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

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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: $\...

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13 views

### Sufficient conditions for a conormal vector to be regular for an orbit stratification

Let a complex reductive group $G$ act on a $\mathbb{C}^{n}$ with finitely many orbits. Let $\mathcal{S}$ be the stratification of $\mathbb{C}^{n}$ according to these orbits. Let $(x,\xi) \in T_S^{*}\...

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32 views

### Show that a very regular kernel $k(x,y)$ has operator $K : \mathcal{E}'(\Omega) \to \mathcal{D}'(\Omega)$ which is pseudo-local

I am reading Francois Treves' Introduction to pseduodifferential and Fourier integral operators, vol. I. I am having trouble understanding the proof of Lemma 2.1, which is stated as follows.
Let $\...

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113 views

### Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following:
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...

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318 views

### 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, http://www.sciencedirect.com/science/...

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303 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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39 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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**1**answer

111 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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**1**answer

226 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

143 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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83 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 I)^{-...

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770 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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123 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

231 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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178 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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153 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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105 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 $\mathcal{D}_X^{\infty}\...

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**1**answer

132 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 f(s\...

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**1**answer

154 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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157 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)
http://link.springer.com/...

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107 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 $k\...

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### 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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85 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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123 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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108 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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148 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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### 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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207 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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468 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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### 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 $f_{\pi}: T^*Y \times_Y X \to T^*Y$...

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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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105 views

### 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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240 views

### 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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291 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 ...