Convegno
“WORKSHOP ON GLOBAL AND MICROLOCAL ANALYSIS”

We study the Cauchy problem \begin{equation} \label{CP} \begin{cases} P(t,x,D_t,D_x)u(t,x) =f(t,x) \\ u(0,x)=g(x) \end{cases}, \qquad (t,x) \in [0,T] \times \mathbb{R}, \end{equation} for $p$-evolution operators of the form $$P(t,x,D_t,D_x)= D_t + a_p(t) D_x^p + \sum_{j=1}^{p-1} a_j(t,x)D_x^j, \qquad (t,x) \in [0,T]\times \mathbb{R},$$ where $a_p \in C([0,T], \mathbb{R})$ and $a_j \in C([0,T], C^\infty(\mathbb{R}; \mathbb{C})), j=0,\ldots,p-1,$ in the Gevrey functional setting. When the coefficients $a_j(t,x), j=0,\ldots,p-1,$ of the lower order terms are complex-valued, it is possible to obtain well-posedness results in Gevrey spaces under suitable decay assumptions on $a_j$ for $|x| \to \infty.$ In the first part of the talk, we present a well-posedness result for $3$-evolution equations obtained in [1]. In the second part we discuss necessary conditions for Gevrey well-posedness in the case of $p$-evolution equations for an arbitrary positive integer $p$, see [2]. The results presented in the talk are obtained in collaboration with Alexandre Arias Junior (Universit\`{a} di Torino) and Alessia Ascanelli (Universit\`{a} di Ferrara). References: [1] A. Arias Junior, A. Ascanelli, M. Cappiello, \textit{Gevrey well-posedness for $3$-evolution equatons with variable coefficients}, 2022. To appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. DOI: 10.2422/2036-2145.202202\_011, https://arxiv.org/abs/2106.09511; [2]A. Arias Junior, A. Ascanelli, M. Cappiello, {\it On the Cauchy problem for $p$-evolution equations with variable coefficients: a necessary condition for Gevrey well-posedness}. Preprint (2023), https://arxiv.org/abs/2309.05571

We illustrate the asymptotic behaviour of the eigenvalue counting function for self-adjoint, positive, elliptic linear operators, defined through classical weighted symbols of order (1,1), on an asymptotically Euclidean manifold X. We first prove a two term Weyl formula, improving previously known remainder estimates. Subsequently, we show that, under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. This is joint work with Moritz Doll.

In 1978 Metivier showed that a differential operator with real-analytic coefficients is elliptic if and only if any non-analytic Gevrey vector is a Gevrey function of the same order. In this talk we generalize Metivier's Theorem to Denjoy-Carleman classes given by weight sequences. In particular we show that if $\mathcal{E}^{\{\mathbf{M}\}}$ is a Denjoy-Carleman class such that the associated Borel map is surjective, then there is a vector $u$ of class $\{\mathbf{M}\} $ for any non-elliptic differential operator with real-analytic coefficients, which is not an element of $\mathcal{E}^{\{\mathbf{M}\}}$. This is joint work with Gerhard Schindl.

Magic angles are a topic of current interest in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. In this talk, we shall discuss a simple operator describing the chiral limit of twisted bilayer graphene, whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. By adapting analytic hypoellipticity results of Kashiwara, Trepreau, Sjöstrand, and Himonas, we show that the corresponding eigenfunctions decay exponentially in suitable geometrically determined regions, as the angle of twisting decreases. This is joint work with Maciej Zworski.

It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the Chevalley-Eilenberg complex. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. In this talk, we discuss how to obtain an injective homomorphism between the cohomology spaces associated with left-invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex. We discuss some cases in which the homomorphism is surjective, such as the Dolbeault cohomology and certain elliptic and CR structures. The results provide new insights regarding the general theory of involutive structures as, for example, they reveal algebraic obstructions for solvability for the associated differential complexes.

This talk will be an overview of some relations between the geometry and analysis of CR structures in three dimensions. The subject began with Poincare's observation that in 2-dimensional complex space there are more real hypersurfaces through a given point than there are local biholomorphisms leaving that point fixed. Elie Cartan then computed the geometric invariants that solve the local equivalency problem. Hans Lewy studied Cartan's work, in particular the partial differential operator on the hyperquatric and its relation to the underlying geometry. This led to his famous counterexample - a linear partial differential equation with no solution. The talk will conclude with the realization problem and a false analogy.

Given a function $f\in L^2(\mathbb{R})$, we consider means and variances associated to $f$ and its Fourier transform $\hat{f}$, and explore their relations with the Wigner transform $W(f)$, obtaining, as particular cases, a simple new proof of Shapiro's mean-dispersion principle, as well as a stronger result due to Jaming and Powell. Uncertainty principles for orthonormal sequences in $L^2(\mathbb{R})$ involving linear partial differential operators with polynomial coefficients and the Wigner distribution, or different Cohen class representations, are obtained, and an extension to the case of Riesz bases is studied. This is a joint work with Chiara Boiti (Università degli Studi di Ferrara) and Alessandro Oliaro (Università di Torino)

Time-frequency localization operators (with Gaussian window) were introduced by Daubechies in 1988. Since then they have been studied intensively, in particular regarding boundedness, compactness, Schatten properties and estimates for the eigenvalues. However, sharp estimates for the norm of these operators are still few. In this talk I will present a classical result by Lieb and two new result that give sharp estimates for the norm of localization operators under the assumption that the weight function belongs to one or more $L^p$ spaces.

We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey-setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated to an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the PhD-thesis of Javier Jiménez-Garrido. This is joint work with David Nicolas Nenning from the University of Vienna.

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb-Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential oerators originally proved by Grubb-Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents.