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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, wave front sets, and oscillatory integrals.

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

84 questions
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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 compact manifold $X$. If $A$ is invertible, is it…
Appliqué
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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 and Simon's Methods of Mathematical Physics, the…
Nik Quine
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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 continuous linear map $K : C_c^\infty(\Omega) \to…
JZS
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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 following conditions: $X = C^\infty_c(\mathbb…
Appliqué
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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 literature are very welcome, but also, how would one…
user103756
4
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1 answer

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, the solution operator to a family of…
Nick Alger
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How does one show that the operator whose kernel is properly supported is a smoothing operator?

Proposition 1.7 (Properly supported smoothing operators). $L^{-\infty}=$ smoothing operators. Given $A \in L^{-\infty}$ with properly supported amplitude $a \in S^{-\infty}\left(\Omega \times \Omega \times \mathbb{R}^n\right)$ $$ A…
Pambra iskra
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Expected Squared Derivative of White Noise?

Let $W_t$ denote the continuous time Wiener process at time $t$. Then its time derivative $\xi(t):=\frac{d W_t}{dt}$ is white noise, and may be "seen" in the weak sense via inner products $\langle \xi, \psi\rangle = \langle W', \psi\rangle = \int…
Nathan Wycoff
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Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| \frac{\partial^{\alpha+\beta}}{\partial \xi^{\alpha} \partial x^{\beta}} p(x,\xi)…
saz
  • 123,507
4
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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 trying to understand why not. I realize that the proof…
anonymous
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Proof of first step theorem 7.7.1. in Lars Hormander's "The Analysis of Linear Partial Differential Operators I"

I'm struggling to follow the "obvious" k=0 step in the proof of this theorem: THEOREM 7.7.1. Let $K\subset\mathbb{R}^n$ be a compact set, $X$ an open neighborhood containing $K$ and $j, k$ non-negative integers. If $u\in C^{k}_c(K), f\in…
Nathanael Schilling
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Product of distributions under wavefront set condition is zero

Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\bullet$. If $\operatorname{WF}(u) \cap…
Ceka
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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. The context is the following. Let $\Omega$ be an…
rebo79
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Wavefront set of coherent state

Given that $$\psi_{(x_0,\xi_0)}(x)=(\pi h)^\frac{-n}{4}e^{\frac{i}{h}(x-\frac{x_0}{2}).\xi_0}e^{-\frac{1}{2h}(x-x_0)^2}$$ Then the wavefront set $\text{WS}(\psi_{(x_0,\xi_0)})=\{(x_0,\xi_0)\}$ In the textbook of semi-classical analysis they give…
Hoang Skadi
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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 write $$ a(x,\xi) \sim \sum_{k = 0}^\infty…
Jacob Denson
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