Questions tagged [pseudo-differential-operators]

This tag is for questions regarding to pseudo-differential operators, which are generalizations of differential operators and Fourier multipliers. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.

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On class of Hörmander symbols. [duplicate]

In An Introduction to Pseudo-Differential Operators by Wong, the Hörmander's class is defined as every function $\sigma(x,\xi)\in\mathcal{C}^{\infty}(\mathbb{R}^n\times \mathbb{R}^n)$ such that for ...
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Lawson's Spin Geometry book - pieces that do not fit

This is from Lawson's book "Spin Geometry", there is a problem here, there are pieces that do not fit. In (4.3) and (4.4) shows how to construct the operator Q from P and its symbol. For ...
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Reference needed: Symbol sequence for pseudodifferential operators

In Higsons's book Analytic K-Homology there is a section (subsection (b) in 2.8 "Geometric Examples of Extensions", starting from page 46) which discusses the following exact sequence called ...
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Good books and lecture notes to learn pseudo-differential operators and spectral theory

I am looking for a list of good books and lecture notes to learn pseudo-differential operators and spectral theory (for infinite dimensions.) I am familiar with introductory functional analysis, ...
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Extending symbol of pseudodifferential operator defined outside of a ball

While reading Wong's book "An introduction to pseudo-differential operators" (3rd edition), I came across the following statement in the exercises : Let $\sigma \in C^\infty(R^n\times R^n) $...
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Pseudodifferential operator properly supported

Be $P$ a pseudodifferential operator in $\Omega$, it have properly supported when both $P$ and $P^*$ have the for each compact $K \subset \Omega$ there is a compact $K^′ \subset \Omega$ such that ...
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Simple expression for $(-\Delta)^{\frac{3}{2}}$ in the spatial domain?

I can write the harmonic and biharmonic operators as: $$-\Delta = -\sum_i \partial_{ii}, \quad (-\Delta)^2 = \sum_i\sum_j \partial_{ii}\partial_{jj}$$. Is there such a simple expression for $(-\Delta)^...
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Generalization of the Leibniz rule to negative power differential operator $\partial_x^{-s}\,,\, s\geq 0$

I'm trying to understand how the operator $\partial_x^{-s}\,,\, s\geq 0$ acts on a function, and on the product of two functions. In addition, do we have $$ \partial_x^{-1}(uv)=\partial_x^{-1}(u)v+u\...
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Spectrum of the Weyl quantized operator $\mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}}\right)$

Consider a 1D phase space whose generic points are denoted as $(x,p)$. We know that the Weyl quantization $\mathrm{Op}\left(\frac{x^2+p^2}{2}\right)$ is the harmonic oscillator Hamiltonian, whose ...
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Contorlled at infinity by one-point compactification

My question is based on a lecture note on pseudo-differential operator by Melrose. Let $X$ a compact manifold with boundary and $U$ an open manifold which is not compact, and let $U\to X$ a smooth ...
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Friedrichs Inequality

I'm a little confused with the following proof of Friedrichs inequality in Lawson-Michelsohn's book “spin geometry”, page 194, Theorem 5.4: I don't understand why the last inequality $C(\|\varphi u\|_{...
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Definition of a weak solution in a general context.

I am trying to find in some book or text the general definition of weak solution for a pseudo differential equation. In the book An introduction of the pseudo differential operators by Wong M. the ...
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Defect measure associated to a sequence of exponentials

In my road to understand microlocal defect measures, at the beginning of Gerard's article Microlocal defect measures, there is an statement about (an example of) defect measures where I am struggling. ...
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Composition of two differential operators . Laplacian operator.

I have a question with the composition of two operators. To contextualize. Let $p_1 = p_1 (x,D_x):= 1-\partial_ {x}^2 $ be the operator $ p_1: D(p_1)\subset L^2 \to L^2 $ where $$ D(p_1) = \left\{u \...
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Asymptotic expansions with compactly supported terms and smoothing operators

Suppose that we have a pseudodifferental operator $A\in \Psi^m(\mathbb{R}^n)$ with symbol $a\in S^m(\mathbb{R}^n\times\mathbb{R}^n)$, and $a$ has an asymptotic expansion $a\sim \sum\limits_{j=0}^\...
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Doubt in notation in Pseudodifferential and Singular Integral Operators: An Introduction with Applications, Abel

In Pseudodifferential and Singular Integral Operators: An Introduction with Applications by Abel, page 47 Query. How is the function $\left\{\mathcal{X}_{\epsilon}(y,\eta)\right\}_{0<\epsilon<1}...
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composition/multiplication of pseudo-differential operators

I am very new to this and I am stuck at a, supposedly, easy problem: Let $V: \mathbb R^n \to \mathbb R$ be smooth and set $ p(x,\xi) = |\xi|^2 + V(x) + i, \quad p(x,\xi) = (|\xi|^2 + V(x) + i)^{-1} $ ...
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Isomorphism of Differential operator involving the Laplacian

I have to show that $(1+\Delta)^s:H^k(U)\rightarrow H^{k-2s}(U)$ is an isomorphism where $\Delta$ is the Laplacian on $U\subset\mathbb{R}^n$ where $U$ has compact support and $H^k(U),H^{k-s}(U)$ are ...
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Asymptotic Expansions of Symbols Vs Asymptotic Expansion of Pseudodifferential Operators

Let $a_k(x,\xi)$ be a family of symbols on $\mathbf{R}^d \times \mathbf{R}^d$, where $a_k$ has order $\alpha_k$, and $\lim_{k \to \infty} \alpha_k = -\infty$. Then for another symbol $a(x,\xi)$, we ...
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Theorem 3.12 Lawson's Spin Geometry book

Theorem 3.12: Let $\phi\colon U \rightarrow V$ be a diffeomorphism between open subset of $\mathbb{R}^n$. Then for each compact subset $K\subset U$, $\phi$ induces a map $\phi_{*}\colon \Psi DO _{K,m} ...
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What is a Symbol (as in "Symbol Calculus")?

I have been reading through several papers on Deformation Quantization and the terms "symbol" and "symbol calculus" keep cropping up. I am somewhat well acquainted with most of the ...
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Product with a smooth function, Sobolev regularity and pseudo-differential operator

I am working on microlocal Sobolev spaces using Hörmander's books, and while trying to write down the details of a proof, I have faced the following problem. It looks easy, but I can't find a correct ...
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Comparison of quantization procedures

Let $M$ be a compact manifold and let $U\subset M$ be an open set with a coordinate chart $\Phi:U\to V$ with $V\subset \mathbb{R}^d$. Suppose that $w\in C_c^\infty(U)$ is a smooth function supported ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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"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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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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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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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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1 answer
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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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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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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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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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2 votes
1 answer
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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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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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2 votes
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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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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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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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