2020/2021
Organizers: V. Casarino, B. Gariboldi, S. Meda and A. Monguzzi
Organizers: V. Casarino, B. Gariboldi, S. Meda and A. Monguzzi
The Hardy space H^2 made its way into signal theory since Wiener's time, and it belongs to the standard toolbox of all engineers who deal with signals. We will see how H^2 and its related function spaces H^1, $H^\infty$, and $BMOA$ arise from basic practical problems, and how multiplication, Toeplitz, and Hankel operators enter the picture. Feedback systems will take us at the front step of Pick interpolation. The aim is advertising a possible intuition of a beautiful chapter of pure mathematics.
This talk is an account of my work on Morrey spaces with Marcel Rosenthal in recent years.
There are several definitions for weighted Morrey spaces. We obtain boundedness results in all of them for operators satisfying the assumptions of the usual extrapolation theorem, that is, we get weighted Morrey estimates from weighted Lebesgue estimates with A_p weights. The results can be applied to a variety of operators and together with the norm estimates, our technique also provides the
definition of the operator by embedding.
Recently we obtained results for a more general class of weighted Morrey spaces from an extension of the usual Muckenhoupt condition to the Morrey setting, involving the Khöthe dual of the space. In some
cases the conditions characterize the weighted inequalities of maximal operators.
(slides)
14/10/20 - Tommaso Bruno (Ghent University) - Factorization properties of smooth functions and vectors
Given a module M over a non-unital algebra A, we say that M has the weak factorization property if M = span{A · M}, while it has the strong factorization property if M = A · M. In this talk we shall review old and recent results about strong and weak factorizations of smooth functions and smooth vectors of Lie group representations. We shall also discuss open problems and current lines of research.
21/10/20 - Giuseppe Negro (University of Birmingham) - Sharp estimates for the wave equation via the Penrose transform
In 2004, Foschi found the best constant, and the extremizing functions, for the Strichartz inequality for the wave equation with data in the Sobolev space $\dot{H}^{1/2}\times\dot{H}^{-1/2}(\mathbb R^3)$. He also formulated a conjecture, concerning the extremizers to this Strichartz inequality in all spatial dimensions d ≥ 2. We disprove such conjecture for even d, but we provide evidence to support it for odd d. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform.
(slides)
28/10/20 - Stefano Decio (NTNU) -Nodal sets of Steklov eigenfunctions
Steklov eigenfunctions are harmonic functions in a bounded domain whose normal derivative at the boundary is proportional to the function itself. We study the zero sets of such functions: we show that there are many zeros near the boundary and we discuss lower and upper bounds for the Hausdorff measure of the zero set; several questions remain unanswered. Extensive comparisons with the (slightly) better understood case of eigenfunctions of the Laplace-Beltrami operator will also be provided.
11/11/20 - Ujué Etayo (TUGraz) - A Bombieri-type inequality for Weierstrass sigma functions
The Bombieri inequality is a classic inequality in number theory, see [B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery, Products of polynomials in many variables, Journal of Number Theory, 36(2):219– 245, 1990]. The original statement says that given two homogeneous polynomials on N variables P,Q respectively of degree m and n, then
$\frac {m!n!}{(m+n)!}}\|P\|^{2}\,\|Q\|^{2}\leq \|P\cdot Q\|^{2}\leq \|P\|^{2}\,\|Q\|^{2},$
where the norm is the Bombieri-Weyl norm. This inequality admits a rewriting in terms of integrals on the sphere, a property exploited in [U. Etayo. A sharp bombieri inequality, logarithmic energy and well conditioned polynomials, 2019]. In a joint work with Joaquim Ortega-Cerdà and Haakan Hedenmalm, we use this new definition to generalize the inequality to other Riemannian manifolds, in particular the torus $\mathbb{C}/\Lambda$.
(slides)
25/11/20 - Leonardo Colzani (Università di Milano-Bicocca) -A chapter in the history of trigonometric series: Trigonometric expansions of Bernoulli polynomials
The title is self explanatory.
02/12/20 - José Luis Romero (University of Vienna) - Sampling, density, and equidistribution
The sampling problem concerns the reconstruction of every function within a given class from their values observed only at certain points (samples). A density theorem gives necessary or sufficient conditions for such reconstruction in terms of an adequate notion of density of the set of samples. The most classical density theorems, due to Shannon and Beurling, involve bandlimited functions (that is, functions whose Fourier transforms are supported on the unit interval) and provide a precise geometric characterization of all configurations of points that lead to reconstruction. I will present modern variants of these results and their applications in other fields of analysis.
(slides)
13/01/21 - Alessio Martini (University of Birmingham) - Spectral multipliers for sub-Laplacians: recent developments and open problems
I will present some old and new results about the Lp functional calculus for sub-Laplacians L. It has been known for a long time that, under quite general assumptions on the sub-Laplacian and the underlying sub-Riemannian structure, an operator of the form F(L) is bounded on Lp, 1 < p < ∞, whenever the multiplier F satisfies a scale-invariant smoothness condition of sufficiently larger order. The problem of determining the minimal smoothness assumptions, however, remains widely open and will be the focus of our discussion.
27/01/21 - Marco Peloso (Università di Milano) -Sampling, interpolation and function theory
In this talk first I will review the notions of sampling and interpolation of holomorphic functions, and try to illustrate their interplay with some problems in function theory, operator theory and harmonic analysis. In the second part of the talk I will indicate some possible developments, recent results and open problems.
10/02/21 - Giona Veronelli (Università di Milano-Bicocca) - Sobolev spaces on manifolds with lower bounded curvature.
There are several notions of Sobolev spaces on a Riemannian manifold: from the operator theory viewpoint it is natural to consider Sobolev functions defined by taking the Lp norms of functions and of powers of their Laplacian. Instead, the regularity theory of elliptic equations involves Sobolev functions defined via the Lp norm of all the derivatives up to a certain order. Moreover, Sobolev spaces can be characterized via compactly supported smooth approximations. In this talk, we will focus on non-compact manifolds with lower bounded curvature. We will discuss some results giving the (non)-equivalence between the different Sobolev spaces. In particular, we will highlight the role played in the theory by the Calderon-Zygmund inequality and the Bochner formulas, and we will sketch how to exploit singular metric spaces (e.g. Alexandrov or RCD) as a tool to construct smooth counterexamples.
24/02/21 - Fulvio Ricci (Scuola Normale Superiore) - Multi-parameter structures
In this talk we give a survey on a certain number of multi-parameter structures, on R^n and on nilpotent groups, that have been introduced in the last 20 years. They include flag and multi-norm structures. These structures are intermediate between the one-parameter dilation structures of standard Calderon-Zygmund theory and the full n-parameter product structure. Each structure has its own type of maximal functions, singular integral operators, square functions, Hardy spaces.
10/03/21 - Maria Vallarino (Politecnico di Torino) - Analysis on trees with nondoubling flows
The classical Calderon–Zygmund theory was developed in the Euclidean space and, more generally, on spaces of homogeneous type, which are measure metric spaces with the doubling property. In this talk we consider trees endowed with flow measures, which are nondoubling measures of at least exponential growth. In this setting, we develop a Calderon–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in the classical setting. This is a joint work with Matteo Levi, Federico Santagati and Anita Tabacco.
24/03/21 - Gian Maria Dall'Ara (Indam/Scuola Normale Superiore) - L^p mapping problems for Bergman projections
This is for the most part a survey talk. I will discuss various aspects of the following problem: for which values of p is the Bergman projection of a given domain in Cn bounded on Lp? The answer depends heavily on the complex geometry of the domain. We will discuss the problem in one and several variables, its connection with the theory of conformal mappings and that of singular integrals, highlighting many open problems.
31/03/21 - Loredana Lanzani (Syracuse University) - The commutator of the Cauchy-Szegő projection for domains in $C^n$ with minimal smoothness
Let D ⊂ C^n be a bounded, strongly pseudoconvex domain whose boundary bD satisfies the minimal regularity condition of class C^2. We characterize boundedness and compactness in L^p(bD, ω), for 1 < p < ∞, of the commutator [b, Sω] where Sω is the Cauchy–Szegő (orthogonal) projection of L^2(bD, ω) onto the holomorphic Hardy space H^2(bD, ω) and the measure ω belongs to a family (the “Leray Levi-like” measures) that includes induced Lebesgue measure σ. We next consider a much larger family of measures {Ωp} modeled after the Muckenhoupt Ap-weights for σ: we define the holomorphic Hardy spaces H^p(bD, Ωp) and we characterize boundedness and compactness of [b, SΩ2] in L^2(bD, Ω2). Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy–Szegő kernel that are not available in the settings of minimal regularity of bD and/or Ap-like measures. This is joint work with Xuan Thinh Duong, Ji Li and Brett D. Wick.
07/04/21 - Giacomo Gigante (Università di Bergamo) - On the Schwarz alternating method
By means of a few examples and simulations, we would like to give an idea of the Schwarz alternating method for the solution of boundary value problems. We will review some of the historic milestones, and explain how Fourier Analysis may contribute to the topic.
21/04/21 - Ivan S. Trapasso (Università di Genova) - Dispersion, spreading and sparsity of Gabor wave packets
Sparsity properties for phase-space representations of several types of operators (including pseudodifferential, metaplectic and Fourier integral operators) have been extensively studied in recent articles, with applications to the analysis of dispersive evolution equation. It has been proved that such operators are approximately diagonalized by Gabor wave packets - equivalently, the corresponding phase-space representations (Gabor matrix/kernel) can be thought of as sparse infinite-dimensional matrices. While wave packets are expected to undergo some spreading and dispersion phenomena, there is no record of these issues in the aforementioned estimates. We recently proved refined estimates for the Gabor matrix of metaplectic operators, also of generalized type, where sparsity, spreading and dispersive properties are all simultaneously noticeable. We also provide applications to the propagation of singularities for the Schr ̈odinger equation; in this connection, our results can be regarded as a microlocal refinement of known estimates. The talk is based on joint work with Elena Cordero and Fabio Nicola.
(slides)
05/05/21 - Joan Verdera (Universitat Autònoma de Barcelona) - Boundary regularity of vortex patches for some transport equations
A vortex patch is a weak solution of the vorticity form of the planar Euler equation which is the characteristic function of a domain depending on time. Then vorticity is 1 in the domain at time t and 0 outside. A famous theorem of Chemin from the 90s, based on paradifferential calculus, states that if the domain has smooth boundary at time 0 then it has smooth boundary for any time. We have found a new proof of this theorem which works for many other transport equations and is basically classical analysis. I will discuss the context and the analytic tools we use: commutators, Whitney’s extension Theorem, singular integrals... This is joint work with Cantero, Mateu and Orobitg.
(slides)
12/05/21 - Jim Wright (University of Edinburgh) - Some remarks on pointwise ergodic theorems
In this talk we revisit some results of Bourgain from the 80’s on pointwise ergodic theorems with respect to certain arithmetic sets of integers as well as the multiparameter pointwise ergodic theorem of Dunford and Zygmund from 1951. We develop a perspective to put these results in a single framework.
19/05/21 - Marco Fraccaroli (Universität Bonn) - Duality for outer L^p spaces
The theory of L^p spaces for outer measures, or outer L^p spaces, was developed by Do and Thiele to encode the proof of boundedness of certain multilinear operators in a streamlined argument. Accordingly to this purpose, the theory was developed in the direction of the real interpolation features of these spaces, while other questions remained untouched. For example, the outer L^p spaces are defined by quasi-norms generalizing the classical mixed L^p norms on sets with a Cartesian product structure. Therefore, it is natural to ask whether in arbitrary settings the outer L^p quasi-norms are equivalent to norms. In this talk, we will answer this question, with a particular focus on certain settings on the upper half space R^d × (0, ∞) related to the work of Do and Thiele.
09/06/21 - Angela Pasquale (Université de Lorraine) - Symmetry breaking operators for real reductive dual pairs
A symmetry breaking operator is an intertwining operator from an irreducible representation of a group to an irreducible representation of a subgroup. Symmetry breaking operators are intrinsically related to branching problems, which justifies their name. In Howe’s theory of theta correspondence, the space of the symmetry breaking operators from the Weil representation of the metaplectic group to an irreducible representation of a reductive dual pair is one dimensional. An explicit construction of a symmetry breaking operator in this space gives an additional insight into the theta correspondence. In this talk, which is based on an ongoing project with Mark McKee and Tomasz Przebinda (University of Oklahoma), I will present the general context of this problem and some results for the case of real reductive dual pairs with one member compact.
16/06/21 - Gianmarco Brocchi (University of Birmingham) - Sparse T1 theorems
Many operators in analysis are non-local, in the sense that a perturbation of the input near a point modifies the output everywhere; consider for example the operator that maps the initial data to the corresponding solution of the heat equation. Sparse Domination consists in controlling such operators by a sum of positive, local averages. This allows to derive plenty of estimates, which are often optimal. For example, it has been shown that Calderon-Zygmund operators and square functions admit such a domination even under minimal T1 hypotheses. In this talk we introduce the concept of sparse domination and present a sparse T1 theorem for square functions, discussing the new difficulties and ideas in this case. Time permitting, we will see how sparse domination can be applied in very different context.
30/06/21 - Filippo De Mari (Università di Genova) - Views on the Radon Transform
I will recall and introduce some of the many existing Radon transforms, focusing in particular on the setup of G-dual pairs (X, Ξ) introduced by Helgason more than fifty years ago, where G is a locally compact group that acts transitively both on X and Ξ. I will then present some results obtained in collaboration with G. S. Alberti, F. Bartolucci, E. De Vito, M. Monti and F. Odone which bring into play (square integrable) representations. If the functions to be analyzed live on X and the quasi regular representation of G on L^2(X) and L^2(Ξ) are square integrable, then it is possible to write a nice inversion formula for the Radon transform associated to the families of submanifolds of X that are prescribed by the object Ξ which is dual to X. This formula hinges on a unitarization of the Radon transform that may be proved in a rather general setup if the quasi regular representations of G on L^2(X) and L^2(Ξ) are irreducible, and on an intertwining property of the Radon transform. The former result is inspired by work of Helgason. Some examples are discussed, mostly the guiding case related to shearlets that points in the direction of possible practical inversion techniques.