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Questions tagged [microlocal-analysis]

Microlocal analysis involves using Fourier transform techniques to study linear and nonlinear PDEs. Topics include pseudo-differential operators, Fourier integral operators, and oscillatory integrals.

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Integration by parts for an oscillatory integral

I want to understand a certain integration by parts argument in Treves' book Introduction to Pseudodifferential and Fourier Integral Operators (It appears in the proof of Theorem 4.1). Here is a self-...
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1answer
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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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About FBI transform

I read a statment on a book saying that the FBI transform $$\mathcal{F}_u(x,\xi)=\int_{\mathbb{R}^m} e^{i\xi.(x-y)-|\xi||x-y|^2}u(y)\,dy ,\; (x,\xi)\in \mathbb{R}^m \times \mathbb{R}^m$$ is nonlinear. ...
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Reed and Simon definition of product of distributions

Let $\mathcal{D}$ denote the space of $C^{\infty}$, compactly supported functions on $\mathbb{R}^{d}$, and let $\mathcal{D}'$ denote its dual (i.e. the space of distributions). In volume II of Reed ...
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Solution operator to scalar hyperbolic equation on Sobolev space $H^s$

If i have the solution operator $e^{itA}$ to the following equation $$\frac{du}{dt}=iA(x,D)u$$ where $A\in OPS^1$ is a peudo differential operator of order 1. Then how could one show that the solution ...
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177 views

Convolution Theorem for Distributions

I am searching for a version of the convolution theorem for functions (i.e. $\hat{f \cdot g} = \hat f \star \hat g$) that also applies to Distributions/tempered Distributions. Is it automatically true,...
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Show that functions $a \in C^\infty(\overline{\mathbb{B}^n})$ with $a|_{\partial{\overline{\mathbb{B}^n}}} = 0$ belong to $S^{-1}(\mathbb{R}^n)$

Let $\mathbb{B}^n$ denote the open unit ball in $\mathbb{R}^n$. I am reading a paper which claims that functions $a \in C^\infty(\overline{\mathbb{B}^n})$ such that $a|_{\partial{\overline{\mathbb{B}^...
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Proof of the Pseudolocal property of a pseudo differential operator

Let $a\in S^{\infty}, u\in\mathcal{S}', \Omega=\mathbb{R}^n-singsupp(u)$. Then $\phi u\in C_0^\infty, \forall \phi \in C_0^\infty(\Omega)$ and for any $\psi\in C_0^\infty(\Omega)$ one can find a $\phi\...
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1answer
29 views

Construct $\psi \in C^\infty_0(\mathbb{R})$ so that $\psi > 0$ on $[-M,M]$ and $\partial_x (x \psi) \ge 0$

Suppose we are given a closed interval $[-M,M] \subseteq \mathbb{R}$, $M > 0$. I would like to know whether there exists a function $\psi \in C^\infty(\mathbb{R}; [0,1])$ such that $\psi$ ...
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Wavefront Set of the Heaviside Function

If we define the Heaviside function in the standard way $H(x)=\begin{cases} \ 1 & x\geq 0 \\ \ 0& x<0 \end{cases}$ Then I want to find the Wavefront set where I am using the definition ...
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Determining smooth domain for self-convolution of distribution.

Suppose I have a distribution, $T_\varphi\in \mathcal{\mathcal{D}}'(\mathbb{R^n})$, which can be represented by a locally-integrable function, $\varphi:\mathbb{R}^n\to \mathbb{R}$. i.e., for any test ...
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1answer
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Example of a Distribution where $(x, \xi) \in WF(u)$ but $(x, -\xi) \notin WF(u)$

I am searching for an example of a Distribution $u \in \mathcal{D'}(\mathbb{R}^2)$ where $(x, \xi) \in WF(u)$ but the opposite direction $(x, -\xi) \notin WF(u)$ is not. A quick Google-Search didn't ...
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H^s order of jump discontinuities

Consider the 2D step function $f(x,y)$ which is 1 for $y>0$ and $0$ for $y\leq 0$. I want to determine the $H^s$ order of the jump along the $x$ axis. We know the singular support of $f$ is the ...
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Maths and Economics

The function $f:\mathbb{R} \to \mathbb{R}$ defined by $$ f(x)=\begin{cases} 1\,,\qquad &\text{if $|x| \le 1$}\\ 2x\,, &\text{if $|x|>1$}. \end{cases} $$ Is this function convex and ...
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1answer
210 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 ...
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If $F(x,a) \in C^\infty$ and $DF_x(x_0, a_0)$ is nonsingular , is $x \to F(x,a)$ a diffeomorphism for all $a$ near $a_0$?

Let $X \subseteq \mathbb{R}^n$, $A \subseteq \mathbb{R}^p$ be open sets. Suppose that there is an injective $C^\infty$ map $\varphi : A \to X$. Furthermore, suppose that we have a $C^\infty$ map $F : ...
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Wave front set of vector-valued Dirac delta distribution

Context: I am reading a physics paper Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime which applies the notion of the wave front set to operator-valued ...
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2answers
262 views

How to calculate the wave front set for the characteristic function of a 2-dimensional ball?

I've been trying to show that the wave front set for the characteristic function of the open ball, $B(0,1)$, is given by the boundary normal vectors $\{(x,\xi) \in S^1 \times \mathbb{R}^2-\{0\}$: $x \|...
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Textbook/monograph for microlocal analysis

I want to grasp the theory of microlocal analysis and apply this theory to some PDEs in $R^n$. But most textbooks I found put much priority on manifolds. Sadly, I know little about them and don't ...
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Continuity of smoothing operators

This question comes from Grigis-Sjorstrand's "Microlocal Analysis." Let $X,Y$ be open subets of Euclidean space, and let $K \in C^\infty(X\times Y)$, and consider the associated operator $A$, defined ...
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1answer
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Show that a regularizing operator $K : C_c(\Omega) \to \mathcal{D}'(\Omega)$ has kernel $k \in C^\infty(\Omega \times \Omega)$.

I am reading Francois Treves' Introduction to pseudodifferential and Fourier integral operators, vol. I. Let $\Omega \subseteq \mathbb{R}^n$ be open. On page 11, Treves defines what it means for a ...
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Characteristic Variety of the Principal Symbol solves PDE system?

In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see, https://en.wikipedia.org/wiki/...
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279 views

wave front set - directions of singularities

I am learning about the wave front set of a distribution but am having difficulty understanding some details, which to me seem counter intuitive. We know the fourier transform of a smooth function ...
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77 views

References for microlocal versions of some theorems

I am trying to introduce myself into Microlocal Analysis. In particular, motivated by some results in Inverse Problems, I would like to find good references for the microlocal versions of Helgason and ...
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2answers
219 views

Non-compactness of the resolvent

Consider a complete non-compact Riemannian manifold $M$ and the resolvent of the Laplacian $(-\Delta + \lambda I)^{-1}$. It is known that the resolvent is in general not a compact operator. I am ...
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Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda I)^{-...
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Wavefront set of a smooth function

I'm trying to show that the wavefront set of a function $f \in C^{\infty}(\mathbb{R}^{n})$ is empty, $WF(f) = \emptyset$. Can anyone help me prove this? This is my first exposition into wavefront ...
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1answer
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Example of pseudodifferential operators that smooth out the singularity of delta function

What is one example of pseudodifferential operator $P$ that smooth out the singularity of delta function, i.e. $P$ s.t. $P \delta(x) \in C^{\infty}(\mathcal{R})$?
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219 views

Are pseudo(micro)-local operators pseudodifferential?

$\DeclareMathOperator{supp}{supp} \DeclareMathOperator{sing}{sing}$Let $\Omega$ be a domain with compact closure in $\mathbb R^n$. Consider a linear operator $A \colon X \to X$ satisfying one of the ...
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When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral $$...
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2answers
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semiclassical principal symbol

What is the semiclassical principal symbol $\sigma_h$ of the operator $h^2\Delta-1$ (here $\Delta=-\sum_j\partial^{2}_{x_j}$)? $h^2\Delta-1$ is a second order semiclassical partial differential ...
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Applications of Microfunctions

Can anyone suggest good (a) uses/applications or (b) construction of micro-functions (introduced by Mikio Sato in 1971) in analysis? I am trying to understand the subject better. Suggestions of ...
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1answer
160 views

Implicit function theorem for pseudo-differential operators

Is there something analogous to the regularity results of the implicit function theorem, but for pseudodifferential operators? I'm looking for something to the effect of, "under certain conditions, ...