2022/2023
Organizers: V. Casarino, B. Gariboldi, A. Martini and A. Monguzzi
Organizers: V. Casarino, B. Gariboldi, A. Martini and A. Monguzzi
05/10/22 - Izabella Łaba (University of British Columbia) - Favard length estimates via cyclotomic divisibility
The Favard length of a planar set E is the average length of its one-dimensional projections. It is well known (due to Besicovitch) that if E is a purely unrectifiable planar self-similar set of Hausdorff dimension 1, then its Favard length is 0. Consequently, if Eδ is the δ-neighbourhood of E, then the Favard length of Eδ goes to 0 as δ → 0. A question of interest in geometric measure theory, ergodic theory and analytic function theory is to estimate the rate of decay, both from above and below. Partial results in this direction have been proved by many authors, including Mattila, Nazarov, Perez, Volberg, Bond, Bateman, and myself. In addition to geometric measure theory, this work has involved methods from harmonic analysis, additive combinatorics, and algebraic number theory. I will review the relevant background, and then discuss my recent work with Caleb Marshall on upper bounds on the Favard length for 1-dimensional planar Cantor sets with a rational product structure. This improves on my earlier work with Bond and Volberg, and incorporates new methods introduced in my work with Itay Londner on integer tilings.
19/10/22 - Eugenia Malinnikova (Stanford University - NTNU) - Some inequalities for Laplace eigenfunctions and their gradients
We will survey some recent results on restrictions of Laplace eigenfunctions and present new norm inequalities for the eigenfunctions and their gradients obtained in a joint work with Stefano Decio. The guiding principle, which goes back to the works of Donnelly and Fefferman, is that eigenfunctions with eigenvalue E^2 behave like (harmonic) polynomials of degree E.
02/11/22 - Błażej Wróbel (University of Wrocław) - Dimension-free L^p estimates for odd order maximal Riesz transforms in terms of the Riesz transforms
In recent joint work with Maciej Kucharski and Jacek Zienkiewicz (Wroclaw) we proved a dimension-free L^p estimate for the vector of maximal Riesz transforms of odd order in terms of the corresponding Riesz transforms. Analogous dimension-free estimates were also obtained for single Riesz transforms of odd orders with an improved estimate of the constants in terms of p. I will discuss how our work relates to previous results of J. Mateu, J. Orobitg, C. Perez, and J. Verdera. I will also sketch key steps of our arguments.
16/11/22 - Giancarlo Travaglini (University of Milano - Bicocca) - Irregularities of distribution
The term “Irregularities of distribution” appeared for the first time in the title of 1954 K. Roth’s seminal paper and referred to a conjecture of J. van der Corput on the non-existence of a “good” way to choose an infinite sequence in the unit interval. Roth approached van der Corput’s conjecture by checking the quality of any choice of N points in the 2-dimensional torus with respect to arbitrary squares therein, and proving a logarithmic lower bound for the discrepancy. Later W. Schmidt, H. Montgomery and J. Beck independently proved that the discrepancy is at least a power of N when squares are replaced with disks. We construct a family of intermediate cases and we show that positive curvature plays no role in this problem which reduces to a careful study of the decay of certain Fourier transforms. We shall also describe two related d-dimensional problems (from joint works with Luca Brandolini and Leonardo Colzani).
30/11/22 - Sandra Pott (Lund University) - What is good about the Bergman projection?
In a series of papers around 2010, A. Aleman and O. Constantin showed that the Bergman projection on the unit disk behaves much better than the Szeg ̋o projection, if one considers the associated Hankel operators in an infinite-dimensional setting, or more generally, an infinite-dimensional, operator-weighted setting. In this talk, we will investigate why this is so, and make a connection to the theory of sparse, Lerner-type operators. This talk is based on joint work with Adem Limani (Lund University).
14/12/22 - Wolfgang Woess (Graz University of Technology) - Polyharmonic functions on the Poincaré disk
A polyharmonic function of order n on a Euclidean domain is a function which is annihilated by the nth iterate (power) of the Laplacian. There is a large body of literature. Motivated by work on polyharmonic functions on trees [Cohen, Colonna, Gowrisankaran and Singman], [Picardello and W.] and [Sava-Huss and W.], we study λ-polyharmonic functions on the hyperbolic disk, i.e., functions annihilated by the nth iterate of (λI − ∆h), where ∆h is the hyperbolic Laplacian. We derive a boundary integral representation of those functions, introduce and study the polyspherical functions, and present boundary convergence results.
11/01/23 - Rajula Srivastava (Max Planck Institute Bonn - University of Bonn) - The Korányi Spherical Maximal Function on Heisenberg groups
In this talk, we discuss the problem of obtaining sharp L^p → L^q estimates for the local maximal operator associated with averaging over dilates of the Koranyi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Koranyi sphere (in particular, the flatness at the poles) and an “imbalanced” scaling argument encapsulated by a new type of Knapp example, which we shall describe in detail.
25/01/23 - David Beltran (University of Valencia) - Endpoint sparse domination for oscillatory Fourier multipliers
Sparse domination was first introduced in the context of Calderon–Zygmund theory. Shortly after, the concept was successfully extended to many other operators in Harmonic Analysis, although many endpoint situations have remained unknown. In this talk, we will present new endpoint sparse bounds for oscillatory and Miyachi-type Fourier multipliers using Littlewood–Paley theory. Furthermore, the results can be extended to more general dilation-invariant classes of multiplier transformations via Hardy space techniques, yielding results, for instance, for multi-scale sums of radial bump multipliers.
08/02/23 - Francesca Bartolucci (ETH Zürich) - What's new in wavelet phase retrieval?
22/02/23 - Dorothee Frey (Karlsruhe Institute of Technology) - Wave equations with low regularity coefficients
In this talk we discuss fixed-time L^p estimates and Strichartz estimates for wave equations with low regularity coefficients. It was shown by Smith and Tataru that wave equations with C^{1,1} coefficients satisfy the same Strichartz estimates as the unperturbed wave equation on Rn, and that for less regular coefficients a loss of derivatives in the data occurs. We improve these results for Lipschitz coefficients with additional structural assumptions. We show that no loss of derivatives occurs at the level of fixed-time L^p estimates, and that existing Strichartz estimates can be improved. The permitted class in particular excludes singular focussing effects. We also discuss perturbation results, and a recently introduced class of function spaces adapted to Fourier integral operators.
08/03/23 - Diogo Oliveira e Silva (Instituto Superior Técnico Lisboa) - Exponentials rarely maximize Fourier extension inequalities for cones
This talk is based on recent joint work with G. Negro, B. Stovall and J. Tautges. Global maximizers for the L^2 Fourier extension inequality on the cone in R^{1+d} have been characterized in the lowest-dimensional cases d ∈ {2, 3}. We prove that these functions are critical points for the L^p to L^q Fourier extension inequality if and only if p = 2. We also establish the existence of maximizers and the precompactness of L^p-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in R^{1+d}. In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. The proof uses tools from the calculus of variations, bilinear restriction theory, conformal geometry and the theory of special functions.
22/03/23 - Nir Lev (Bar-Ilan University) - Tiling by translates of a function
I will discuss tilings of the real line by translates of a function f, that is, systems {f(x − λ), λ ∈ Λ} of translates of f that form a partition of unity. Which functions f can tile by translations, and what can be the structure of the translation set Λ? I will survey the subject and present some recent results.
05/04/23 - Gabriella Kuhn (University of Milano - Bicocca) - Boundary representations of free groups
Let Γ be a finitely generated free group acting on its boundary Ω endowed with a quasi-invariant measure ν. The quasi-regular representation of Γ on L^2(Ω, dν) is often called a boundary representation. In this talk we shall investigate the case of Markov multiplicative mesures and we show how one can construct a huge family of irreducible representations for each measure. We also show how to relate these constructions to the representations studied by Tim Steger and the author.
19/04/23 - Dmitriy Bilyk (University of Minnesota) - Minimizers of discrete measures and energy integrals
We shall survey various results and conjectures about energy minimization problems that arise in different fields: electrostatics, discrete and metric geometry, discrepancy theory and uniform distribution, signal processing etc. While in many natural examples optimizing the energy imposes uniform distribution, we shall pay special attention to the opposite effect – when minimizers exhibit clustering or discretization, or are supported on small or lower dimensional subsets. We shall also touch upon energies that depend on interactions of three or more particles, rather than just pairwise interactions, and describe difficulties that arise in this setting.
10/05/23 - Mattia Calzi (University of Milano) - Schrödinger Operators on Lie Groups
Let G be a (connected) Lie group, endowed with a left-invariant sub-Riemannian structure, and let L be the associated left-invariant sub-Laplacian. Given a function V ∈ L^1_loc(G) bounded from below, we consider the Schr ̈odinger operator L + V as a self-adjoint operator on L^2(G). We shall present some necessary and sufficient conditions for the discreteness of the spectrum of L + V , and provide characterizations when V is polynomial or belongs to suitable local Muckenhoupt classes. This is joint work with T. Bruno.