Advances in Microlocal and Time-Frequency Analysis

Frontmatter

Anisotropic Gevrey-Hörmander Pseudo-Differential Operators on Modulation Spaces

Abstract

We show continuity properties for the pseudo-differential operator \(\operatorname {Op} (a)\) from \(M(\omega _0\omega ,\mathscr B )\) to \(M(\omega ,\mathscr B )\), for fixed s, σ ≥ 1, \(\omega ,\omega _0\in \mathscr P _{s,\sigma }^0\) (\(\omega ,\omega _0\in \mathscr P _{s,\sigma }\)), \(a\in \Gamma ^{\sigma ,s}_{(\omega _0)}\) (\(a\in \Gamma ^{\sigma ,s;0}_{(\omega _0)}\)), and \(\mathscr B\) is an invariant Banach function space.

Ahmed Abdeljawad, Joachim Toft

Hardy Spaces on Weighted Homogeneous Trees

Abstract

We consider an infinite homogeneous tree \(\mathcal V\) endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space \((\mathcal V,d,\mu )\) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H ¹(μ) on \((\mathcal V,d,\mu )\) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H ¹(μ).

Laura Arditti, Anita Tabacco, Maria Vallarino

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

Abstract

We consider the initial value problem for the plate equation with (t, x) −depending complex valued lower order terms. Under suitable decay conditions as |x|→∞ on the imaginary part of the subprincipal term we prove energy estimates in weighted Sobolev spaces. This provides also well posedness of the Cauchy problem in the Schwartz space \(\mathcal {S}(\mathbb R^n)\) and in \(\mathcal {S}^\prime (\mathbb R^n)\).

Alessia Ascanelli

Cone-Adapted Shearlets and Radon Transforms

Abstract

We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the one-dimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform.

Francesca Bartolucci, Filippo De Mari, Ernesto De Vito

Linear Perturbations of the Wigner Transform and the Weyl Quantization

Abstract

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.

Dominik Bayer, Elena Cordero, Karlheinz Gröchenig, S. Ivan Trapasso

About the Nuclearity of and

Abstract

We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space \(\mathcal {S}_{(M_p)}\) to be nuclear. As a consequence, we obtain that for a weight function ω satisfying the mild condition: 2ω(t) ≤ ω(Ht) + H for some H > 1 and for all t ≥ 0, the space \(\mathcal {S}_\omega \) in the sense of Björck is also nuclear.

Chiara Boiti, David Jornet, Alessandro Oliaro

Spaces of Ultradifferentiable Functions of Multi-anisotropic Type

Abstract

The paper deals first with the relationship between multi-anisotropic Gevrey spaces and Denjoy-Carleman spaces and then it introduces a class of ultradifferentiable functions unifying these both spaces.

Chikh Bouzar

Comparison Principle for Non-cooperative Elliptic Systems and Applications

Abstract

In this paper are given some sufficient conditions for validity of the comparison principle for linear and quasi-linear non-cooperative elliptic systems. Existence of classical solutions is proved as well.

Georgi Boyadzhiev, Nikolay Kutev

On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation

Abstract

In the present paper we consider the Dirichlet problem for the n-dimensional Helmholtz equation. In particular we deal with the problem of representability of the solutions by means of simple layer potentials. The main result concerns the solvability of a boundary integral equation of the first kind. Such a result is here obtained by using the theories of differential forms and reducible operators.

Alberto Cialdea, Vita Leonessa, Angelica Malaspina

Decay Estimates and Gevrey Smoothing for a Strongly Damped Plate Equation

Gevrey Spaces Meet Math Everywhere!

Abstract

In this note, we study a damped plate equation. On the one hand, the action of the damping creates a smoothing effect in Gevrey classes, on the other hand, it dissipates the energy of the solution.

Marcello D’Abbicco

Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation

Abstract

We study the Cauchy problem for the linear generalized double dispersion equation and derive long time decay estimates for the solution in L ^p spaces and in real Hardy spaces.

Marcello D’Abbicco, Alessandra De Luca

On Density Operators with Gaussian Weyl Symbols

Abstract

The notion of reduced density operator plays a fundamental role in quantum mechanics where it is used as a tool to study statistical properties of subsystems. In the present work we review this notion rigorously from a mathematical perspective using pseudodifferential theory, and we give a new necessary and sufficient condition for a Gaussian density operator to be separable.

Maurice A. de Gosson

On the Solvability of a Class of Second Order Degenerate Operators

Abstract

In this paper we will be concerned with the problem of solvability of second order degenerate operators that are not of principal type. We will describe some recent results we have obtained about local solvability in the Sobolev spaces of a class of degenerate operators which is an elaboration of the class considered by Colombini-Cordaro-Pernazza (in turn, an elaboration of the adjoint of the Kannai operator).

Serena Federico, Alberto Parmeggiani

Small Data Solutions for Semilinear Waves with Time-Dependent Damping and Mass Terms

Abstract

We consider the following Cauchy problem for a wave equation with time-dependent damping term b(t)u _t and mass term m(t)²u, and a time-dependent non-linearity h = h(t, u):

$$\displaystyle \begin {cases} u_{tt}-\Delta u+b(t)u_t+m^2(t)u=h(t,u), & t\geq 0, \ x\in \mathbb R^n,\\ u(0,x)=f(x), \quad u_t(0,x)=g(x). \end {cases} $$

Here, we consider an effective time-dependent damping term and a time-dependent mass term, in the case in which the mass is dominated by the damping term, i.e. m(t) = o(b(t)) as t →∞. Under suitable assumptions on the non-linearity h = h(t, u) (Hypothesis 1.3), we prove the global existence of small data solutions in a supercritical range \(p>\bar p\), assuming small data in the energy space (f, g) ∈ H ¹ × L ².

Giovanni Girardi

Integrating Gauge Fields in the ζ-Formulation of Feynman’s Path Integral

Abstract

In recent work by the authors, a connection between Feynman’s path integral and Fourier integral operator ζ-functions has been established as a means of regularizing the vacuum expectation values in quantum field theories. However, most explicit examples using this regularization technique to date, do not consider gauge fields in detail. Here, we address this gap by looking at some well-known physical examples of quantum fields from the Fourier integral operator ζ-function point of view.

Tobias Hartung, Karl Jansen

A Class of Well-Posed Parabolic Final Value Problems

Abstract

This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax–Milgram operators in vector distribution spaces are the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.

Jon Johnsen

Localization of a Class of Muckenhoupt Weights by Using Mellin Pseudo-Differential Operators

Abstract

Let Γ be a finite or infinite interval of \(\mathbb R\), p ∈ (1, ∞), and let w ∈ A _p( Γ) be a Muckenhoupt weight. Relations of the weighted singular integral operator \(wS_{\mathbb{R}_+}w^{-1}I\) on the space \(L^p(\mathbb{R}_{+})\) and Mellin pseudo-differential operators with non-regular symbols are studied. A localization of a class of Muckenhoupt weights to power weights at finite endpoints of Γ, which is related to the Allan-Douglas local principle, is obtained by using quasicontinuous functions and Mellin pseudo-differential operators with non-regular symbols.

Yu. I. Karlovich

Carleman Regularization and Hyperfunctions

Abstract

Carleman regularization is a method to give a meaning to Fourier-type integrals which are highly divergent in a classical sense. We use it to give a local representation of hyperfunctions in terms of such integrals. While such representations are not unique, uniqueness can be achieved in terms of Dolbeault type cohomology with coefficients in L ² spaces with weights.

Otto Liess

Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space

Abstract

We consider the strictly hyperbolic Cauchy problem

$$\displaystyle \begin {cases} D_t^m u - \sum \limits _{j = 0}^{m-1} \sum \limits _{|\gamma |+j = m} a_{m-j,\,\gamma }(t,\,x) D_x^\gamma D_t^j u = 0,\\ D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots ,\,m, \end {cases} $$

for \((t,\,x) \in [0,\,T]\times \mathbb {R}^n\) with coefficients belonging to the Zygmund class \(C_\ast ^s\) in x and having a modulus of continuity below Lipschitz in t. Imposing additional conditions to control oscillations, we obtain a global (on [0, T]) L ² energy estimate without loss of derivatives for \(s \geq \max \{1+\varepsilon ,\,\frac {2m_0}{2-m_0}\}\), where m ₀ is linked to the modulus of continuity of the coefficients in time.

Daniel Lorenz

Quantization and Coorbit Spaces for Nilpotent Groups

Abstract

We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product, making it via the exponential diffeomorphism a copy of its unique connected simply connected nilpotent Lie group. Using harmonic analysis tools, we emphasize the role of a Weyl system, of the associated Fourier-Wigner transformation and, at the level of symbols, of an important family of exponential functions. Such notions also serve to introduce a family of phase-space shifts. These are used to define and briefly study a new class of coorbit spaces of symbols and its relationship with coorbit spaces of vectors, defined via the Fourier-Wigner transform.

M. Măntoiu

On the Measurability of Stochastic Fourier Integral Operators

Abstract

This work deals with the measurability of Fourier integral operators (FIOs) with random phase and amplitude functions. The key ingredient is the proof that FIOs depend continuously on their phase and amplitude functions, taken from suitable classes. The results will be applied to the solution FIO of the transport equation with spatially random transport speed as well as to FIOs describing waves in random media.

Michael Oberguggenberger, Martin Schwarz

Convolution and Anti-Wick Quantisation on Ultradistribution Spaces

Abstract

We present recent advances in convolution theory for the quasi-analytic and non quasi-analytic ultradistribution spaces and generalised Gelfand-Shilov spaces. Additionally, we consider the existence of convolution of non-quasianalytic ultradistribution with the Gaussian kernel \(e^{s|x|{ }^2}\), \(s\in \mathbb R\backslash \{0\}\), and identify the largest subspace of non-quasi-analytic ultradistributions for which this convolution exists. This gives a way to extend the definition of Anti-Wick quantisation for symbols that are not necessarily tempered ultradistributions. Finally, we discuss convolution in quasi-analytic classes with \(e^{s|\cdot |{ }^q}\), q > 1, \(s\in \mathbb R\).

Stevan Pilipović, Bojan Prangoski

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear Schrödinger Equation

Abstract

This paper deals with the nonlinear Schrödinger equation (NLSE) with logarithmic and power nonlinearities as well as with several generalizations of NLSE—Biswas-Milovic̆ equation and others. Solutions of special form are written explicitly via hyperbolic, Jacobi elliptic, Weierstrass and Legendre elliptic functions of the three kinds. Generalized (distribution) solutions for the logarithmic Schrödinger equation containing one or two delta potentials are constructed too. The Biswas-Milovic̆ equation is ill-posed in the periodic Sobolev space H ^s with respect to the space variable x for s < 0.

Petar Popivanov, Angela Slavova

Dirichlet-to-Neumann Operator and Zaremba Problem

Abstract

Starting with the Zaremba problem for the Laplacian, a boundary value problem with jumping conditions from Dirichlet to Neumann data or also with discontinuous Dirichlet- or Neumann data, a reduction to the boundary in terms of Boutet de Monvel’s calculus gives rise to an interface problem which can be interpreted as a boundary value problem on the Neumann side for the Dirichlet-to-Neumann operator. This is a first order elliptic classical pseudo-differential operator on the boundary without the transmission property at the interface. A specific choice of edge quantization admits an interpretation within the edge calculus, and we apply the formalism of the edge algebra together with interface conditions.

B.-W. Schulze

Extended Gevrey Regularity via the Short-Time Fourier Transform

Abstract

We study the regularity of smooth functions whose derivatives are dominated by sequences of the form \(M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}\), τ > 0, σ ≥ 1. We show that such functions can be characterized through the decay properties of their short-time Fourier transforms (STFT), and recover (Cordero et al., Trans. Am. Math. Soc., 367 (2015), 7639–7663; Theorem 3.1) as the special case when τ > 1 and σ = 1, i.e. when the Gevrey type regularity is considered. These estimates lead to a Paley-Wiener type theorem for extended Gevrey classes. In contrast to the related result from Pilipović et al. (Sarajevo Journal of Mathematics, 14 (2) (2018), 251–264; J. Pseudo-Differ. Oper. Appl. (2019)), here we relax the assumption on compact support of the observed functions. Moreover, we introduce the corresponding wave front set, recover it in terms of the STFT, and discuss local regularity in such context.

Nenad Teofanov, Filip Tomić

Wiener Estimates on Modulation Spaces

Abstract

We characterise modulation spaces by Wiener estimates on the short-time Fourier transforms. We use the results to refine some formulae for periodic distributions with Lebesgue estimates on their coefficients.

Joachim Toft

The Gabor Wave Front Set of Compactly Supported Distributions

Abstract

We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set.

Patrik Wahlberg

Correction to: On the Solvability of a Class of Second Order Degenerate Operators

Serena Federico, Alberto Parmeggiani

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