Questions tagged [micro-local-analysis]

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3
votes
1answer
110 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
1
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0answers
29 views

Is the cone $\Sigma(T)$ orthogonal to the singular support of a distribution?

Hello I am totally new to microlocal analysis and I have a question. Is the cone $\Sigma(T)$ orthogonal to the singular support of a distribution? $$\xi \notin \Sigma(T) \iff \exists V\ conic\ ...
7
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0answers
149 views

Smoothness of solution map for PDE

I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
4
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0answers
210 views

Properties of microlocalization

Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
1
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0answers
50 views

Extension of a compactly supported pseudo-differential operator

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends ...
2
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0answers
51 views

Wave equation with data on null surfaces

Consider the solid cone $C$ as the region inside $z=1-\sqrt{x^2+y^2}$ and bounded by $0\leq z\leq 1$. Now let us define $$\Omega= \{z=\frac{1}{2}\} \cap C\quad \text{and}\quad \Sigma= (\partial C \cap ...
5
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0answers
261 views

How to understand the Fourier-Sato transform and microlocalization functors?

Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, ...
1
vote
1answer
285 views

Kernel of the composition of operators

Let $X \subset \mathbb{R}^{n}$, $Y \subset \mathbb{R}^{m}$, and $Z \subset \mathbb{R}^{p}$ be open subsets, and let $K_P \in C_0^\infty(X \times Y)$ and $K_Q \in C_0^{\infty}(Y \times Z)$. Then, $K_P$ ...
5
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1answer
172 views

Wavefront set of characteristic function of rough set

It is a standard exercise to show that if $X\subseteq\mathbb{R}^n$ has smooth boundary, then the characteristic function $1_X$ has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}...
7
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2answers
326 views

Weyl law for (non-semiclassical) Schrodinger operator

The Weyl law for a semiclassical Schrodinger operator $$ A_h\ := \ -h^2\Delta+V(x) $$ on an $d$-dimensional complete Riemannian manifold $M$ says that the number $N(A_h,1)$ of eigenvalues of $A_h$ ...
3
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0answers
86 views

Entire analytic functions with entire analytic Fourier transform, and corresponding distributions

I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...
10
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2answers
592 views

How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
0
votes
1answer
87 views

Associating a pseudo-differential operator to the symbol in the SG setting

We all know that given a symbol $a(x,\xi) \in S^{\mu,\rho}(\mathbb{R}^n,\mathbb{R}^n)$, a pseudo-differential operator can be defined as \begin{equation} Op(a)u(x)=(2\pi)^{-n}\int \int e^{i(x-x')\cdot ...
10
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2answers
325 views

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
4
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1answer
233 views

Wavefront set and Duhamel's principle

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal ...
4
votes
1answer
147 views

Interesting (non) examples of singular support

I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...
1
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0answers
200 views

Support of a microlocal defect measure

I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
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0answers
93 views

Propagation of singularities and the Schrodinger equation

I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation $$(i \partial_t-p(x,D))...
4
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0answers
148 views

One-parameter unitary group preserving invariant domain of infinitesimal generator

Let $\mathcal{H}$ be a separable Hilbert space (e.g. $L^{2}(\mathbb{R}^{d}))$, and let $\mathcal{D}_{1}\subset\mathcal{H}$ be a dense subspace (e.g. $\mathcal{S}(\mathbb{R}^{d})$). Suppose that an ...
3
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0answers
332 views

Propagation of Singularities

I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the ...
1
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0answers
32 views

Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem

This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without ...
10
votes
1answer
301 views

Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?

Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$. Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \...
5
votes
2answers
376 views

Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
13
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4answers
725 views

Is every continuous microlocal operator a pseudo-differential operator?

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave ...
1
vote
1answer
214 views

Microlocal proof of Wigner semicircle theorem?

Something I really enjoy about Tao's writing is that he proves the same theorem over and over. While I complain a bit sometimes about clarity, this is a heuristic that I very much believe in. This ...
1
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0answers
129 views

Interpretation of Smoothing Operators as $\Psi$DO's

In most of the computations I've seen involving pseudodifferential operators (referred to as $\Psi$DO's from now on) we do not have true equality. Instead we often only have equivalence of operators, ...
3
votes
1answer
287 views

Characterisation of the wavefront set

I am totally new to microlocal analysis, and have been studying Jared Wunsch's notes. I have been puzzling over the properties of the wavefront set. Let $X$ be a compact Riemannian manifold, and $\...
2
votes
1answer
481 views

Schwartz kernel theorem

I would like to understand how the Schwartz kernel theorem works for some more difficult cases and therefore would like to discuss an example from scratch: Let the Dirichlet Laplacian on the half-...
1
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0answers
45 views

Show that if $a$ belongs to the symbol class $S^m_{\rho , \delta}$ for $\rho > 1$, then in fact $a \in S^{-\infty}$

I'm trying to learn more tools from microlocal analysis by reading Grigis and Sjostrand's Microlocal analysis for differential operators. In the first chapter, they define the standard $C^\infty$ ...
22
votes
7answers
2k views

Applications of Microlocal Analysis?

What examples are there of striking applications of the ideas of Microlocal Analysis? Ideally i'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...
1
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0answers
51 views

Normal form of Principal type $\Psi$DO's

Suppose we have a pseudo differential operator of principal type with a complex symbol and such that the poisson bracket of the real and imaginary parts on the characteristic set is non-negative.I ...
2
votes
1answer
305 views

Inverse of pseudo differential operator

Let $\operatorname{Op}_h(x,D)(a)$ denote the Weyl-quantisation of a symbol $a$. Is there an explicit way to invert this pseudo-differential operator in an asymptotic series? By this I mean, can we ...
3
votes
1answer
256 views

Help understand a calculation involving RHom of sheaves on manifolds

I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask. It is this paper, example 3.10 , page 25 arxiv.org/pdf/1005.1517v4....
1
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0answers
22 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^{*}\...
2
votes
1answer
178 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 $\...
3
votes
1answer
374 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 ...
8
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3answers
816 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/...
2
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0answers
82 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 ...
3
votes
1answer
388 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 ...
4
votes
1answer
411 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
173 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
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0answers
101 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)^{-...
2
votes
1answer
378 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 \; ;...
6
votes
2answers
4k 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
175 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
561 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
238 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
122 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}\...
3
votes
1answer
253 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
174 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/...