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6
votes
3answers
277 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, ...
1
vote
0answers
36 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 ...
2
votes
1answer
100 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 ...
3
votes
1answer
207 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 ...
1
vote
1answer
137 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 ...
2
votes
0answers
75 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 ...
2
votes
1answer
294 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 \; ...
3
votes
2answers
535 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 ...
1
vote
0answers
116 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 ...
3
votes
1answer
213 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 ...
2
votes
2answers
172 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 ...
2
votes
0answers
100 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 ...
3
votes
1answer
144 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 ...
2
votes
0answers
151 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) ...
1
vote
1answer
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 ...
3
votes
1answer
146 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 ...
0
votes
1answer
147 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 ...
1
vote
1answer
81 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 ...
3
votes
1answer
121 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 ...
3
votes
1answer
131 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 ...
3
votes
0answers
104 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 ...
2
votes
0answers
142 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 ...
3
votes
1answer
200 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 $u$ ...
5
votes
1answer
446 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 ...
1
vote
0answers
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 ...
2
votes
0answers
78 views

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 ...
1
vote
0answers
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 ...
6
votes
0answers
234 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 ...
6
votes
1answer
689 views

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 ...
0
votes
1answer
288 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 ...
9
votes
2answers
1k views

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: ...