The micro-local-analysis tag has no usage guidance.

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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 for example, ...

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33 views

### Combining microlocal Helgason's support and Holmgren's theorem to prove injectivity of limited-angle Radon transform

This questions is slightly related to Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems, in which I asked for some references. Now I ...

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90 views

### Wave front set from the FBI or Segal-Bergman transform (and a motivation)

In the André Martinez's notes "Introduction to microlocal and semiclassical analysis" the Wave Front Set is defined as the complement of the set of points having neighborhoods where the FBI transform ...

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203 views

### Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems

I would like to find good references for the theorems mentioned above in the title. I am reading chapter VIII of Hörmander's classic, but I wonder whether there is something more up-to-date.
My ...

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**1**answer

135 views

### Ask the validity of Tauberian lemma in Sogge's book

In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):
Lemma. Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume ...

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73 views

### Resolvent estimate of hyperbolic Laplacian [closed]

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 ...

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**1**answer

288 views

### Pseudo-differential evolution equation

I'm looking for results (or some ideas) on the following kind of pseudo-differential evolution equation:
$$
\frac{\partial u(t,x)}{\partial t} = \int_{-\infty}^{t} B(t-s,x)\, A(x,D_{x})u(s,x)\,ds \; ...

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402 views

### Fourier transform of compactly supported distribution is smooth

My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...

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112 views

### Exact Functors from Perverse Sheaves

Pretty sure this is a simple question, but there's something I'm missing here.
For context, I'm reading 'An Elementary Construction of Perverse Sheaves' by MacPherson and Vilonen. A key aspect of the ...

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**1**answer

193 views

### Application of Egorov's Theorem for Pseudodifferential Operators

Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...

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169 views

### Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...

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98 views

### $D^{\infty}$ modules on analytic spaces

In Mebkhout's paper on Local Cohomology of Analytic Spaces, the following theorem is stated:
Let $X$ be a complex smooth manifold and $Y$ is an analytic subspace of $X$. Then ...

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**1**answer

135 views

### Elliptic Fourier integral operators

I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.
But ...

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149 views

### geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
...

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106 views

### wavefront is a coisotropic

I'm thinking about the following example: Let $u_k:=e^{i(kx_1+k^2x_2)}$, where $k\in\mathbb{N}$ and $x_1,x_2\in\mathbb{R}$. Then for the sequence $h_k:=1/(k\sqrt{1+k^2})$ ($h_k\rightarrow0$ as ...

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145 views

### Real-analytic variant of theorem 4.2.5 of Duistermaat's “FIO”, 1996

Theorem 4.2.5 of Duistermaat's "Fourier Integral Operators", 1996, states:
Let $A \in I^m(X,Y,C)$ be an elliptic Fourier Integral Operator of order $m$, associated to a bijective canonical ...

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**1**answer

142 views

### 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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76 views

### Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$

Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...

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110 views

### Support-preserving pseudodifferential operators

Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the ...

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130 views

### Interpretation of the integral “with respect to a plane wave” in terms of Radon transform

This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform:
$\newcommand{\R}{\mathbb{R}}$
$$
Rf(\varphi,s)=\int_{-\infty}^\infty ...

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102 views

### What's a good resource for Hormander symbols of type (1/2, 1/2)?

I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...

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141 views

### A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...

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197 views

### C^\infty versus semiclassical wavefront sets

Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ ...

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433 views

### propagation of singularities & the Schrodinger equation

I've been thinking about the following propagation of singularities result:
Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol ...

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166 views

### Characteristic variety of a D-module along smooth pullback

All varieties are over the complex numbers. Given a smooth variety $X$, write $T^* X$ for its cotangent bundle.
For a morphism of smooth varieties $f: X \to Y$ write $f_{\pi}: T^*Y \times_Y X \to ...

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### the relation between a continuous family of distributions and a distribution of 2 variables

Let X,Y be smooth manifolds and let $f:X \to C^{-\infty}(Y)$ be a continuous map, where $ C^{-\infty}(Y)$ is the space of generalized functions on $Y$ equipped with the weak topology. By Schwartz ...

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### base change for distributions

For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...

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### Non-characteristic maps (ala D-modules)

I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is ...

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### Pseudo-differential operators which are independent of lower order perturbations

In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real ...

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286 views

### From microlocal to local

Assume $u\in L^2(\mathbb{R}^n)$ and let $(x_0, \xi _0) \in T^\ast \mathbb{R}^n = \mathbb{R}^n_x \times \mathbb{R}^n_\xi $. Assume I can find $a\in C^\infty (T^\ast \mathbb{R}^n)$ which is also bounded ...

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### Characterization of inverse differential operators

If I have a partial differential operator $p(D)$, where $p$ is a polynomial with constant coefficients and $D$ is the derivative in Euclidean space. Its inverse is easily described in Fourier space: ...