Pseudodifferential Operators Research Articles

Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes

Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes

Perturbation of an α-stable type stochastic process by a pseudo-gradient

A Markov process defined by some pseudo-differential operator of an order $1\lt \alpha \lt 2$ as the process generator is considered. Using a pseudo-gradient operator, that is, the operator defined by the symbol $i\lambda |\lambda {|^{\beta -1}}$ with some $0\lt \beta \lt 1$, the perturbation of the Markov process under consideration by the pseudo-gradient with a multiplier, which is integrable at some large enough power, is constructed. Such perturbation defines a family of evolution operators, properties of which are investigated. A corresponding Cauchy problem is considered.

Energy Landscapes and Non-Archimedean Pseudo- Differential Operators as Tools for Studying the Spreading of Infectious Diseases in a Situation of Extreme Social Isolation

In this article, we introduce a new type of pseudo-differential equations naturally connected with non-archimedean pseudo-differential operators and whose symbols are new classes of negative definite functions in the p-adic context and in arbitrary dimension. These equations are proposed as a mathematical models to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation.

On Ultradifferentiable Regularity of Perturbations by Lower Order Terms of Globally $$C^\infty$$ Hypoelliptic Ultradifferentiable Pseudodifferential Operators

On Ultradifferentiable Regularity of Perturbations by Lower Order Terms of Globally $$C^\infty$$ Hypoelliptic Ultradifferentiable Pseudodifferential Operators

Quantum properties of classical Pearson random variables

This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable [Formula: see text] with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of [Formula: see text] as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable [Formula: see text] and characterize type-[Formula: see text] and type-[Formula: see text] real-valued random variables. We prove that a necessary condition for [Formula: see text] to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type.

A RECOGNITION METHOD OF R-PEAKS ON ELECTROCARDIOGRAMS BASED ON WAVELET TRANSFORM WITH PSEUDO-DIFFERENTIAL OPERATORS

In this paper, we propose a recognition method of R-peaks on electrocardiograms (ECGs) based on wavelet transform with pseudo-differential operators. It is well known that the accurate recognition of R-peaks is highly importance for diagnosis of cardiac diseases and autonomic ataxia. However, the existing results for detection of R-peaks are not always accurate and can have missed peaks or false. Difficulties in accurate R-peaks detection is caused by presence of various noises in ECGs and the physiological variability of the QRS complex. From the above, we propose a more flexible and adaptive recognition method of R-peaks. In order to develop the proposed detection method, noises, artifacts, and baseline variation in ECGs are firstly suppressed by using the low-pass/high-pass filters, moving average, and MaMeMi filter. Next, the time-frequency domain's energy distribution is computed by using wavelet transform with pseudo-differential operators. Furthermore, we introduce a time-series index, -Normalized Spectrum Index ( f^p-NSI) obtained by scalograms based on the wavelet transform with pseudo-differential operators. Finally, R-peaks are recognized by taking the threshold toward the results of f^p-NSI. In this paper, we present the proposed recognition method of R-peaks on ECGs, and the effectiveness (accuracy) of the proposed method is evaluated.

Reducibility of a class of operators induced by the dispersive third order Benjamin-Ono equation

We prove the reducibility of a class of quasi-periodically time dependent linear operators, which are derived from linearizing the dispersive third order Benjamin–Ono (BO) equation on the circle at a small amplitude quasi-periodic function, with a diophantine frequency vector ω∈O0⊂Rν. It is shown that there exists a set O∞⊂O0 of asymptotically full Lebesgue measure such that for any ω∈O∞, the operators can be reduced to the ones with constant coefficients by some linear transformations depending on time quasi-periodically. These transformations include a change of variable induced by a diffeomorphism of the torus, the flow of some partial differential equations and a pseudo-differential operator of order zero. We first reduce the linearized operator of order three to the one with constant coefficients plus a remainder of order zero, and then a perturbative reducibility scheme is performed. The major difficulties encountered are brought by the non-smooth character of the dispersive relation in view of the presence of the Hilbert operator H. We look for several appropriate transformations which are real, reversibility-preserving and satisfy the sharp tame bounds which are used for the reducibility. This work will be the first fundamental step in proving the existence of time quasi-periodic solutions for the dispersive third order BO equation.

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.

A groupoid approach to the Wodzicki residue

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis [33] and Guillemin [19]. In this article we give a definition of the Wodzicki residue, using the language of r-fibered distributions from [24], [30], in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition [25] for contact and codimension 1 foliation Heisenberg manifolds.

Besov wavefront set

We develop a notion of wavefront set aimed at characterizing in Fourier space the directions along which a distribution behaves or not as an element of a specific Besov space. Subsequently we prove an alternative, albeit equivalent characterization of such wavefront set using the language of pseudodifferential operators. Both formulations are used to prove the main underlying structural properties. Among these we highlight the individuation of a sufficient criterion to multiply distributions with a prescribed Besov wavefront set which encompasses and generalizes the classical Young’s theorem. At last, as an application of this new framework we prove a theorem of propagation of singularities for a large class of hyperbolic operators.

Lipschitz continuity of spectra of pseudodifferential operators in a weighted Sjöstrand class and Gabor frame bounds

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions.

Sharp spectral stability for a class of singularly perturbed pseudo-differential operators

Let a(x,\xi) be a real Hörmander symbol of the type S_{0,0}^0(\mathbb{R}^{d}\times \mathbb{R}^d) , let F be a smooth function with all its derivatives globally bounded, and let K_\delta be the self-adjoint Weyl quantization of the perturbed symbols a(x+F(\delta{} x),\xi) , where |\delta|\leq 1 . First, we prove that the Hausdorff distance between the spectra of K_\delta and K_{0} is bounded by \sqrt{|\delta|} , and we give examples where spectral gaps of this magnitude can open when \delta\neq 0 . Second, we show that the distance between the spectral edges of K_\delta and K_0 (and also the edges of the inner spectral gaps, as long as they remain open at \delta=0 ) are of order |\delta| , and give a precise dependence on the width of the spectral gaps.

On L2-boundedness of pseudo-multipliers associated to the Grushin operator

In this article we define analogues of pseudo-differential operators associated to the joint functional calculus of the Grushin operator using their spectral resolution, and study Calderón–Vaillancourt-type theorems for these operators.

On the rôle of singular functions in extending the probabilistic symbol to its most general class

The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving Hunt behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle.

Control Estimates for 0th-Order Pseudodifferential Operators

Abstract We introduce the control conditions for 0th-order pseudodifferential operators $\textbf{P}$ whose real parts satisfy the Morse–Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation $(i\partial _{t}-\textbf{P})u=f$ when $\textbf{P}$ has a damping term that satisfies the control condition and $f\in C^{\infty }$. This is motivated by the study of a microlocal model for the damped internal waves.

Wigner Equations for Phonons Transport and Quantum Heat Flux

Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the acoustic, optical and Z phonons are deduced. The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal’s calculus and its properties, the pseudo-differential operators are expanded up to the second order in ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}. An energy transport model is obtained by using the moment method with closure relations based on a quantum version of the Maximum Entropy Principle by employing a relaxation time approximation for the production terms of energy and energy flux. An explicit form of the thermal conductivity with quantum correction up to ħ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar ^2$$\\end{document} order is obtained under a long-time scaling for the most relevant phonon branches.

Pseudo-differential calculi and entropy estimates with Orlicz modulation spaces

We deduce continuity properties for pseudo-differential operators with symbols in Orlicz modulation spaces when acting on other Orlicz modulation spaces. In particular we extend well-known results in the literature. For example we generalize the classical result by Cordero and Nicola that if1p′+1q′⩽1p1+1p2,1p′+1q′⩽1q1+1q2,pj,qj⩽q′,q⩽p and a∈Mp,q, then the pseudo-differential operator Op(a) is continuous from Mp1,q1 to Mp2′,q2′.We also show that the entropy functional Eϕ possess suitable continuity properties on a suitable Orlicz modulation space MΦ satisfying Mp⊆MΦ⊆M2, though Eϕ is discontinuous on M2=L2.

Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \emph{even} symbol $p(x,\xi)$ on $\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\Delta )^a$. Let $\Omega \subset \mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\Omega)$ defined under the exterior condition $u=0$ in $\mathbb{R}^n\setminus\Omega$. When $p(x,\xi)$ and $\Omega$ are $C^\infty $, it is known that the eigenvalues $\lambda_j$ (ordered in a nondecreasing sequence for $j\to \infty$) satisfy a Weyl asymptotic formula \begin{equation*} \lambda _j(P_{D})=C(P,\Omega )j^{2a/n}+o(j^{2a/n}) \text {for $j\to \infty $}, \end{equation*} with $C(P,\Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\tilde P=P+P'+P”$, where $P'$ is an operator of order $<\min\{2a, a+\frac 12\}$ with certain mapping properties, and $P”$ is bounded in $L_2(\Omega )$ (e.g. $P”=V(x)\in L_\infty(\Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.

Notes on pseudodifferential operators commutators and Lipschitz functions

Abstract This article focuses on the boundedness of the commutators generated by pseudodifferential operators with Lipschitz functions and obtains a sufficient condition such that these operators are bounded from L p ( R n ) {L}^{p}\left({{\bf{R}}}^{n}) into the homogeneous Triebel-Lizorkin space F ˙ p γ , ∞ ( R n ) {\dot{F}}_{p}^{\gamma ,\infty }\left({{\bf{R}}}^{n}) and from L p ( R n ) {L}^{p}\left({{\bf{R}}}^{n}) into L q ( R n ) {L}^{q}\left({{\bf{R}}}^{n}) .

Truncation quantization in the edge calculus

Pseudodifferential operators on the half-space associated with classical symbols of order zero without transmission property are shown to belong to the so-called edge algebra.