Questions tagged [pseudo-differential-operators]

This tag is for questions regarding to the Pseudo-differential operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.

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43 views

Justifying Interchange of Integral

I am trying to show that if $P$ is a pseudo-differential operator with symbol given by $p(x,\xi)$ i.e. the operator $$P:\mathcal S\rightarrow \mathcal S$$ defined by $$Pf(x)=\int_{\mathbb R^n}e^{ix\...
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10 views

Laplacian raised to positive integer

In books about Pseudo-differential operators, they use many times $\triangle^k u$ but i have a question, what means this really option A $\displaystyle{\sum_{|\alpha|=k}}{\, \frac{k!}{\alpha!} \left({\...
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35 views

How and when is the inverse Laplacian well-defined as a pseudo-differential operator?

I recently came across an interesting (mis-)use of formal equivalencies. First, the uncontroversial bits. By the Fourier derivative theorem, it is straightforward that the 2D Laplace operator can be ...
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59 views

What is the meaning of $\flat$ and $\sharp$ in this smoothing operator? $M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $

I found this: $$M(x,\xi) = M^\sharp(x,\xi)+M^\flat(x,\xi),$$ in this paper, where $$M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $$ and i'm not sure whether i understand it ...
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27 views

Convolution algebra of a groupoid

In this paper in order to explain conormal distribution the authors use the fact that “the elements of the convolution algebra of a groupoid are sections of a density bundle $\Omega ^{1/2}$ rather ...
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15 views

Wave front set of characteristic functions?

There are some problems to calculate the wave front set of $\chi_\Omega$, where $\Omega$ is a region with smooth boundary $\partial \Omega$; $\Omega=\{(x,y)\in \mathbb{R}^2| x^3\ge y^2\}$. What I ...
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12 views

Is there any compactly supported function in the Gelfand-Shilov space $\Sigma_1(=S^{(1)})$?

Good evening, I have a question related to the Gelfand-Shilov spaces as defined in https://arxiv.org/pdf/1505.04096.pdf (see 1.2, pag.2). I seemed to understand that there is no non-zero function of $...
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28 views

Two Definition of Elliptic Symbols

There two definition of elliptic symbol. A smooth matrix function $p(x,\xi)$ is a elliptic symbol of order $m\in\mathbb{R}$ if exist a constant $c>0$ such that for all $|\xi|>c$ we have $p(x,\xi)...
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35 views

Is there some classification of idempotent pseudo-differential operators?

Let $\pi:E\rightarrow M$ be a complex vector bundle over a smooth compact manifold $M$. Let $\operatorname{Psd}(E)$ be the associative $\mathbb{Z}$-graded complex algebra of pseudo-differential ...
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1answer
79 views

Is it possible to construct a vector space on convolution

Let $S$ be a set of functions on $R$ such that for any two functions $f$ and $g$ in $S$, the convolution: $$ (f{\ast}g)(x)={\int}f(y)\ g(x-y)\ dy $$ exists. Since the Dirac delta is technically not a ...
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20 views

How can I show that the integral mentioned is divergent integral (the integral is in $\mathbb{R}^{n}$)

I am studying the book "An Introduction to Pseudo-Differential Operators" by M.W. Wong - Chapter 8, The Product of Two Pseudo-Differential Operators. I'm trying to verify that the integral ...
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1answer
68 views

Confusion about definition (in ΨDO theory) of Sobolev spaces on open sets in Euclidean space

I'm reading Pseudodifferential Operators by M. E. Taylor, where the author talks about $H^s(\Omega)$ for $s\in\mathbb{R}$ and $\Omega\subset\mathbb{R}^n$ an open set (for example, in the statement of ...
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30 views

Composition of (Hörmander) symbols with smooth functions between cones

I am working with Hörmander's Fourier integral operators I and have some problems understanding the proof of proposition 1.1.7. It's about symbols $a \in S^m_{\rho,\delta}(X\times \mathbb{R}^n)$ ...
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57 views

Fractional derivative via Continuous Functional Calculus

If we study C* algebras, at one point or another we are exposed to the idea of the Continuous functional Calculus, i.e. if $\mathcal{Y}$ is a unital C* algebra, $\xi \in \mathcal{Y}$ is normal, that ...
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28 views

Why is the Order of a Classical Symbol Well-defined?

A function $a$ defined on $T^*\mathbb{R}^n$ is called a classical symbol of order $m$ provided it is smooth and there exists $\tilde{a}\in C^\infty(\mathbb{R}^n\times S^{n-1}\times\mathbb{R}_+)$ so ...
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Applications of equations $B\partial_t u=Au$ where $B$ and $A$ are differential operators and $B$ invertible.

I was wondering if there are any nice applications of equations of the flavour $$ B\partial_t u=Au, $$ where $A$ and $B$ are differential operators and $B$ is invertible. E.g. On suitable domains such ...
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34 views

Example of a symbol in $S^m_{\frac{1}{2},0}$ Hormander class.

Let $\Phi$ be a smooth function satisfying \begin{equation} 1 \leq \Phi(|x|) \lesssim \langle x \rangle =(1+|x|^2)^{1/2} \quad x \in \mathbb{R}^n, \end{equation} and \begin{equation} |\partial_x^...
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26 views

Fourier Transform (Wong's book)

I'm with problems to prove an equality that appears in some proofs on book that I'm reading (An Introduction to Pseudo-Differential Operators). We define the fourier transform of a function $f$ in $L^...
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78 views

Why does the equation hold?

Why is $$\exp(i\tau\Delta/4\pi)(xp)=x\exp(i\tau\Delta/4\pi)(p)+i\tau \exp(i\tau\Delta/4\pi)(p')/2\pi, $$ where $p$ is a polynomial. $\Delta$ denotes the Laplacian operator. I do not understand the ...
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54 views

Question about Folland PDE book Chapter 8

Page 284-285: Let $\Omega\subset {\mathbb R}^n$ be a domain. Let $a(x, \xi, y)$ be an "amplitude", i.e. a smooth function on $\Omega\times{\mathbb R}^n\times\Omega$ so that there is $m$, and for ...
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70 views

Leading Symbol of Pseudo-Differential Operators

I have seen various definitions of symbols used to define pseudo-differential operators. The class of symbols I am working with is $S^d(U)=\{p(x,\xi)\in C^{\infty}\left (U\times \mathbb R^m ,M_{k\...
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1answer
49 views

Heine-Borel property of symbol spaces

I am interested in whether the uniform symbol spaces $S^m_\infty$ from the theory of pseudo-differential operators have the Heine-Borel property (i.e. every bounded set is relatively compact). To ...
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78 views

“Crazy” Attempt to Understand Second-Order DEs on Manifolds

(This is kind-of a crazy ramble; it's kind-of not a questions and kind-of not a request for comments, but it's more of a request for comments.) We all know what the second derivative of a real-valued ...
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1answer
111 views

Show $(1+|\xi|^2)^{m/2}\in S^m$ (pseudo-differential operators)

I am trying to show that $p(\xi)=(1+|\xi|^2)^{m/2}\in S^m$, i.e., that for all $m\in\mathbb{R}$ and all multi indices $\alpha,\beta$ $$|D^{\beta}_{x} D^{\alpha}_{\xi}p(\xi)|\leq C_{\alpha,\beta}(1+|\...
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1answer
144 views

Distribution Theory

I am using http://www.math.mcgill.ca/gantumur/math581w12/downloads/pseudodiff.pdf as a self study on Pseudo-Differential Operators and Distribution Theory. In example 3 on page 6 the following ...
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1answer
130 views

Symbol of a pseudo-differential operator. Hormander property and principal symbol

Suppose I have a pseudo-differential operator on $\mathbb R$ whose symbol is $a(x,\xi) = e^{-\xi^2/2}$ (notice, no dependence on $x$) Question 1: Are the following statements correct? $a$ is an ...
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1answer
105 views

Pseudo-differential operators - Motivation for definition of symbol

I'm just beginning the study of pseudo-differential operators with focus on symbols in $S_{1,0}^m$ and I'm having a hard time grasping why we require a symbol $p(x,\xi)$ to satisfy $$|D_{x}^{\beta}D_{...
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36 views

Extending a pseudo-differential operator on $L^2(\mathbb{R}, e^x\,dx)$

Let $A$ be a pseudo-differential operator with symbol $a(x,\xi)=e^{-x}\psi(\xi)$ where $\psi$ is a continuous negative definite function. It is known that $A$ is the generator of a self-similar Markov ...
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14 views

Generalized functions transforming as 1/4-densities under diffeomorphisms

I am looking for generalized functions (including pseudodifferential operators) that satisfy the following identity ($x,y \in R^d$): $f(h(x),h(y))=\left(\frac{1}{\det(D_xh) \det(D_yh)}\right)^{1/4} f(...
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1answer
97 views

What does constant along characteristic mean

When we have linear or quasilinear first order pde $$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$ And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$ I ...
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1answer
82 views

About pseudo-differential operators

Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is ...
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1answer
53 views

The symbol of a pseudifferential operator: how to reconstruct the symbol from an operator?

To simplify let us assume that $U \subset \mathbb{R}^n$ is an open set. Let $P$ be a (scalar) psuedodifferential operator of order $d$ and $p$ be its symbol. Therefore it may be expressed as follows: ...
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40 views

Understanding Y U Egorov's theorem

I was reading the paper "Hörmander, L., Fourier integral operators. I, Acta Math. 127, 79-183 (1971). ZBL0212.46601." and on page 3, there is a statement saying the following thing: Can someone ...
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1answer
22 views

Differentiation of an operator raised to a variable power

I am studying first order systems of the form \begin{equation} L=\partial_t+K(t,x,D_x)\text{ where }D_x=-i\partial_x \end{equation} There is a change of variable and operator of our concern becomes ...
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1answer
104 views

Pseudo Differential Operator example

I was reading a introduction about Pseudo differential operators, and the definition of that type of operator was: $$|\partial_x^\alpha \partial_\xi^\beta f(x,\xi) | \le C_{\alpha,\beta}(1 + |\xi|)^{...
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131 views

Adjoint of a symbol of a pseudo differential operator

This is a passage from Hormanders book, the analysis of linear partial differential operators volume 3. If the adjoint of the symbol $a(x,\xi)$ is $b(x,\xi)=(2\pi)^{-n}\int e^{-i y \cdot \eta}\bar{a}...
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Is this operator pseudodifferential or trace-class?

Let $M$ be a closed manifold and $D$ a Dirac operator on $M$. Then (the closure of) $D^2+1$ has a bounded inverse $$(D^2+1)^{-1}:L^2\rightarrow H^2,$$ where $H^2$ is the second Sobolev space. In ...
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39 views

Reference request: trace class operator

Suppose $D$ is an elliptic differential operator on a non-compact manifold $M$ of order $1$. Then $D$ has a pseudodifferential parametrix $Q$ of order $-1$ such that $$DQ-1=S_0,\qquad QD-1=S_1$$ are ...
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Expressing an operator as a sum of the identity and a compact operator

My problem concerns with the unique solvability of a linear system of integral equations. In my problem, I was able to write the system in matrix form: $$ M \begin{align} \begin{bmatrix} ...
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2answers
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Existence of solutions of $u_t = Lu$ for pseudo-differential operators $L$?

I'm wondering if it is known if existence results for the equation $$\left\{\begin{array}{ll} \partial_t u = Lu & \text{on }(0,T)\times\mathbb{R}^n\\ u(0,-) = \varphi & \text{on }\mathbb{R}^n\...
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52 views

Difference of two pseudodifferential operators of the same order

Let $M$ be a manifold. Does there exist a theorem that says something like: "given two pseudodifferential operators $P$ and $Q$ on $M$ of the same order $k$, there exists an $L^2$-bounded operator $B$ ...
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523 views

Dense subspaces of $L^\infty(\Omega\times\Omega)$

Let $\Omega\subset\mathbb R$ be open and bounded. For continuous functions $C(\Omega\times\Omega)$, the Stone-Weierstrass theorem shows that the products $a(x)b(y)$ of univariate continuous functions ...
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179 views

The inverse of a $\Psi$DO is a $\Psi$DO

The following question looks quite simple, but unfortunately I was not able to find an answer in the literature so far. Let $A \in OPS^m(X)$, $m \in \mathbb R$, be a pseudodifferential operator on a ...
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44 views

Elliptic pseudodifferential operator has at most finite number of non-positive eigenvalues

Let $P \in OPS^m(M)$ ($m$ can be negative of positive) be a self-adjoint elliptic pseudodifferential operator on a compact manifold $M$. How to show that $P$ has only finitely many non-positive ...
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145 views

Is the Fourier transform of a compactly supported distribution a symbol?

I am learning some pseudo-differential operator and wondering is the Fourier transform of a compactly supported distribution a symbol, which gives a way to interpret the Fourier Inversion for those ...
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23 views

How can we prove that the exponential function shifts an arbitrary non-chaotic differential operator by a constant?

if $f(D)y$ is an arbitrary differential operator, of the basic differential $D \equiv \frac{d}{dx}$, acting on any generic infinitely differentiable function $y(x)$ of some real independent variable $...
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1answer
163 views

Is strongly elliptic operator necessarily properly elliptic?

Let us consider $$ A(D)u:=\sum_{|p|\leq l} a_p D^p u, $$ where $a_p$ may be complex, and define $$ A_0(\xi)= \sum_{|p|=l} a_p \xi^p $$ Def 1. The operator $A$ is called strongly elliptic if $$ A_0(\...
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1answer
159 views

BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

Let me illustrate my question by starting with the simplest possible example: Let us consider $P := - \mathrm{d}^2/\mathrm{d}x^2$, an elliptic partial differential operator on $\mathbb{R}$; let us ...
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415 views

Every Pseudo-Differential Operator is Pseudo-Local operator.

This is the theorem 8.9 of the book Introduction to PDE, Folland, G. B. I'm trying to complete the proof details. I would like to know how to justify the "Afirmation", with more rigor. Every pseudo-...
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1answer
241 views

Extension of Pseudodifferential Operator to the real line

When defining Pseudodifferential-Operators on manifolds, there is the subtlety that one theoretically has to consider all possible charts instead of just a chosen atlas. I am trying to understand how ...