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Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation

In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...
monotone operator's user avatar
1 vote
1 answer
250 views

Wick product of free fields and wave front sets in the sense of Lars Hörmander

Let $\phi$ be the neutral, massive and free scalar field in $\mathbb{R}^4$. That is, $\phi$ is a tempered distribution whose values are unbounded operators on the Bosonic Fock space. Note that the ...
Isaac's user avatar
  • 3,113
3 votes
1 answer
185 views

Equivalent Littlewood-Paley-type decompositions

The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that \begin{...
vmist's user avatar
  • 939
3 votes
0 answers
47 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
2 votes
0 answers
111 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
1 vote
0 answers
31 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
2 votes
0 answers
140 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
1 vote
2 answers
207 views

Hörmander’s propagation of singularities in two variables

I am trying to apply the propagation of singularities theorem to a distribution $u \in D’(M \times M)$ that verifies $Pu = f$, with $P$ a linear differential operator and $f \in D’(M \times M)$, as ...
zarathustra's user avatar
1 vote
0 answers
95 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
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0 votes
2 answers
219 views

Well-defined distribution and its singular support

Let $f$ be a smooth function on $X$, an open subset of $\mathbb{R}^n$, with $Im(f) \geq 0$. Let us fix an $\epsilon > 0$. Let $T_{\epsilon} := \frac{1}{f(x)+i\epsilon} $ in $D’(X)$. Now if we ...
zarathustra's user avatar
2 votes
1 answer
140 views

Microlocal approach to definition of product of distributions

My question may be simple to an expert, but I'm not: Let's consider $u \in C^{s}(\mathbb{R}^d)$ be a Hölder function sor some $s\in [0,1/2)$ which we may take very close to $0$. Of course, $u^2 \in C^{...
J.Mayol's user avatar
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3 votes
0 answers
51 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
  • 673
4 votes
1 answer
317 views

Sharpest version of semiclassical Calderon-Vaillancourt theorem

Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
Yonah Borns-Weil's user avatar
6 votes
0 answers
199 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
3 votes
0 answers
98 views

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
  • 1,077
4 votes
0 answers
69 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
  • 489
2 votes
0 answers
165 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
3 votes
2 answers
329 views

Reference for commutator estimate

I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S_{1,0}^{m_1}$ and let $q = q(\...
Gary Moon's user avatar
  • 673
0 votes
0 answers
103 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
4 votes
0 answers
147 views

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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3 votes
1 answer
244 views

do hyperfunction solutions always exist?

I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...
Avi Steiner's user avatar
  • 3,039
3 votes
0 answers
140 views

"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
  • 2,768
21 votes
1 answer
2k views

Algebraic microlocal analysis and nonlinear PDE

Though originating in the study of linear partial differential equations, microlocal analysis has become an invaluable tool in the study of nonlinear pde. Of particular importance has been the ...
Gary Moon's user avatar
  • 673
8 votes
0 answers
684 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
  • 191
2 votes
0 answers
66 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
4 votes
0 answers
140 views

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
0 votes
0 answers
243 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
1 vote
0 answers
309 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
3 votes
0 answers
45 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
  • 4,113
4 votes
0 answers
205 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
  • 8,653
5 votes
1 answer
2k views

Does the 'reproducing kernel formula' for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$

We refer to the 'reproducing convolution formula with a kernel' for an open bounded domain $U$ of $R^n$, $n \geq 2$ discussed in the paper of G. Talenti (Annali de Matematica, Dec 1976) on Best ...
Nagaraj Iyengar's user avatar
9 votes
0 answers
359 views

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
5 votes
1 answer
220 views

Why do people study Weyl asymptotics and partial-spectral-projections?

The major focus of the research that my advisor has me doing centers around the idea of asymptotic behavior of partial-spectral-projections on compact manifolds. In a few sentences, here is the ...
Patch's user avatar
  • 367
1 vote
0 answers
285 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
122 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
4 votes
1 answer
198 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 ...
Dima's user avatar
  • 335
1 vote
0 answers
46 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\ ...
NSR's user avatar
  • 97
8 votes
0 answers
242 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 ...
Quarto Bendir's user avatar
4 votes
0 answers
333 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 ...
mfox's user avatar
  • 283
1 vote
0 answers
105 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
  • 491
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
  • 4,113
6 votes
1 answer
1k 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, ...
Mathmank's user avatar
  • 272
1 vote
1 answer
810 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, $K_P$ ...
Math's user avatar
  • 491
5 votes
1 answer
366 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\}...
Yonah Borns-Weil's user avatar
10 votes
2 answers
862 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 $A_h$ ...
Maxim Braverman's user avatar
3 votes
0 answers
212 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
12 votes
2 answers
1k 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 ...
john mangual's user avatar
  • 22.7k
0 votes
1 answer
163 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 ...
Rahul Raju Pattar's user avatar
11 votes
2 answers
435 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 ...
Matt Rosenzweig's user avatar
3 votes
1 answer
326 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 ...
F.M.R.'s user avatar
  • 43