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2013 | Book

Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds Read chapter Read first chapter

Pseudo-Differential Operators, Generalized Functions and Asymptotics

Editors: Shahla Molahajloo, Stevan Pilipović, Joachim Toft, M. W. Wong

Publisher: Springer Basel

Book Series : Operator Theory: Advances and Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume consists of twenty peer-reviewed papers from the special session on pseudodifferential operators and the special session on generalized functions and asymptotics at the Eighth Congress of ISAAC held at the Peoples’ Friendship University of Russia in Moscow on August 22‒27, 2011. The category of papers on pseudo-differential operators contains such topics as elliptic operators assigned to diffeomorphisms of smooth manifolds, analysis on singular manifolds with edges, heat kernels and Green functions of sub-Laplacians on the Heisenberg group and Lie groups with more complexities than but closely related to the Heisenberg group, Lp-boundedness of pseudo-differential operators on the torus, and pseudo-differential operators related to time-frequency analysis. The second group of papers contains various classes of distributions and algebras of generalized functions with applications in linear and nonlinear differential equations, initial value problems and boundary value problems, stochastic and Malliavin-type differential equations. This second group of papers are related to the third collection of papers via the setting of Colombeau-type spaces and algebras in which microlocal analysis is developed by means of techniques in asymptotics. The volume contains the synergies of the three areas treated and is a useful complement to volumes 155, 164, 172, 189, 205 and 213 published in the same series in, respectively, 2004, 2006, 2007, 2009, 2010 and 2011.

Table of Contents

Frontmatter
Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds
Abstract
In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader’s convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.
Anton Savin, Boris Sternin
The Singular Functions of Branching Edge Asymptotics
Abstract
We investigate the structure of branching asymptotics appearing in solutions to elliptic edge problems. The exponents in powers of the half-axis variable, logarithmic terms, and coefficients depend on the variables on the edge and may be branching.
B.-W. Schulze, L. Tepoyan
The Heat Kernel and Green Function of the Sub-Laplacian on the Heisenberg Group
Abstract
We give a construction of the heat kernel and Green function of a hypoelliptic operator on the one-dimensional Heisenberg group \(\mathbb{H}\), the sub-Laplacian \(\mathcal{L}\). The explicit formulas are developed using Fourier–Wigner transforms, pseudo-differential operators of the Weyl type, i.e., Weyl transforms, and spectral analysis. These formulas are obtained by first finding the formulas for the heat kernels and Green functions of a family of twisted Laplacians \({L}_{\tau}\) for all non-zero real numbers \({\tau}\). In the case when \({\tau=1, {L}_{1}}\) is just the usual twisted Laplacian.
Xiaoxi Duan
Metaplectic Equivalence of the Hierarchical Twisted Laplacian
Abstract
We use a metaplectic operator to prove that the hierarchical twisted Laplacian L m is unitarily equivalent to the tensor product of the one-dimensional Hermite operator and the identity operator on L 2(ℝm+11), and we use this unitary equivalence to show that L m is globally hypoelliptic in the Schwartz space and in the Gelfand–Shilov spaces.
Shahla Molahajloo, Luigi Rodino, M. W. Wong
The Heat Kernel and Green Function of a Sub-Laplacian on the Hierarchical Heisenberg Group
Abstract
We give the hierarchical Heisenberg group underpinning the hierarchical twisted Laplacian discovered recently. This hierarchical twisted Laplacian is obtained by taking the inverse Fourier transform of a sub-Laplacian with respect to a subcenter of the hierarchical Heisenberg group. Using parametrized versions of Wigner transforms and Weyl transforms, we give formulas for the heat kernels and Green functions of the parametrized hierarchical twisted Laplacians. Taking the Fourier transform of the parametrized heat kernels so obtained, we give explicit formulas for the heat kernel and Green function of the hierarchical sub-Laplacian on the hierarchical Heisenberg group.
Shahla Molahajloo, M. W. Wong
L p -bounds for Pseudo-differential Operators on the Torus
Abstract
We establish L p bounds for a class of periodic pseudo-differential operators corresponding to symbols with limited regularity on the torus \({\mathbb{T}}^{n}\). The analysis is carried out using global representation of the symbols on \({\mathbb{T}}^{n} \times \mathbb{Z}^n\).
Julio Delgado
Multiplication Properties in Gelfand–Shilov Pseudo-differential Calculus
Abstract
We consider modulation space and spaces of Schatten–von Neumann symbols where corresponding pseudo-differential operators map one Hilbert space to another. We prove Hölder–Young and Young type results for such spaces under dilated convolutions and multiplications. We also prove continuity properties for such spaces under the twisted convolution, and the Weyl product. These results lead to continuity properties for twisted convolutions on Lebesgue spaces, e.g., L p (ω) is a twisted convolution algebra when 1 ≤ p ≤ 2 and appropriate weight ω.
Joachim Toft
Operator Invariance
Abstract
Linear time invariant (LTI) systems have produced a rich set of ideas including the concepts of convolution, impulse response function, causality, and stability, among others. We discuss how these concepts are generalized when we consider invariance other than time shift invariance. We call such systems linear operator invariant systems because the invariance is characterized by an operator. In the standard case of LTI systems the relation between input and output function in the Fourier domain is multiplication. We generalize this and show that multiplication still holds in the operator transform domain. Transforming back to the time domain defines generalized convolution.
Leon Cohen
Initial Value Problems in the Time-Frequency Domain
Abstract
We transform an initial value problem for a stochastic differential equation to the time-frequency domain. The result is a deterministic time-frequency equation whose forcing term incorporates the given set of initial values. The structure and solution of the time-frequency equation reveal the spectral properties of the nonstationary random process solution to the stochastic differential equation. By applying our method to the Langevin equation, we obtain the exact time-frequency spectrum for an arbitrary initial value.
Lorenzo Galleani
Polycaloric Distributions and the Generalized Heat Operator
Abstract
The aim of this paper is to introduce the notion of p-caloric distributions with respect to the generalized heat operator and to prove a representation formula. Based on the representation formula for p-caloric distributions and using the parametrix of the generalized heat operator we shall give two extensions of Poisson formula. Finally, we shall define the generalized iterated heat operator of order \(\lambda\,\,\in\,\,\mathrm C,{\rm Re}\, \lambda\,{<}\,{0}\) by means of the kernel distribution of its parametrix.
Viorel Catană
Smoothing Effect and Fredholm Property for First-order Hyperbolic PDEs
Abstract
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers (parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.
I. Kmit
A Note on Wave-front Sets of Roumieu Type Ultradistributions
Abstract
We study wave-front sets in weighted Fourier–Lebesgue spaces and corresponding spaces of ultradistributions. We give a comparison of these sets with the well-known wave-front sets of Roumieu type ultradistributions. Then we study convolution relations in the framework of ultradistributions. Finally, we introduce modulation spaces and corresponding wave-front sets, and establish invariance properties of such wave-front sets.
Karoline Johansson, Stevan Pilipović, Nenad Teofanov, Joachim Toft
Ordinary Differential Equations in Algebras of Generalized Functions
Abstract
A local existence and uniqueness theorem for ODEs in the special algebra of generalized functions is established, as well as versions including parameters and dependence on initial values in the generalized sense. Finally, a Frobenius theorem is proved. In all these results, composition of generalized functions is based on the notion of c-boundedness.
Evelina Erlacher, Michael Grosser
Asymptotically Almost Periodic Generalized Functions
Abstract
The paper introduces an algebra of asymptotically almost periodic generalized functions of Colombeau type and gives their main properties
Chikh Bouzar, Mohammed Taha Khalladi
Wave Equations and Symmetric First-order Systems in Case of Low Regularity
Abstract
We analyse an algorithm of transition between Cauchy problems for second-order wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the standard conditions guaranteeing well-posedness. The typical operations involved in rewriting equations into systems are then neither defined classically nor consistently extendible to the distribution theoretic setting. However, employing the nonlinear theory of generalized functions in the sense of Colombeau we arrive at clear statements about the transfer of questions concerning solvability and uniqueness from wave equations to symmetric hyperbolic systems and vice versa. Finally, we illustrate how this transfer method allows to draw new conclusions on unique solvability of the Cauchy problem for wave equations with non-smooth coefficients.
Clemens Hanel, Günther Hörmann, Christian Spreitzer, Roland Steinbauer
Concept of Delta-shock Type Solutions to Systems of Conservation Laws and the Rankine–Hugoniot Conditions
Abstract
To solve nonlinear systems of conservation laws, we need a proper concept of weak solution. The aim of this paper is to explain how to derive integral identities for defining δ-shock type solutions in the sense of Schwartzian distributions. We consider two types of systems to compare our definitions. System (1.3) is a standard system admitting delta-shocks and our definition is given by the identities (2.9). System (3.1) is non-typical, and in addition to the identities (3.8), we need to use relation (3.7). We restrict ourselves to the consideration of δ-shocks concentrated only on the surface of codimension 1. Our approach can be used to derive integral identities for other type systems.
V. M. Shelkovich
Classes of Generalized Functions with Finite Type Regularities
Abstract
We introduce and analyze spaces and algebras of generalized functions which correspond to Hölder, Zygmund, and Sobolev spaces of functions. The main scope of the paper is the characterization of the regularity of distributions that are embedded into the corresponding space or algebra of generalized functions with finite type regularities.
Stevan Pilipović, Dimitris Scarpalézos, Jasson Vindas
The Wave Equation with a Discontinuous Coefficient Depending on Time Only: Generalized Solutions and Propagation of Singularities
Abstract
This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth coefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the one-dimensional wave equation with a discontinuous coefficient depending on time. After demonstrating the existence and uniqueness of generalized solutions in the sense of Colombeau to the problem, we investigate the phenomenon of propagation of singularities, arising from delta function initial data, for the case of a piecewise constant coefficient. We also provide an analysis of the interplay between singularity strength and propagation effects. Finally, we show that in case the initial data are distributions, the Colombeau solution to the model problem is associated with the piecewise distributional solution of the corresponding transmission problem.
Hideo Geguchi, Günther Hörmann, Michael Oberguggenberger
Generalized Solutions of Abstract Stochastic Problems
Abstract
The Cauchy problem \({u}{^\prime}(t)={Au}(t)\,+\,B{\mathbb{W}}(t),t\,\geq\,0,{u(0)}=\zeta\) with singular white noise\(\mathbb{W}\) and A not necessarily generating a C0–semigroup is studied in spaces of distributions. Spaces of generalized with respect to both time variable t and random variable ω are built. Existence and uniqueness of generalized solutions in the obtained spaces is proved.
I. V. Melnikova, M. A. Alshanskiy
Nonhomogeneous First-order Linear Malliavin Type Differential Equation
Abstract
In this paper we solve a nonhomogeneous first-order linear equation involving the Malliavin derivative operator with stochastic coefficients by use of the chaos expansion method. We prove existence and uniqueness of a solution in a certain weighted space of generalized stochastic distributions and represent the obtained solution in the Wiener-ItÔ chaos expansion form.
Tijana Levajković, Dora Seleši
Metadata
Title
Pseudo-Differential Operators, Generalized Functions and Asymptotics
Editors
Shahla Molahajloo
Stevan Pilipović
Joachim Toft
M. W. Wong
Copyright Year
2013
Publisher
Springer Basel
Electronic ISBN
978-3-0348-0585-8
Print ISBN
978-3-0348-0584-1
DOI
https://doi.org/10.1007/978-3-0348-0585-8

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