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.
87
questions
1
vote
0answers
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 $$Pf(x)=\int_{\mathbb R^n}e^{ix\...
0
votes
0answers
10 views
Laplacian raised to positive integer
In books about Pseudo-differential operators, they use many times
$\triangle^k u$
but i have a question, what means this really
option A
$\displaystyle{\sum_{|\alpha|=k}}{\, \frac{k!}{\alpha!} \left({\...
1
vote
0answers
35 views
How and when is the inverse Laplacian well-defined as a pseudo-differential operator?
I recently came across an interesting (mis-)use of formal equivalencies. First, the uncontroversial bits.
By the Fourier derivative theorem, it is straightforward that the 2D Laplace operator can be ...
1
vote
0answers
59 views
What is the meaning of $\flat$ and $\sharp$ in this smoothing operator? $M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $
I found this:
$$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) $$
and i'm not sure whether i understand it ...
0
votes
0answers
27 views
Convolution algebra of a groupoid
In this paper in order to explain conormal distribution the authors use the fact that “the elements of the convolution algebra of a groupoid are sections of a density bundle $\Omega ^{1/2}$ rather ...
0
votes
0answers
15 views
Wave front set of characteristic functions?
There are some problems to calculate the wave front set of $\chi_\Omega$, where
$\Omega$ is a region with smooth boundary $\partial \Omega$;
$\Omega=\{(x,y)\in \mathbb{R}^2| x^3\ge y^2\}$.
What I ...
0
votes
0answers
12 views
Is there any compactly supported function in the Gelfand-Shilov space $\Sigma_1(=S^{(1)})$?
Good evening, I have a question related to the Gelfand-Shilov spaces as defined in https://arxiv.org/pdf/1505.04096.pdf (see 1.2, pag.2).
I seemed to understand that there is no non-zero function of $...
1
vote
0answers
28 views
Two Definition of Elliptic Symbols
There two definition of elliptic symbol.
A smooth matrix function $p(x,\xi)$ is a elliptic symbol of order $m\in\mathbb{R}$ if exist a constant $c>0$ such that for all $|\xi|>c$ we have $p(x,\xi)...
0
votes
0answers
35 views
Is there some classification of idempotent pseudo-differential operators?
Let $\pi:E\rightarrow M$ be a complex vector bundle over a smooth compact manifold $M$. Let $\operatorname{Psd}(E)$ be the associative $\mathbb{Z}$-graded complex algebra of pseudo-differential ...
1
vote
1answer
79 views
Is it possible to construct a vector space on convolution
Let $S$ be a set of functions on $R$ such that for any two functions $f$ and $g$ in $S$, the convolution:
$$
(f{\ast}g)(x)={\int}f(y)\ g(x-y)\ dy
$$
exists. Since the Dirac delta is technically not a ...
0
votes
0answers
20 views
How can I show that the integral mentioned is divergent integral (the integral is in $\mathbb{R}^{n}$)
I am studying the book "An Introduction to Pseudo-Differential Operators" by M.W. Wong - Chapter 8, The Product of Two Pseudo-Differential Operators. I'm trying to verify that the integral ...
2
votes
1answer
68 views
Confusion about definition (in ΨDO theory) of Sobolev spaces on open sets in Euclidean space
I'm reading Pseudodifferential Operators by M. E. Taylor, where the author talks about $H^s(\Omega)$ for $s\in\mathbb{R}$ and $\Omega\subset\mathbb{R}^n$ an open set (for example, in the statement of ...
0
votes
0answers
30 views
Composition of (Hörmander) symbols with smooth functions between cones
I am working with Hörmander's Fourier integral operators I and have some problems understanding the proof of proposition 1.1.7. It's about symbols $a \in S^m_{\rho,\delta}(X\times \mathbb{R}^n)$ ...
0
votes
1answer
57 views
Fractional derivative via Continuous Functional Calculus
If we study C* algebras, at one point or another we are exposed to the idea of the Continuous functional Calculus, i.e. if $\mathcal{Y}$ is a unital C* algebra, $\xi \in \mathcal{Y}$ is normal, that ...
1
vote
0answers
28 views
Why is the Order of a Classical Symbol Well-defined?
A function $a$ defined on $T^*\mathbb{R}^n$ is called a classical symbol of order $m$ provided it is smooth and there exists $\tilde{a}\in C^\infty(\mathbb{R}^n\times S^{n-1}\times\mathbb{R}_+)$ so ...
0
votes
0answers
13 views
Applications of equations $B\partial_t u=Au$ where $B$ and $A$ are differential operators and $B$ invertible.
I was wondering if there are any nice applications of equations of the flavour
$$
B\partial_t u=Au,
$$
where $A$ and $B$ are differential operators and $B$ is invertible. E.g. On suitable domains such ...
0
votes
0answers
34 views
Example of a symbol in $S^m_{\frac{1}{2},0}$ Hormander class.
Let $\Phi$ be a smooth function satisfying
\begin{equation}
1 \leq \Phi(|x|) \lesssim \langle x \rangle =(1+|x|^2)^{1/2} \quad x \in \mathbb{R}^n,
\end{equation}
and
\begin{equation}
|\partial_x^...
1
vote
0answers
26 views
Fourier Transform (Wong's book)
I'm with problems to prove an equality that appears in some proofs on book that I'm reading (An Introduction to Pseudo-Differential Operators).
We define the fourier transform of a function $f$ in $L^...
0
votes
1answer
78 views
Why does the equation hold?
Why is $$\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, $$ where $p$ is a polynomial. $\Delta$ denotes the Laplacian operator. I do not understand the ...
1
vote
0answers
54 views
Question about Folland PDE book Chapter 8
Page 284-285:
Let $\Omega\subset {\mathbb R}^n$ be a domain.
Let $a(x, \xi, y)$ be an "amplitude", i.e. a smooth function on $\Omega\times{\mathbb R}^n\times\Omega$ so that there is $m$, and for ...
4
votes
0answers
70 views
Leading Symbol of Pseudo-Differential Operators
I have seen various definitions of symbols used to define pseudo-differential operators. The class of symbols I am working with is
$S^d(U)=\{p(x,\xi)\in C^{\infty}\left (U\times \mathbb R^m ,M_{k\...
2
votes
1answer
49 views
Heine-Borel property of symbol spaces
I am interested in whether the uniform symbol spaces $S^m_\infty$ from the theory of pseudo-differential operators have the Heine-Borel property (i.e. every bounded set is relatively compact).
To ...
2
votes
0answers
78 views
“Crazy” Attempt to Understand Second-Order DEs on Manifolds
(This is kind-of a crazy ramble; it's kind-of not a questions and kind-of not a request for comments, but it's more of a request for comments.)
We all know what the second derivative of a real-valued ...
2
votes
1answer
111 views
Show $(1+|\xi|^2)^{m/2}\in S^m$ (pseudo-differential operators)
I am trying to show that $p(\xi)=(1+|\xi|^2)^{m/2}\in S^m$, i.e., that for all $m\in\mathbb{R}$ and all multi indices $\alpha,\beta$
$$|D^{\beta}_{x} D^{\alpha}_{\xi}p(\xi)|\leq C_{\alpha,\beta}(1+|\...
0
votes
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 ...
2
votes
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 ...
2
votes
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
$$|D_{x}^{\beta}D_{...
0
votes
0answers
36 views
Extending a pseudo-differential operator on $L^2(\mathbb{R}, e^x\,dx)$
Let $A$ be a pseudo-differential operator with symbol $a(x,\xi)=e^{-x}\psi(\xi)$ where $\psi$ is a continuous negative definite function. It is known that $A$ is the generator of a self-similar Markov ...
1
vote
0answers
14 views
Generalized functions transforming as 1/4-densities under diffeomorphisms
I am looking for generalized functions (including pseudodifferential operators) that satisfy the following identity ($x,y \in R^d$):
$f(h(x),h(y))=\left(\frac{1}{\det(D_xh) \det(D_yh)}\right)^{1/4} f(...
2
votes
1answer
97 views
What does constant along characteristic mean
When we have linear or quasilinear first order pde
$$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$
And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$
I ...
2
votes
1answer
82 views
About pseudo-differential operators
Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is ...
1
vote
1answer
53 views
The symbol of a pseudifferential operator: how to reconstruct the symbol from an operator?
To simplify let us assume that $U \subset \mathbb{R}^n$ is an open set. Let $P$ be a (scalar) psuedodifferential operator of order $d$ and $p$ be its symbol. Therefore it may be expressed as follows:
...
2
votes
0answers
40 views
Understanding Y U Egorov's theorem
I was reading the paper
"Hörmander, L., Fourier integral operators. I, Acta Math. 127, 79-183 (1971). ZBL0212.46601." and on page 3, there is a statement saying the following thing:
Can someone ...
0
votes
1answer
22 views
Differentiation of an operator raised to a variable power
I am studying first order systems of the form
\begin{equation}
L=\partial_t+K(t,x,D_x)\text{ where }D_x=-i\partial_x
\end{equation}
There is a change of variable and operator of our concern becomes ...
0
votes
1answer
104 views
Pseudo Differential Operator example
I was reading a introduction about Pseudo differential operators, and the definition of that type of operator was:
$$|\partial_x^\alpha \partial_\xi^\beta f(x,\xi) | \le C_{\alpha,\beta}(1 + |\xi|)^{...
1
vote
0answers
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}...
4
votes
0answers
57 views
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 ...
1
vote
0answers
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 ...
2
votes
0answers
94 views
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:
$$
M \begin{align}
\begin{bmatrix}
...
1
vote
2answers
48 views
Existence of solutions of $u_t = Lu$ for pseudo-differential operators $L$?
I'm wondering if it is known if existence results for the equation $$\left\{\begin{array}{ll}
\partial_t u = Lu & \text{on }(0,T)\times\mathbb{R}^n\\
u(0,-) = \varphi & \text{on }\mathbb{R}^n\...
0
votes
0answers
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 $B$ ...
3
votes
1answer
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 ...
8
votes
1answer
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 ...
4
votes
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 $...
1
vote
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(\...
2
votes
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 ...
4
votes
0answers
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-...
2
votes
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 ...
