2021/2022

We investigate non-centered Hardy-Littlewood maximal operators related to the exponential measure dμ(x) = exp(−|x1| − . . . − |xd|) dx in R^d, d ≥ 2. The mean values are taken over Euclidean balls or cubes (l^∞ balls) or l^1 balls. In the cases of cubes and l^1 balls we prove the L^p-boundedness for p > 1 and disprove the weak type (1, 1) estimate. The same is proved in the case of Euclidean balls, under the restriction d ≤ 4 for the positive result.

The Vapnik–Chervonenkis (VC) dimension was invented in 1970 to study learning models. This notion has since become one of the cornerstones of modern data science. This beautiful idea has also found applications in other areas of mathematics. In this talk we are going to describe how the study of the VC-dimension in the context of families of indicator functions of spheres centered at points in sets of a given Hausdorff dimension (or in sets of a given size inside vector spaces over finite fields) gives rise to interesting, and in some sense extremal, point configurations.

Suppose you are given a real-valued function f(x) and want to compute a local average at a certain scale. What we usually do is to pick a nice probability measure u, centered at 0 and having standard deviation at the desired scale, and convolve. Classical candidates for u are the characteristic function or the Gaussian. This got me interested in finding the ”best” function u – this problem comes in two parts: (1) describing what one considers to be desirable properties of the convolution and (2) understanding which functions satisfy these properties. I tried a basic notion for the first part, ”the convolution should be as smooth as the scale allows”, and ran into fun classical Fourier Analysis that seems to be new: (a) new uncertainty principles for the Fourier transform, (b) that potentially have the characteristic function as an extremizer, (c) which leads to strange new patterns in hypergeometric functions and (d) produces curious local stability inequalities. Noah Kravitz and I managed to solve two specific instances on the discrete lattice completely, this results in some sharp weighted estimates for polynomials on the unit interval – both the Dirichlet and the Fejer kernel make an appearance. The entire talk will be completely classical Harmonic Analysis, there are lots and lots of open problems.

We discuss theories of duality which are applicable to the multijoint and joint problems, which are themselves discrete formulations of multilinear and linear Kakeya problems. This is joint work in part with Timo Hanninen and Stefan Valdimarsson, and in part with Michael Tang.

This will be a survey of two classes of problems in analysis: measuring the size of projections of sets, and incidences of circles in the plane. I will discuss some landmark results and recently discovered connections between the two.

The Kakeya (maximal) conjecture concerns how collections of long, thin tubes which point in different directions can overlap. Such geometric problems underpin the behaviour of various important oscillatory integral operators and, consequently, understanding the Kakeya conjecture is a vital step towards many central problems in harmonic analysis. In this talk I will discuss work with K. Rogers and R. Zhang which apply tools from the theory of semialgebraic sets to yield new partial results on the Kakeya conjecture. Also, more recent work with J. Zahl has used these methods to improve the range of estimates on the Fourier restriction conjecture.

In this talk, which will be based on joint research with S. Buschenhenke and A.Vargas, I intend to discuss some of the new challenges that arose in our studies of Fourier restriction estimates for hyperbolic surfaces, compared to the case of elliptic surfaces. Given the complexity of the bilinear, and even more so of the polynomial partitioning approach, I shall mainly focus on those parts of these methods which required new ideas, so that a familiarity with these methods will not be expected from the audience.

We consider a uniformly elliptic operator LA in divergence form associated with a matrix A with real, bounded, and possibly non-symmetric coefficients. If a proper L^1-mean oscillation of the coefficients of A satisfies suitable Dini-type assumptions, we prove the following: if μ is a compactly supported Radon measure in R^{n+1}, n ≥ 2, the L^2(μ)-operator norm of the gradient of the single layer potential Tμ associated with LA is comparable to the L^2-norm of the n-dimensional Riesz transform Rμ, modulo an additive constant. This makes possible to obtain direct generalizations of some deep geometric results, initially proved for the Riesz transform, which were recently extended to Tμ under a Holder continuity assumption on the coefficients of the matrix A. This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Given a self-adjoint differential operator with continuous spectrum acting on a Hilbert space H , its resonances are the poles of a meromorphic extension across the spectrum of its resolvent acting on a dense subspace of H in which the operator is no longer self-adjoint. They can be thought of as replacements of eigenvalues for problems on noncompact domains. In this talk we will first explore two well-understood cases: the Laplacian on Euclidean spaces and the Laplace-Beltrami operator on rank one Riemannian symmetric spaces of the noncompact type, two settings where a notion of Fourier analysis is available. In both cases, the Laplacian comes from the action of the Casimir operator through the left regular representation of the underlying group, and the Plancherel formula provides a direct integral decomposition of such representation. Elaborating from this point of view, in collaboration with A. Pasquale and T. Przebinda we started to develop methods to study resonances in more general settings. As an example of such methods, in the talk we will consider some instances of Capelli operators and see how one can exploit Howe’s theory for reductive dual pairs. We will also consider the problem of identifying the representations that are naturally attached to the resonances in these settings.

The 2−plane transform is the operator that maps a function to its averages along affine 2−planes. We consider the operator obtained by restricting the allowed directions of the 2−planes to those normal to a fixed surface S (quadratic, for simplicity) of codimension 2. By duality and discretisation, L^p → L^q estimates for such an operator imply Kakeya-type estimates for the supports of Fourier-transformed wave-packets adapted to the surface S (wave-packet decompositions being a powerful tool in proving Fourier Restriction results). We connect this operator to Gressman’s theory of affine invariant measures by showing that if the surface is well-curved `a la Gressman (meaning, the affine invariant surface measure on S is non-vanishing) then the restricted 2−plane transform is L^p → L^q bounded in the maximal range of (p, q) exponents allowed. The proof relies on a characterisation of well-curvedness in Geometric Invariant Theory terms, which will be discussed. Joint work with S. Dendrinos and A. Mustata.