Class Of Pseudodifferential Operators Research Articles

On decay properties for solutions of the Zakharov–Kuznetsov equation

This work mainly focuses on spatial decay properties of solutions to the Zakharov–Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition u0 verifies 〈σ⋅x〉ru0∈L2(σ⋅x≥κ), for some r∈N, κ∈R, being σ be a suitable non-null vector in the Euclidean space, then the corresponding solution u(t) generated from this initial condition verifies 〈σ⋅x〉ru(t)∈L2σ⋅x>κ−νt, for any ν>0. Additionally, depending on the magnitude of the weight r, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power r>0 not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data u0 has a decay of exponential type on a particular half space, that is, ebσ⋅xu0∈L2(σ⋅x≥κ), then the corresponding solution satisfies ebσ⋅xu(t)∈Hpσ⋅x>κ−t, for all p∈N, and time t≥δ, where δ>0. To our knowledge, this is the first study of such property. As a further consequence, we also obtain well-posedness results in anisotropic weighted Sobolev spaces in arbitrary dimensions.Finally, as a by-product of the techniques considered here, we show that our results are also valid for solutions of the Korteweg–de Vries equation.

Special affine Fourier transform of tempered distributions and pseudo-differential operators

The primary aim is to develop a new class of pseudo-differential operators by incorporating the Special Affine Fourier Transform (SAFT) and some of its fundamental properties in Schwartz and tempered distribution space. The secondary aim is to explore the utility of the SAFT in constructing a new generalized heat equation and derive its analytical solution. Further, we have investigated particular cases of the proposed generalized heat equation. Furthermore, we have visually illustrated solutions of this equation via MATLAB R2023b.

Pseudo-differential extension for graded nilpotent Lie groups

Classical pseudo-differential operators of order zero on a graded nilpotent Lie group G form a ^* -subalgebra of the bounded operators on L^2(G) . We show that its \mathrm{C}^* -closure is an extension of a noncommutative algebra of principal symbols by compact operators. As a new approach, we use the generalized fixed point algebra of an \mathbb{\R}_{>0} -action on a certain ideal in the \mathrm{C}^* -algebra of the tangent groupoid of G . The action takes the graded structure of G into account. Our construction allows to compute the \mathrm{K} -theory of the algebra of symbols.

On diffeological principal bundles of non-formal pseudo-differential operators over formal ones

We describe the structure of diffeological bundle of non formal classical pseudo-differential operators over formal ones, and its structure group. For this, we give results on diffeological principal bundles with (a priori) no local trivialization including an Ambrose-Singer theorem, use the smoothing connections alrealy exhibited by the author in previous works, and finish with open questions.

Global functional calculus, lower/upper bounds and evolution equations on manifolds with boundary

Given a smooth manifold M (with or without boundary), in this paper we establish a global functional calculus, without the standard assumption that the operators are classical pseudo-differential operators, and the Gårding inequality for global pseudo-differential operators associated with boundary value problems. The analysis that we follow is free of local coordinate systems. Applications of the Gårding inequality to the global solvability for a class of evolution problems are also considered.

Asymptotics of singular values for quantised derivatives on noncommutative tori

We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus Tθd of the form TIθ−1. Here T stands for classical pseudodifferential operators on quantum torus of order 0, and Iθ−1 is the Riesz potential of order −1. As an application, we obtain Weyl type asymptotics for the quantised derivative ▪ of an operator x from the homogeneous Sobolev space W˙d1(Tθd). The asymptotic coefficient is equivalent to the norm of ▪ in the principal ideal Ld,∞, as well as the norm of x in W˙d1(Tθd). We precise and rectify earlier results in our previous paper (Comm. Math. Phys. 371 (2019), no. 3, 1231–1260) on quantum integration formula for ▪.

Inverse Random Potential Scattering for Elastic Waves

This paper is concerned with the inverse elastic scattering problem for a random potential in three dimensions. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance operator is a classical pseudodifferential operator. Given the potential, the direct scattering problem is shown to be well-posed in the sense of distributions by studying the equivalent Lippmann–Schwinger integral equation. For the inverse scattering problem, we demonstrate that the microlocal strength of the random potential can be uniquely determined with probability one by a single realization of the high frequency limit of the averaged compressional or shear backscattered far-field pattern of the scattered wave. The analysis employs the integral operator theory, the Born approximation in the high frequency regime, the microlocal analysis for the Fourier integral operators, and the ergodicity of the wave field.

Some further classes of pseudo-differential operators in the p-adic context and their applications.

The purpose of this article is to study new non-Archimedean pseudo-differential operators whose symbols are determined from the behavior of two functions defined on the p-adic numbers. Thanks to the characteristics of our symbols, we can find connections between these operators and new types of non-homogeneous differential equations, Feller semigroups, contraction semigroups and strong Markov processes.

A regularity result for the bound states of N-body Schrödinger operators: blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the L^2-eigenfunctions of Schrödinger-type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is , where is the usual Euclidean distance to the union of the set of collision planes {mathcal {F}}. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification {overline{X}} of the underlying space X and we first blow up the spheres {mathbb {S}}_Y subset {mathbb {S}}_X at infinity of the collision planes Y in {mathcal {F}} to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes {mathcal {F}}. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher-order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators.

Continuity of pseudodifferential operators with nonsmooth symbols on mixed-norm Lebesgue spaces

Mixed-norm Lebesgue spaces found their place in the study of some questions in the theory of partial differential equations, as can be seen from recent interest in the continuity of certain classes of pseudodifferential operators on these spaces. In this paper, we use some recent advances in the pseudodifferential calculus for nonsmooth symbols to prove the boundedness of pseudodifferential operators with such symbols on mixed-norm Lebesgue spaces.

GENERALIZED SPACES OF S AND S′ TYPES

In paper the topological structure of generalized spaces of $ S $ type and the basic operations in such spaces was investigated. The question of quasi-analyticity (non-quasi-analyticity) of generalized spaces of $ S $ type was studied. Some classes of pseudodifferential operators, properties of Fourier transformation of generalized functions from spaces of type $S'$, convolutions, convoluters and multipliers was investigated.

On a class of pseudodifferential operators on the product of compact Lie groups

AbstractIn this paper, a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in his paper of 1975, is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen in their book of 2010 (see also V. Fischer's paper of 2015), in that it exploits the harmonic analysis of the groups involved.

An Inverse Random Source Problem for the Biharmonic Wave Equation

This paper is concerned with an inverse source problem for the stochastic biharmonic wave equation. The driven source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The well-posedness of the direct problem is examined in the distribution sense, and the regularity of the solution is discussed for the given rough source. For the inverse problem, the strength of the random source, involved in the principal symbol of its covariance operator, is shown to be uniquely determined by a single realization of the magnitude of the wave field averaged over the frequency band with probability one. Numerical experiments are presented to illustrate the validity and effectiveness of the proposed method for the case that the random source is white noise.

Generic Transporters for the Linear Time-Dependent Quantum Harmonic Oscillator on ℝ

Abstract In this paper we consider the linear, time-dependent quantum Harmonic Schrdinger equation ${\textrm {i}} \partial _t u= \frac {1}{2} ( - \partial _x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb {R}$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, self-adjoint, and $2\pi $ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Fréchet space of symbols. This shows that generic, classical pseudodifferential, $2\pi $-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre’s positive commutator theory.

Inverse Source Problems for the Stochastic Wave Equations: Far-Field Patterns

This paper addresses the direct and inverse source problems for the stochastic acoustic, biharmonic, electromagnetic, and elastic wave equations in a unified framework. The driven source is assumed to be a centered generalized microlocally isotropic Gaussian random field, whose covariance and relation operators are classical pseudodifferential operators. Given the random source, the direct problems are shown to be well-posed in the sense of distributions and the regularity of the solutions are given. For the inverse problems, we demonstrate by ergodicity that the principal symbols of the covariance and relation operators can be uniquely determined by a single realization of the far-field pattern averaged over the frequency band with probability one.

Localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on a symbol ς and two windows functions and . We first present some basic properties of the windowed Opdam–Cherednik transform. Then, we use modulation spaces associated with the Opdam–Cherednik transform as appropriate classes for symbols and windows and study the boundedness and compactness of the localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces. Finally, we show that these operators are in the Schatten–von Neumann class.

THE NONLOCAL PROBLEM FOR FRACTAL DIFFUSION EQUATION

Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.

Fredholm conditions for operators invariant with respect to compact Lie group actions

Let G be a compact Lie group acting smoothly on a smooth, compact manifold M, let P∈ψ m (M;E 0 ,E 1 ) be a G–invariant, classical pseudodifferential operator acting between sections of two G-equivariant vector bundles E i →M, i=0,1, and let α be an irreducible representation of the group G. Then P induces a map π α (P):H s (M;E 0 ) α →H s-m (M;E 1 ) α between the α-isotypical components. We prove that the map π α (P) is Fredholm if, and only if, P is transversally α-elliptic, a condition defined in terms of the principal symbol of P and the action of G on the vector bundles E i .

Quadratic-phase Fourier transform of tempered distributions and pseudo-differential operators

In this work, the quadratic-phase Fourier transform (QPFT) on Schwartz space is defined and its necessary results are derived, including adjoint formula, Parseval identity, and continuity property. Furthermore, the continuity property on tempered distribution space is discussed, and also an example of the QPFT is provided. The present analysis is applied to develop a new class of pseudo-differential operators and prove an important theorem on Schwartz space. The boundary value problems of generalized partial differential equations (i.e. generalized telegraph, wave, and heat equations) are discussed using QPFT, and graphical plots are provided.

On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators

We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP) hierarchy solutions in the algebra of $\F Cl(S^1,\K^n)$ of formal classical pseudo-differential operators. The classical algebra $\Psi DO(S^1,\K^n)$ where the KP hierarchy is well-known appears as a subalgebra of $\F Cl(S^1,\K^n).$ We investigate algebraic properties of $\F Cl(S^1,\K^n)$ such as splittings, r-matrices, extension of the Gelfand-Dickii bracket, almost complex structures. Then, we prove the existence and uniqueness of the KP hierarchy solutions in $\F Cl(S^1,\K^n)$ with respect to extended classes of initial values. Finally, we extend this KP hierarchy to complex order formal pseudo-differential operators and we describe their Hamiltonian structures similarly to previously known formal case.