set of N points in R^d (d ≥ 2) and let E ⊆ R^d. We want to estimate the quality of the distribution of these points with respect to a probability measure μ supported in E. We consider a reasonably large family R of measurable sets and, for R ∈ R, we introduce the discrepancy D_N (R) = card (P_N ∩ R) − Nμ (R) . We prove a few theorems which extend several known results.
22/11/23 - María Cristina Pereyra (University of New Mexico) - (Variable) Haar multipliers revisited
In this talk I will discuss variable Haar multipliers which are akin to pseudo-differential operators where the trigonometric functions have been replaced by the Haar functions. On the real line, the symbol of these operators is a function of both the space variable x ∈ R and the “frequency variable” which in this case is encoded in the dyadic intervals I ∈ D, denoted by s : R× D → C. When the symbol is independent of the space variable, s(x, I) = s_I , the operator corresponds to a constant Haar multiplier, when s_I = ±1 it is the well know martingale transform. When the symbol is independent of the “frequency variable”, s(x, I) = s(x), the operator corresponds to multiplication by the function s. We will be mostly concerned with symbols of the form s_t(x, I) = σ_I (w(x)/⟨w⟩_I )^t where t ∈ R, σ_I = ±1, w is a weight, and ⟨w⟩_I denotes the integral average of w over the interval I, the corresponding operators are variable Haar multipliers that we call t-Haar multipliers. Unlike other dyadic operators that can be cast in this framework (like Petermichl’s shift operators) these operators are generally not of Calderon-Zygmund type. We will be concerned with their boundedness properties on Lebesgue spaces and weighted Lebesgue spaces. The recent one and two weight results are joint work with Daewon Chung, Weiyan (Claire) Huang, Jean Carlo Moraes, and Brett Wick.
06/12/23 - Mariusz Mirek (Rutgers University) - Polynomial Progressions in Topological Fields and Their Applications to Pointwise Convergence Problems
We will discuss multilinear variants of Weyl’s inequality for the exponential sums arising in pointwise convergence problems related to the Furstenberg-Bergelson-Leibman conjecture. We will also illustrate how to use the multilinear Weyl inequality and variety of harmonic analysis tools to derive quantitative bounds (in the spirit of Peluse and Prendiville) in Szemeredi’s theorems for polynomial progressions in topological fields.
10/01/24 - Joaquim Ortega-Cerdà (University of Barcelona) - Hypercontractive inequalities of complex polynomials
If we endow the polynomials in one complex variable with the Bombieri norm, we have a Hilbert space of holomorphic functions with reproducing kernel. We will show that every convex functional of the pointwise norm of the normalized functions in the space achieves its extreme at the normalized reproducing kernels. This allows us to give a new, very elementary proof, of a physical conjecture about the behavior of the Wehrl entropy for Bloch coherent states, which was initially proved by Lieb and Solowej. We will also obtain results analogous to the Faber-Krahn inequality in the context of polynomials. This is a joint work with N. Fabio, A. Kulikov and P. Tilli. If time allows it, we will present some more recent results regarding the stability of such type of inequalities.
24/01/24 - Francesco Di Plinio (University of Napoli - Federico II) - Weighted T(1) theorems on Sobolev spaces and applications
The sharp weighted norm inequality for the Beurling transform, due to Petermichl and Volberg, leads to borderline injectivity of the Beltrami resolvent. When the dilatation coefficient belongs to the Sobolev class on a bounded domain D with suitable boundary regularity, quantitative Sobolev estimates for the Beltrami resolvent are instead related to weighted Sobolev norms of the compression to D of the Beurling transform. These norms are connected to the boundary regularity of D by a testing type theorem for singular integrals on domains. In this talk, we describe a wavelet representation formula and the ensuing general testing type characterization of the weighted Sobolev space boundedness of singular integrals on domains, which is sharp and, in this generality, new on Euclidean space as well, recovering the Beurling transform result as a very special case. Joint work with W. Green and B. Wick.
07/02/24 - Michele Villa (University of Oulu) - Faber-Krahn inequalities and Carleson's conjecture in higher dimensions
In this talk I will report on a joint work with Ian Fleschler and Xavier Tolsa on higher dimensional analogues of the Carleson’s epsilon square conjecture. In particular, we characterise cone points of domains in Euclidean space via a novel ”spherical” square function. Beyond this purely geometric result, we explore amusing connections with quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula.
21/02/24 - José Conde Alonso (University of Madrid-Autónoma) - From multipliers on noncommutative groups to bad student products: harmonic analysis to study the geometry of groups
A Schur multiplier is a linear operator associated to a - possibly infinite - matrix M that we call symbol. It acts on other matrices by entry wise multiplication. The boundedness of Schur multipliers in the Schatten-von Neumann classes is an interesting question in operator algebra that was already considered by Grothendieck. In this talk, we will explain how smoothness of the matrix M (seen as a function of two variables) is key to finding sufficient conditions for the boundedness of the corresponding Schur multiplier and other operators of the same flavor. This will come from a surprising connection with Fourier multipliers called transference: the boundedness of a Schur multiplier with symbol M(x, y) on L^p is equivalent to that of a Fourier multiplier with symbols m(x − y). Based on joint works with Adrian M. Gonzalez Perez, Javier Parcet and Eduardo Tablate.
06/03/24 - Nikolaos Chalmoukis (University of Milano-Bicocca) - Convergence of ergodic means of Hilbert space shift operators
In this talk we study the convergence of the ergodic means of an abstract Hilbert space shift operator. Shift operators are the principle examples of non-surjective isometries and have a long history in hard analysis and operator theory. It is a well known fact that if the Koopman operator of a measure preserving system is unitarily equivalent to a Hilbert space shift operator, then the system is strong mixing and therefore ergodic. A classical example of a measure preserving system with this property is the Bernoulli shift and some endomorphisms of the d-dimensional torus. In the particular case of toral endomorphisms we prove almost sharp versions of our results in the speed of convergence. This is a joint work with L. Colzani, B. Gariboldi and A. Monguzzi.
20/03/24 - Maria Reguera (University of Málaga) - Weak weighted estimates for the square function
Perhaps one of the last problems to be understood in the area of weighted sharp estimates is that of the weak weighted estimates for the square function. We will discuss the conjecture and report on recent progress on a relevant problem in the matrix weight setting. This is joint work with Sandra Pott and Gianmarco Brocchi.
10/04/2024 - Valentina Ciccone (University of Bonn) - Endpoint estimates for higher-order square functions and Marcinkievicz multipliers
The Littlewood-Paley square function formed by rough frequency projections adapted to a lacunary partition of the real (frequency) line is a classical object in analysis and it is a bounded operator on L^p for 1 < p < ∞. The same holds for Littlewood-Paley square functions formed by rough frequency projections adapted to higher-order lacunary partitions. None of these operators is of weak-type (1, 1). For the case of first-order lacunary partitions, it follows from a more general result of Tao and Wright that the corresponding rough Littlewood-Paley square function maps Llog^{1/2} L to weak L^1 and such a result is sharp, meaning that the exponent 1/2 cannot be replaced by a smaller one. In this seminar, we will first review these objects and some of their aformentioned properties. Then we will discuss some new results concerning the endpoint mapping properties of higher-order lacunary Littlewood-Paley square functions. These properties are established as a consequence of a more general novel endpoint result for higher-order Marcinkiewicz multipliers and, more generally, for a higher-order variant of a class of multipliers introduced by Coifman, Rubio de Francia, and Semmes and further studied by Tao and Wright. The seminar is based on joint work with Odysseas Bakas, Ioannis Parissis, and Marco Vitturi.
24/04/24 - Carlos Beltran (University of Cantabria) - Lower Bound for the Green Energy of Point Configurations in Harmonic Manifolds
The Green function is a natural function defined in M×M where M any compact Riemannian manifold. Its value is infinity in the diagonal and it defines naturally an energy function: for any collection of N points x1, ..., xN , their Green energy is the sum of G(xi, xj ) where G is the Green function and i ̸= j. A fundamental problem in the interface of several areas including Approximation Theory, Discrete Geometry and Potential Theory is to find and/or describe the properties of points minimizing the Green energy, with M the 2-sphere being a highly non trivial yet very much desired space to analyze. In this talk I will present a surprisingly simple approach to the lower bound of this energy, which gives the sharpest known values in all the Harmonic manifolds.
08/05/24 - Bence Borda (TU Graz) - Some applications of harmonic analysis in optimal transport
The Wasserstein metric is a metric on the set of probability measures of a given space, originating from the theory of optimal transport with many applications in both pure and applied mathematics. It has close connections to harmonic analysis both on classical spaces such as the circle or the torus, and on more abstract spaces such as compact Riemannian manifolds or compact Lie groups. The talk will survey some of these connections, including analogues of the Erdos-Turan inequality for the Wasserstein metric. An important application is the quantitative equidistribution properties of (deterministic or random) finite point sets in compact spaces.