Convegno
“SYMPOSIUM IN HARMONIC & COMPLEX ANALYSIS, MICROLOCAL & GEOMETRICAL ANALYSIS AND APPLICATIONS, FOR PHD STUDENTS (SHACAMIGA)”

We consider second-order ergodic Mean-Field Games systems in RN with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. Equilibria solve a system of PDEs where a Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for existence and nonexistence of classical solutions to the MFG system. In the Hardy-Littlewood-Sobolev-supercritical regime, by means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential term. On the other hand, in the Hardy-Littlewood-Sobolev-subcritical regime, using a fixed point argument, we show existence of classical solutions at least for masses smaller than a given threshold value. In the mass-subcritical regime, we show that actually this threshold can be taken to be +∞. Finally, considering the MFG system with a small parameter ε > 0 in front of the Laplacian, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible. First, we obtain existence of classical solutions to potential free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around the minima of the potential.

In this article, we investigate the effects of rotation on the dynamics, by neglecting stratification, in a 2D model where we incorporate the effects of the planetary rotation by adopting the β-plane approximation, which is a simple device used to represent the latitudinal variation in the vertical component of the Coriolis force. We consider the well-known 2D β-plane Navier-Stokes equations (2DβNS) in the statistically forced case. Our problem addresses energy-related phenomena associated with the solution of the equations. To maintain the fluid in a turbulent state, we introduce energy into the system through a stochastic force. In the 2D case, a scaling analysis argument indicates a direct cascade of enstrophy and an inverse cascade of energy. We compare the behaviour of the direct enstrophy cascade with the 2D model lacking the Coriolis force, observing that at small scales, the enstrophy flux from larger to smaller scales remains unaffected by the planetary rotation, confirming experimental and numerical observations. In fact, this is the first mathematically rigorous study of the above equations. In particular, we provide sufficient conditions to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function’s asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. We also prove well-posedness and certain regularity properties necessary to obtain the mentioned results.

In this talk we shall describe the construction of Varopoulos' type extensions of L^p and BMO boundary functions in rough  domains. That is, smooth extensions of functions such that the L^p-norms of their non-tangential maximal function and the Carleson  functional of their gradients can be controlled by the norm of the boundary data. After giving the geometric motivation and a brief survey of known results, we will proceed to present a new and more general approach of constructing Varopoulos' extensions in domains with minor geometrical assumptions for the boundaries.

We consider the quasilinear Cauchy problem (CP) P(t,x,u(t,x),D_t,D_x)u(t,x) = f(t,x), with (t,x)∈[0,T]xR, and initial condition u(0,x) = g(x), x∈R, where P(t,x,u,D_t,D_x) = D_t + a_3(t)D_x^3 + a_2(t,x,u)D_x^2 + a_1(t,x,u)D_x + a_0(t,x,u), a_j(t,x,w), (0≤j≤2), are continuous functions of time t, projective Gevrey regular with respect to the space variable x and holomorphic in the complex parameter w. The coefficient a_3(t) is assumed to be a real-valued continuous function which never vanishes. In this talk we shall discuss how to apply the Nash-Moser inversion theorem in order to obtain local in time well-posedness in projective Gevrey classes for the Cauchy problem (CP).

I will talk about the Baranov-Bessonov-Kapustin conjecture: "let θ be an inner function. Any bounded truncated Toeplitz operator on the model space Kθ admits a bounded symbol only if θ is a one-component inner function." I will present all the objects involved: the model spaces, the one-component inner functions and finally the truncated Toeplitz operators. Eventually, if there is enough time, I will present a possible (in my opinion promising) approach to tackle this problem.

In this talk I will be stating some results about spectral analysis of systems of PDEs. Specifically, a Weyl asymptotic is given for a class of systems containing not only certain quantum optics models such as the Jaynes-Cummings model, which is fundamental in Quantum Optics, but models of geometric differential complexes over R^n, too. Moreover, I discuss a quasi-clustering result for this class of positive systems. Finally, a meromorphic continuation of the spectral zeta function for semiregular Non-Commutative Harmonic Oscillators (NCHO) is given. By “semiregular system” we mean a pseudodifferential systems with a step j in the homogeneity of the jth term in the asymptotic expansion of the symbol. The aforementioned results were obtained jointly with Alberto Parmeggiani.

A function f∈L^2(R) is said to have bandwidth Ω>0, if Ω is the maximal frequency contributing to f. The concept of variable bandwidth arises naturally and it is even more intuitive when we think about music. Indeed, the perceived highest frequency, i.e. the note, is obviously time-varying. This observation provides a reasonable argument for the assignment of different local bandwidths to different segments of a signal when representing it mathematically. However, producing a rigorous definition of variable bandwidth is a challenging task, since bandwidth is global by definition and the assignment of a local bandwidth meets an obstruction in the uncertainty principle. We present a new approach to the study of spaces of variable bandwidth based on time-frequency methods. Our idea is to start with a discrete time-frequency representation that allows us to represent any f as a series expansion of time-frequency atoms with a clear localization both in time and frequency. We may then prescribe a time-varying frequency truncation and, in this way, end up with a space of a given variable bandwidth. For these spaces, we study under which sufficient conditions on a set of points a function can be reconstructed completely from the evaluation of the function at these points. Analyzing some MATLAB experiments, we motivate why these new spaces could be useful for the reconstruction of particular classes of functions.

In this seminar I will introduce the notion of Wodzicki residue, also denoted by non-commutative residue, which was first introduced by Wodzicki in 1984 while studying the meromorphic continuation of the ζ-function for elliptic operators on compact manifold with boundary. The Wodzicki residue was independentely defined by Guillemin in 1985, in the equivalent version of Symplectic residue, in order to find a soft proof of the Weyl formula. It turns out to be the unique trace, up to a multiplication by a constant, on the algebra of classical pseudodifferential operators modulo smoothing operators, provided that the manifold has dimension d>1. In the last years, the interest in the study of Wodzicki residue increased due to its applications both in mathematics (non-commutative geometry) and mathematical physics (relations with Dixmier trace). I will discuss the concept of Wodzicki residue on compact manifold with boundary, for SG-calculus on R^d and for the SG-calculus on manifolds with cylindrical ends. Finally, as a joint work with Professor S. Coriasco, I will present an extension of the non-commutative residue on a certain class of non-compact manifolds called scattering manifolds.

Sampled Gabor phase retrieval --- the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice --- is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in L^2(R), but is not equal to the whole of L^2(R) in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.

In this contributed talk we introduce, in a theoretical representation setting, a necessary and sufficient condition, namely the Rockland condition, for a left-invariant differential operator on a compact Lie group G to be globally hypoelliptic. In particular, we focus on the case of a product of two compact Lie groups G=G1×G2 and we show some examples on T^2 and on T^1×SU(2). It is possible to prove the existence of globally hypoelliptic smooth-coefficient operators that are not locally hypoelliptic. In the end, we present a class of pseudodifferential operators on the product G=G1×G2 and the so called bisingular pseudodifferential calculus, as introduced by L. Rodino in 1975.

We present an explicit area formula to compute the spherical measure of an intrinsic regular graph in an arbitrary homogeneous group. In particular, we assume the intrinsic graph to be intrinsically differentiable at any point with continuous intrinsic differential. This is joint work with V. Magnani.

In this talk, we discuss two classes of spaces of holomorphic functions on the unit disk D. First, the Model Spaces Ku, which arise as the invariant subspaces for the backward shift operator S* on the Hardy space H^2(D), given by S* f(z):=(f(z)-f(0))/z (z∈ D). The second class of spaces that we discuss are the harmonically weighted Dirichlet spaces D(m)$. The space D(m) consists of all analytic functions f on D such that D_m(f) :=∫_D |f'(z)|^2( ∫_{∂D} (1-|z|^2)/|z-\zeta|^2 dm(z)) dA(z) <∞. They are a generalization of the classical Dirichlet space D, and they arise naturally when studying the shift-invariant subspaces of D. After a brief introduction, we discuss sufficient and necessary conditions in order for the embedding Ku ↪ D(m) to hold. This work is related to the boundedness of the derivative operator acting on the model space Ku. This talk is based on joint work with Carlo Bellavita.

See attached file.

It is well-known in the literature on Hardy spaces that the Hardy spaces H^p, 0<p<1, are not normable. However, none of the sources offer proofs of this fact. In 1953, Livingston published an article demonstrating this using a convexity argument based on a theorem by Kolmogorov. In this talk, we will present a direct proof based on a counterexample of the non-normability of the Hardy spaces H^p, 0<p<1. This is a joint work with my thesis advisor Dragan Vukotic.

In a celebrated paper in 1979, Gidas, Ni & Nirenberg proved a symmetry result for a rigidity problem. With minimal hypotheses, the authors showed that positive solutions of semilinear elliptic equations in the unitary ball are radial and radially decreasing. This result had a big impact on the PDE community and stemmed several generalizations. In a recent work in collaboration with Ciraolo, Cozzi & Perugini this problem was investigated from a quantitative viewpoint, starting with the following question: given that the rigidity condition implies symmetry, is it possible to prove that if said condition is "almost" satisfied the problem is "almost" symmetrical? With the employment of the method of moving planes and quantitative maximum principles we are able to give a positive answer to the question, proving approximate radial symmetry and almost monotonicity for positive solutions of the perturbed problem.

We consider the problem of estimating the operator norm of embeddings between certain weighted Paley-Wiener spaces. We discuss some qualitative properties for the extremal problems considered and provide some asymptotic results. For a few cases, we are able to to provide a precise formula for the sharp constant with techniques from the theory of reproducing kernel Hilbert spaces. As an application, these provide sharp constants to higher order Poincare inequalities via the Fourier transform.

The regularity of minimizers of the classical one-phase Bernoulli functional was deeply studied after the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers was investigated as well. We present a regularity result for almost minimizers for a one-phase Bernoulli-type functional in Carnot Groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. Moreover, some recent intrinsic gradient estimates have been employed. Some generalizations will be discussed. Some of the results presented are obtained in collaboration with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University) and will be part of my PhD thesis.

The principal Lyapunov exponent of a dynamical system is a natural measure of the instability of the system. In our work, we computed the supremum of the principal Lyapunov exponent associated to the system dy/dt = A(t)y, y∈R^2, where the function A ranges in L^∞_loc([0,+∞);{A1,A2}), A1,A2∈R^(2x2). This kind of dynamical systems, where the dynamics can be discontinuous with respect to the time variable, are known in literature as switched systems. This computation is reduced to an optimal control problem. Applying Pontryagin Maximum Principle (PMP) to this problem, we were able to find all controls satisfying necessary conditions prescribed by PMP and then we found among them the optimal one. This is a joint work with A. A. Agrachev.

The set of states on CCR(H), the CCR algebra of a separable Hilbert space H, is here looked at as a natural object to obtain a non-commutative version of Freedman’s theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of CCR(H) that are invariant under the action of all automorphisms induced in second quantization by unitaries of H. We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.