Questions tagged [micro-local-analysis]

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Bernoulli-like polynomials

Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then $$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$ $$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$ where $B_n$ is a monic polynomial of degree $n.$ Now ...
Boris Tsygan's user avatar
7 votes
0 answers
618 views

What's the definition of a microlocal sheaf?

I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general. In this paper of ...
EJAS's user avatar
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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 ...
Quarto Bendir's user avatar
6 votes
0 answers
193 views

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 $\...
Ceka's user avatar
  • 501
6 votes
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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 ...
Reladenine Vakalwe's user avatar
4 votes
0 answers
68 views

Local energy estimate in a semiclassical regime

Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
J.Mayol's user avatar
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What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?

As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ ...
Leo Moos's user avatar
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If theorem valid for compactly supported distribution then is it also valid for tempered distribution?

I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution. For instance, Theorem: Any $A \in \Psi^{m}$ ...
Curious student's user avatar
4 votes
0 answers
192 views

Can the existence of geodesics be deduced from properties of the Laplacian?

As I understand it, the semiclassical trace formula in particular relates lengths of geodesics to eigenvalues of the Laplacian. Is it possible to prove that every compact Riemannian manifold has a ...
Vivek Shende's user avatar
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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 ...
mfox's user avatar
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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 ...
Matt Rosenzweig's user avatar
4 votes
0 answers
242 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 ...
Andrew Homan's user avatar
3 votes
0 answers
46 views

microlocalisation of a Lagrangian state

Let $f : \mathbb{R} \to \mathbb{R}$ a smooth function of exponential decay at infinity (I think of something like a Gaussian) ; $g$ a polynomial of $\mathbb{R}$. I would like to have a quantitative ...
Adrien Boulanger's user avatar
3 votes
0 answers
43 views

Conditions ensuring that the paraproduct remainder is well-defined

In short, my question is: are there conditions that one can impose on two tempered distributions $u$ and $v$ that will guarantee that the paraproduct remainder $R(u,v)$ is well-defined and is "...
Gary Moon's user avatar
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Definition clarification: "regular directed distributions"

(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.) In the definition of ...
B.Hueber's user avatar
  • 987
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"higher" micro-support

Recall that for a sheaf $F$ on an analytic manifold $X$ the micro-support consists of those $\omega\in T^*X$ for which there exists a $C^1$ function $f$ defined around $\pi(\omega)$ with $f(\pi(\omega)...
user2520938's user avatar
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3 votes
0 answers
42 views

Partial hypoellipticity

The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\...
Ali's user avatar
  • 4,067
3 votes
0 answers
206 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 ...
Holographer's user avatar
3 votes
0 answers
536 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 ...
Math's user avatar
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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 $f_{\pi}: T^*Y \times_Y X \to T^*Y$...
Reladenine Vakalwe's user avatar
2 votes
0 answers
107 views

Limit of a distribution using Hörmander’s theorem

Let $\alpha \in \mathbb{C}$. I want to prove that $$ (e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \longrightarrow (Q(\xi)-i0)^{-\alpha}, $$ in $D’(\mathbb{R}^n\setminus \left\{0\right\})...
zarathustra's user avatar
2 votes
0 answers
132 views

Reference request: algebraic characteristic 0 version of microlocalization

I am trying to learn about microlocalization and singular supports with the end goal of understanding at least some form of the coherent-constructible correspondence. I am currently powering through ...
Sergey Guminov's user avatar
2 votes
0 answers
156 views

Product of Heavisides: calculus vs Fourier transform vs wavefront set

I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
Evangeline A. K. McDowell's user avatar
2 votes
0 answers
65 views

Asking a reference for a fact about nonlocal operators

Let assume that $u$ is smooth enough and $ -\Delta (u \phi) \in L^1(\Omega)$ for any $\phi \in C_c^{\infty}(\Omega)$. Then it easily follows that $ -\Delta u \in L^1_{\mathrm{loc}}(\Omega)$ by ...
Hheepp's user avatar
  • 361
2 votes
0 answers
57 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 ...
Ali's user avatar
  • 4,067
2 votes
0 answers
302 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 ...
Math's user avatar
  • 481
2 votes
0 answers
106 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 ...
Qwertuy's user avatar
  • 251
2 votes
0 answers
135 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}\...
TFan's user avatar
  • 63
2 votes
0 answers
193 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/...
alphanzo's user avatar
  • 113
2 votes
0 answers
217 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 ...
Sean Gomes's user avatar
1 vote
0 answers
27 views

Construction of a regulariser for the boundary integral operator $\lambda\mathrm{Id} - K'$

$\newcommand\Id{\mathrm{Id}}$Assumptions and Notations : $\Omega$ is a bounded Lipschitz domain in $\mathbb R^2$, $\Gamma$ denotes its boundary and $n$ is the normal vector to the boundary $\Gamma$, ...
SAKLY's user avatar
  • 63
1 vote
0 answers
93 views

Oscillatory integrals and regularity

Let $U\subset\mathbb{R}^{d}$ be open and $N\in\mathbb{N}$. Furthermore, let $a\in\mathcal{S}^{m}_{\rho,\sigma}(U\times\mathbb{R}^{N})$ be a symbol and $\Phi\in C^{\infty}(U\times (\mathbb{R}^{N}\...
B.Hueber's user avatar
  • 987
1 vote
0 answers
300 views

Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?

I have been reflecting on this question, and want to share my thinking thus far. I'd be grateful for the community's inputs. We refer to Morrey's inequality, Theorem 4 on pp 266 of Evan's book on PDE, ...
Nagaraj Iyengar's user avatar
1 vote
0 answers
266 views

History of microlocal analysis

Was the study of pseudo-differential operators the basis for the birth of microlocal analysis? I'm trying to find out how this branch of analysis was developed...
NSR's user avatar
  • 97
1 vote
0 answers
107 views

Wave front set and differential operator

A simple result of microlocal analysis would be, if P is a differential operator then $WF(Pu) \subseteq WF(u)$, where WF denotes the wave front set of the distribution. Can anyone give me an example ...
NSR's user avatar
  • 97
1 vote
0 answers
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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\ ...
NSR's user avatar
  • 97
1 vote
0 answers
100 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 ...
Math's user avatar
  • 481
1 vote
0 answers
200 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))...
Thomas Young's user avatar
1 vote
0 answers
39 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 ...
Umberto Lupo's user avatar
1 vote
0 answers
198 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, ...
Patch's user avatar
  • 367
1 vote
0 answers
97 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 $C^\infty$ ...
JZS's user avatar
  • 469
1 vote
0 answers
64 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 ...
Ali's user avatar
  • 4,067
1 vote
0 answers
31 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^{*}\...
James Mracek's user avatar
1 vote
0 answers
183 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 ...
Alex Zorn's user avatar
  • 902
1 vote
0 answers
125 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 ...
Rami's user avatar
  • 2,571
0 votes
0 answers
101 views

Confusion about notation

I recently came across the following generalization of the Darboux-Weinstein lemma: Let $N$ be a manifold endowed with two symplectic forms $\omega_1, \omega_2$, and let $P$ be a compact submanifold ...
KXJ's user avatar
  • 1
0 votes
0 answers
242 views

Holder continuity relative to Rellic-Kondrachov compactness via the nonlinear Aronsson operator

Connected to the question, Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness? An analysis of the well-known nonlinear Aronsson operator gives $C^{(1, \frac{1}{3})}$ ...
Nagaraj Iyengar's user avatar