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Questions tagged [pseudo-differential-operators]

This tag is for questions regarding to the Pseudo-differential operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.

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Fréchet-valued symbols

Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
Ervin's user avatar
  • 395
2 votes
0 answers
55 views

About Fourier integral operators

Consider the operator $$(A_\Delta\psi)(p) = \int_\Delta \int_{\mathbb{R}^n} e^{ix\cdot(p-k) + i\phi(x,p,k)} a(x,p,k) \psi(k) dk\: dx$$ where $\Delta$ is a Borel or Lebegue set in $\mathbb{R}^n$, in ...
V. Moretti's user avatar
3 votes
0 answers
75 views

Dirichlet-to-Neumann map is analytic

Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
Eduardo Longa's user avatar
0 votes
1 answer
110 views

$L^p$ boundedness for pseudo-differential operators

Let $\rho, \delta, m$ be real parameters such that $0\le \delta\le \rho\le 1, \delta<1$. The set $S^m_{\rho, \delta}(\mathbb R^{2n})$ is defined as the set of smooth functions $a$ on $\mathbb R^n\...
Bazin's user avatar
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2 votes
1 answer
208 views

Leibniz rule for square root of Laplacian

Let $(M,g)$ be a compact Riemannian manifold (e.g. $M=S^3$ the 3-sphere) and let $\Delta$ be the metric Laplacian on $M$. Then $\Delta$ has an $L^2(M)$ basis of eigenfunctions $\pi_m$, $$ \Delta \pi_m ...
onamoonlessnight's user avatar
1 vote
1 answer
206 views

Differential operators in $\Bbb R^n$

Put $P_j=\frac{\partial}{\partial \xi_j}$ et $Q_j=2 i \xi_j$ with$\xi=\left(\xi_1, \ldots, \xi_n\right)$ et $x=\left(x_1, \ldots, x_n\right)$. How to prove : $\exp \left(\sum_{j=1}^n x_j P_j\right)(...
zoran  Vicovic's user avatar
1 vote
1 answer
165 views

Closed form of a Fourier transform

I apologize for not being able to motivate the question below; it would go into technicalities. Let $n=d+1\ge2$ be the space-time dimension, and $$H(y,t):=\left(\frac{t^2}{(t^2+|y|^2)^{1+d/2}}\right)^{...
Denis Serre's user avatar
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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
  • 1,077
0 votes
1 answer
136 views

Differential form of the multidimensional "orthogonal dilation" operator

For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion. ...
Kanghun Kim's user avatar
5 votes
1 answer
308 views

Real-analytic analogue of Schwartz functions

Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
Zislu R.'s user avatar
7 votes
0 answers
74 views

Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
MyShepherd's user avatar
2 votes
1 answer
296 views

Is it possible to define pseudodifferential operator $p(x,T)$ using Cauchy integral formula?

I was wondering how I can define a pseudodifferential operator using Cauchy integral formula. Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as: $$\...
Mirar's user avatar
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2 votes
0 answers
104 views

Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
Mirar's user avatar
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1 vote
0 answers
96 views

Kernel representation of a power of (pseudo-)differential operator

Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ...
Mirar's user avatar
  • 350
2 votes
1 answer
308 views

Bounded operator on $L^2(\Bbb R^2)$

Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$: $$ f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
zoran  Vicovic's user avatar
4 votes
1 answer
256 views

Conormal distributions and the wave front set

Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
Jesús A. Álvarez López's user avatar
2 votes
0 answers
69 views

Is there an easy explanation for the notion of Calderon projectors?

I recently stumbled upon the notion of Calderon projectors, which are standard tools for elliptic boundary value problems. However, when searching on the internet I could not really find any useful ...
B.Hueber's user avatar
  • 1,077
-1 votes
1 answer
87 views

Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$

I am having the following integral: $$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$ where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$, $...
Mr. Proof's user avatar
  • 159
3 votes
0 answers
125 views

Reasons behind different conventions for symbol of operator

I've come across two slightly different conventions for the symbol of a differential operator $D$ (let's say on $\mathbb{R}^n$) and haven't thought much about the motivation behind them until now. The ...
geometricK's user avatar
  • 1,901
3 votes
2 answers
326 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
3 votes
0 answers
296 views

Does convolution by a Schwartz function preserve symbol classes?

I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
Ervin's user avatar
  • 395
3 votes
1 answer
130 views

Positivity of an operator on a compact subset of a manifold

Let $E$ and $F$ be two vector bundles over manifold $X$. Let $P:\Gamma(E)\to \Gamma(F)$ be a self-adjoint differential operator over $X$. Define inner product on the spaces $\Gamma(E)$ of smooth ...
Radeha Longa's user avatar
3 votes
1 answer
147 views

Symbol estimates using metric on the phase space

Consider the symbols introduced by Beals and Fefferman in which the symbol estimates are of the form: \begin{equation}\tag{1} \label{eq1} |\partial_x^\beta\partial_\xi^\alpha a(x,\xi)| \leq C_{\...
Rahul Raju Pattar's user avatar
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
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
1 vote
0 answers
308 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
4 votes
0 answers
287 views

Sobolev spaces and spectral theorem

Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$ where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...
Rahul Raju Pattar's user avatar
1 vote
0 answers
59 views

Regularity and existence linear parabolic fractional equation

\begin{equation} \begin{cases} a(x, t)u_t+(-\Delta)^{\sigma}u+b(x,t)u=f(x,t), & \text{in } \mathbb{R}^n \times [0, T) \\ u(x,0)=u_0(x), & \text{in } \mathbb{R}^n \end{cases} \end{...
Mat's user avatar
  • 11
1 vote
1 answer
188 views

Metric / strong slope restriction of function on unit ball in $\mathbb R^m$

Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...
dohmatob's user avatar
  • 6,824
3 votes
0 answers
90 views

Semiclassical analysis and reflection law

I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
Ivan's user avatar
  • 445
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
1 vote
0 answers
72 views

Well-posedness for a wave equation with degenerate coefficients

Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem: \begin{equation}\label{pf0} \begin{aligned} \begin{cases} t\partial_t(t\partial_t u)-\...
Ali's user avatar
  • 4,113
4 votes
0 answers
163 views

Pseudodifferential Operators and Functional Calculus

I hope this is not too naive a question for MO. I've been taking a mathematical physics course, and was shown how operators like $\sqrt{1-\Delta}$ could be defined by taking multiplication operators ...
Wahome's user avatar
  • 737
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
1 answer
183 views

Pseudo-differential operators and differential operator

I am totally new to pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator. So, I want to show , using the definition of the symbol given by ...
NSR's user avatar
  • 97
1 vote
1 answer
141 views

Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$

How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
Pádua's user avatar
  • 69
2 votes
0 answers
157 views

Principal symbol of a non-local operator and Atiyah–Singer index formula

I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
IgnoranteX's user avatar
1 vote
1 answer
179 views

Reference for singular integral operators such as $(-\Delta)^{-1}$ or $\nabla(-\Delta)^{-1}$

I'm currently working on understanding certain mean-field limits in kinetic theory, and the equations I'm working with are usually of the form $$\partial_t f +v\cdot\nabla_x f \pm c\nabla(-\Delta)^{-1}...
Dominic Wynter's user avatar
4 votes
3 answers
629 views

Question about the regularity of fractional Heat equation

Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation: ‎$‎‎$‎ ‎\...
Hheepp's user avatar
  • 361
3 votes
0 answers
85 views

Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold

Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$. Suppose there is a compact subset $K\...
geometricK's user avatar
  • 1,901
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
1 vote
1 answer
143 views

Extension of pseudodifferential operators to Sobolev spaces

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and define \begin{eqnarray*} \mathcal{S}^m(\Omega ) &=&\left\{ \begin{array}{c} a\in C^\infty(\Omega \times \mathbb{R}^{d}): \hbox{for all ...
Math's user avatar
  • 491
1 vote
0 answers
54 views

The order of regularity improving for elliptic operator with rough coefficients

Let $\frac12<\alpha<1$ and let $L(x)$ be a first order overdetermined elliptic operator with coefficients in $C^\alpha$. We means $L(x)=\sum_{j=1}^nA_j(x)\partial_j$ where $A_j:\mathbb R^n\to\...
Liding Yao's user avatar
2 votes
1 answer
254 views

Metric on the phase space

I am studying PDEs whose symbols satisfy \begin{equation} |\partial^\alpha_\xi\partial^\beta_xp(x,\xi)| \lesssim M(x,\xi)\Psi(x,\xi)^{-|\alpha|}\Phi(x,\xi)^{-|\beta|} \end{equation} for all multi-...
Rahul Raju Pattar's user avatar
2 votes
1 answer
412 views

Question on definition of Dirichlet to Neumann operator

Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with a $C^2-$ boundary $\partial \Omega= \Gamma$. For $f \in H^{1/2}(\Gamma)$, let $F \in H^1(\Omega)$ denote the weak solution of the ...
kaithkolesidou's user avatar
11 votes
1 answer
1k views

Research topics in microlocal analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
Andrew's user avatar
  • 589
1 vote
0 answers
76 views

PDE on an open ball with prescribed value on some open subsets

Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
Overflowian's user avatar
  • 2,533
2 votes
2 answers
728 views

Self-adjoint extensions for pseudo-differential operators

The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that $$ \vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert ...
Bazin's user avatar
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3 votes
0 answers
80 views

Conformal manifolds produce Fredholm modules-pseudodifferential operator

This question is a continuation of the discussion which can be found here. From the exterior derivative one constructs an operator $S$ with the property that the graph of $S$ is the (closure of) the ...
truebaran's user avatar
  • 9,230
3 votes
0 answers
128 views

Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far: 1. There exists a density defined ...
truebaran's user avatar
  • 9,230