Archivio 2026 150 seminari

In this talk we discuss spectral perturbation theory from an abstract perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, in the first part of the talk we discuss a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which depends on the limit eigenspace and of the magnitude of the perturbation through the reso- lution of a minimization problem. In the second part of the talk, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights; secondly, we in- vestigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Rie- mannian manifold changes when a second small manifold is glued on a small portion of it. This is a joint project with Andrea Bisterzo, Roberto Ognibene and Prasun Roychowdhury.

We will consider the coupled system of damped $\sigma$-equation \begin{equation*}\begin{cases} \partial_t^2U + (-\Delta)^\sigma U + D \partial_t U = F(U), & t>0, \ x\in\R^n, \\ (U,\partial_tU)(0,x)=(0,\Phi(x)), & x\in\R^n, \end{cases} \end{equation*} where $\sigma>1$,~$U=(u_1,u_2)\tm$,~$\Phi=(\varphi_1,\varphi_2)\tm$ and \begin{equation*} D U = \begin{pmatrix} a_1(-\Delta)^{\frac{\theta_1}2} & 0 \\ 0 & a_2 (-\Delta)^{\frac{\theta_2}2} \end{pmatrix},\quad \theta_1,\theta_2\in[\sigma,2\sigma], \ a_1,a_2\geq0, \end{equation*} is a damping term in the noneffective case. We will analyze the various scenarios of existence of global-in-time solution that arise due to the semilinear term $F(U)=(f_1(U),f_2(U))\tm$, in the case of a general semilinear coupling \[ F(U)=\begin{pmatrix} c_{11}|u_1|^{p_{11}} + c_{12}|u_2|^{p_{12}}\\ c_{21}|u_1|^{p_{21}} + c_{22}|u_2|^{p_{22}} \end{pmatrix}, \] % or in the case of weak coupling of derivative type \[ F(U)=\begin{pmatrix} |\partial_t u_2|^{p_{2}}\\ |\partial_t u_1|^{p_{1}} \end{pmatrix}. \] Sharp $L^p-L^q$ estimates for the corresponding linear scalar equation and a suitable application of the test function method are crucial for determining the so-called critical curves for the existence of small data solutions.\\ Based on a joint work with Marcello D'Abbicco (University of Bari).

The purpose of this talk is to present some recent advances in the phase space analysis of quadratic differential operators modeled by the Weyl quantization of a complex-valued quadratic symbol. The first part will focus on metaplectic and Schroedinger propagators, where Gabor wave packet decompositions lead to a structured representation which cap- tures insightful phenomena (such as sparsity, dispersion, spreading, confinement). In the second part, we will discuss how these methods provide a unified framework also for non-selfadjoint quadratic models, where heat-type dissipative effects play a crucial role, emphasizing how a phase space perspective effectively connects all the relevant features of the problem.

We consider the NLS equation with combined power-type nonlinearities in the mass-subcritical regime, where a focusing leading term is perturbed by a lower order defocusing term, and we prove small data scattering. The proof relies on the pseudo-conformal transformation in conjunction with a general variational argument used to obtain the positivity of certain modified energies. The smallness assumption is only on the mass of the initial datum, and not on the whole Σ-norm. This is a joint work with J. Bellazzini and V. Georgiev.

I will present a work in progress that I am developping jointly with S. Ciani and I. Skrypnik, about the degenerate parabolic equation of Trudinger type in anisotropic form. After recalling the fundamental solutions of Barenblatt type in the anisotropic regime, and the associated sub-potential estimates, I will present a weak Harnack inequality that is adapted to the intrinsic geometry of the problem, and via this inequality I will prove some rigidity results for the non-negative solutions of the equation. This approach opens to a possible extension of the De Giorgi-Nash-Moser theory in non-isotropic setting.

Segregation problems arise when competing species tend to occupy disjoint regions, leading naturally to a free boundary separating them. In this talk, we will discuss a parabolic model in which the diffusion is governed by fully nonlinear operators. We will describe the main ideas behind the existence of Lipschitz solutions, the free boundary condition, and the regularity of the interface near nondegenerate points. This is ongoing joint work with Emily Casey (U. Minnesota), Cornelia Mihaila (Saint Michael’s College), and Clara Torres-Latorre (ICMAT).

In this talk, we discuss how the time-dependent coefficients in a semilinear wave equation influence the critical exponent, i.e., the threshold value for the power of the nonlinear term that separates the blow-up region from the global existence region.

In this talk, we discuss about the regularity properties of fractional logarithmic p-Laplacian. We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic p-Laplace operator. For a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{{\rm d}}{{\rm d}t}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order s\in (0, 1). The kernel of this operator involves a logarithmic factor and it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional p-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in our setting. We also present an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear case p=2.

The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. They established that any possible singularity of a periodic solution to the Boltzmann equation must be visible macroscopically. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions. In this talk, I will first give an accessible overview of the conditional regularity program by Imbert and Silvestre, highlighting its main ideas and implications. As a first step toward understanding the boundary case, I will then discuss the smooth- ness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection and in-flow condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov’s equation. Finally, I will report on recent joint works with Xavier Ros-Oton and Kyeongbae Kim, in which we establish sharp boundary regularity results for this class of equations.

Lecture 1. l^2 and the shift in l^2. The Hardy space H^2. The shift in H^2. Inner and outer functions; Beurling’s decomposition. Beurling’s theorem. Abstract interpretation of Beurling’s theorem. Shifts in Hilbert spaces. Index of a shift. Equivalence of shifts. Lecture 2. Rota’s universality theorem. Relation with the invariant subspace problem: maximal shift-invariant subspaces. Shift in l^2(H^ 2). The Beurling-Lax theorem. The Beurling-Lax matrix of a shift-invariant subspace of l^2(H^2). Determinantal operators and determinantal subspaces. Lecture 3. Shift-invariant subspaces in l^2(H^2) of finite rank are infinite intersection of determinantal subspaces. Limit of shift-invariant subspaces. Shift-invariant subspaces in l^2(H^2) are limit of infinite intersection of determinantal subspaces. Maximal shift-invariant subspaces in finite direct sum of H^2 and in l^2(H^2).

Lecture I: The invariant form of the Schwarz lemma can be interpreted to say that every holomorphic function from the disk D to itself is distance reducing in the pseudohyperbolic metric, an extremely useful property. Caratheodory (1927) showed how to port the pseudohyperbolic metric to any domain U, in one or several variables, by considering all holomorphic maps from U to D. Kobayashi later (1967) dualized the construction, by considering maps from D to U. Every holomorphic map from U_1 to U_2 is distance reducing with respect to both the the Caratheodory distance and the Kobayashi distance. In 1981, Laszlo Lempert proved the wonderful theorem that on convex domains, both these distances are equal to each other. Lecture II: We will discuss Agler's 1990 operator theory proof of Lempert's theorem, which involves a very careful analysis of two point interpolation problems, and the fact that certain two dimensional representations are contractive if and only if they are completely contractive. Lecture III: If V is a lower dimensional subset of a domain U, it has both its intrinsic Caratheodory and Kobayashi distances, and the ones it inherits from U.When are these the same? This is a complex analogue to asking when a submanifold, or subvariety, of a larger manifold is totally geodesic. The question lies somewhere on the nexus of Pick interpolation problems, von Neumann inequalities, contractive distances, and complex geometry. We will discuss what is known, including some recent results with L. Kosinski in the case that U is the polydisk.

We focus on providing detailed proofs of the quantitative characterizations for a broad class of finite products (words) of these paraproducts act- ing on standard weighted Bergman spaces. These specific cases serve to illustrate the fundamental tools and techniques required to obtain a boundedness characterization for arbitrary words.

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I will discuss how to use the classical Schur's algorithm for analytic functions as a tool for solving inverse spectral problems. The connection between Schur's algorithm and Spectral theory was found very recently. Among other things, it gives sharp spectral stability results for Dirac operators with potentials of class L^2. I will discuss these results and formulate some open questions.

A classical result of Hardy and Littlewood states that the uncentred Hardy–Littlewood maximal operator (UHLMO) on balls is bounded on L p (R n ) for all p > 1 and it is of weak type (1, 1). It is well known that, replacing R n with either homogenous trees of degree at least three or the hyperbolic plane, the UHLMO on balls is bounded on L p if and only if p > 2, and it is of restricted weak type (2, 2). Loosely speaking, the different behaviour of the UHLMO on R n and on the hyperbolic plane depends on the fact that (maximal) fam- ilies of balls with large radii have “worse” overlapping properties in the latter case than in the former. In this talk we consider the UHLMO with respect to half balls, and prove that, perhaps surprisingly, this operator is bounded on L p for all p > 1 and it is not of weak type (1, 1) both on homogeneous trees of degree at least three and on the hyperbolic plane. We shall also discuss generalisations of this result to Damek–Ricci spaces. This is joint work with Nikos Chalmoukis (Milano–Bicocca), Effie Papageorgiou (Paderborn) and Federico Santagati (Politecnico di To- rino).

Motivated by applications in the control theory of infinite-dimensional systems, several authors have investigated the boundedness of the Laplace transform $\mathcal{L}: L^p(0, \infty) \rightarrow L^q( \mathbb{C}_+, \mu)$ in terms of properties of the measure $\mu$, with the case $q=p=2$ being the classical Carleson Embedding Theorem. We will cover the case $p >2$, thus completing the picture to all $p,q$ with $1/p + 1/q \le 1$. In case $p >2$, the Laplace-Carleson Embedding Theorem can also be seen as an appropriate replacement of the Hausdorff-Young inequality. This is joint work with Eskil Rydhe (Lund).

We obtain a characterization of complete interpolating sequences in a class of Fock-type spaces with radial weights for which such sequences exist. Our criterion is formulated in terms of logarithmic separation and controlled perturbations of a reference sequence satisfying an Avdonin-type condition. This provides a geometric description of complete interpolating sequences and extends previous results of Borichev–Lyubarskii and Baranov–Belov–Borichev on Riesz bases of reproducing kernels in Fock-type spaces. It also yields explicit density criteria for sampling and interpolating sequences. This is joint work with Y. Omari

Classical Clark measures are singular measures on the unit circle defined via inner functions that are closely tied to important topics in operator theory and complex analysis (for example, model spaces, compressed shifts, and composition operators). In this talk, we’ll consider an analogous definition for Clark measures associated with two-variable inner functions. For certain classes of such functions, we’ll give exact formulas for these Clark measures, characterize when associated Clark embeddings are unitary, and obtain nice unitary perturbations of pairs of compressed shift operators. This is joint work with John Anderson, Palak Arora, Linus Bergqvist, Joseph Cima, Conni Liaw, and Alan Sola.

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Given an involution on a complex variety, the Smith-Thom inequality says that the total \mathbb{F}_2-Betti number of the fixed locus is no greater than the total \mathbb{F}_2-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. On a Hyper-Kähler manifold X a holomorphic or a anti-holomorphic involution is referred to as a brane involution. While examples of non-compact hyper-Kähler manifolds admitting maximal branes are known, the compact case is more intriguing. In particular, although there exist some K3 surfaces admitting maximal brane involutions, the main result that I will show you is the non-existence of maximal branes on Hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of points on a K3 surface. This talk is based on a joint work with S. Billi, L. Fu and V. Kharlamov.

I will report about my recent joint work with Angel Rios Ortiz on the SYZ conjecture for a special class of singular symplectic varieties. The SYZ conjecture predicts that nef and isotropic line bundles are associated to lagrangian fibrations. After having recalled some generalities about symplectic varieties and the SYZ conjecture, I will state the main result and explain the main ideas behind its proof.