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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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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
473 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
2 votes
1 answer
555 views

Resolvent of the Laplacian as a pseudodifferential operator and its single layer potential

In M.Taylor's book "Partial differential equations II" it is shown that the fundamental solution $E(x,y)$ of the Laplacian equation gives rise to an elliptic pseudodifferential operator $S$ on the ...
Appliqué's user avatar
  • 1,329
2 votes
1 answer
895 views

Generalized Friedrichs Lemma

Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...
Boaz Haberman's user avatar
2 votes
1 answer
309 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
2 votes
1 answer
499 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 ...
Gregory's user avatar
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2 votes
1 answer
301 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 $\...
JZS's user avatar
  • 481
2 votes
1 answer
178 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
user74880's user avatar
2 votes
1 answer
191 views

Paraproduct and Fourier series

I know that it's possible to define the paraproduct $T_a u$ when $a=a(x)\in L^{\infty}$ and $u\in H^s$ and in this case $T_a u\in H^s$. Remark: $T_a u$ is the pseudo differential operator with symbol ...
Phi's user avatar
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2 votes
0 answers
70 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
2 votes
0 answers
106 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
  • 350
2 votes
0 answers
85 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
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2 votes
0 answers
67 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
  • 371
2 votes
0 answers
159 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
2 votes
0 answers
315 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
  • 509
2 votes
1 answer
127 views

Positive form for a homogeneous elliptic pde

I have a pde of the following form: \begin{align} &P(x,D)u = f \text{ on } \Omega, \\ &P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha}, \end{align} where one can assume that $f$ ...
Fedor Goncharov's user avatar
2 votes
0 answers
40 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
2 votes
0 answers
240 views

Proof of Taylor's Schwartz kernel estimate of pseudodifferential operators

I am interested in the proof of the following result which gives an estimate on the Schwartz Kernel of a $\Psi$DO. There is one aspect the proof that is not clear to me which I would like to ask the ...
ClemensB's user avatar
2 votes
0 answers
188 views

Hypoelliptic pseudodifferential operators and Fredholm equations?

I read here the following: The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable ...
PtF's user avatar
  • 383
2 votes
0 answers
472 views

Looking for some "nontrivial" examples of pseudodifferential operators/symbols

I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial. Obviously, the first example of a symbol that most people talk about is just a polynomial in $\...
Patch's user avatar
  • 377
2 votes
0 answers
132 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow. The problem, if stated in as full generality as ...
Ukhrir's user avatar
  • 63
1 vote
1 answer
197 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
378 views

Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
Ali's user avatar
  • 4,135
1 vote
1 answer
207 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
157 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
1 vote
1 answer
1k views

Schwartz kernel

By the Schwartz Kernel Theorem, we know that the kernel of an integral operator belongs to distribution space $\mathscr{S}'(\mathbb{R}^n)$. Moreover, we know that the kernel $K$ is $C^{\infty}$ off ...
user80489's user avatar
1 vote
1 answer
195 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
  • 6,853
1 vote
1 answer
184 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 ...
Henry.L's user avatar
  • 8,071
1 vote
1 answer
209 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 ...
Appliqué's user avatar
  • 1,329
1 vote
1 answer
338 views

On the generalization of the Mittag-Leffler function and fractional derivative

The Mittag-Leffler function $E_{\alpha}(x)$ has an important property: $$ \frac{\partial^{\alpha}}{\partial x^{\alpha}} E_{\alpha}(x^{\alpha}) = E_{\alpha}(x^{\alpha}). $$ I tried to find an ...
Appliqué's user avatar
  • 1,329
1 vote
1 answer
466 views

Elliptic pseudodifferential operator estimate [closed]

If $P$ is an elliptic pseudodifferential operator of order 1 in the sense that its principal symbol is invertible, then we have the a priori estimate $\|u\|_{H^1(U)} \le C (\|Pu\|_{L^2(W)} + \|u\|_{...
flavio's user avatar
  • 450
1 vote
1 answer
183 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
  • 52.3k
1 vote
1 answer
193 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
1 vote
1 answer
852 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
  • 509
1 vote
1 answer
153 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
  • 509
1 vote
1 answer
173 views

Inverse of holomorphic elliptic differential operator

Consider the Beltrami-Laplacian $\Delta$ on $\mathbb{S}^n$ with standard metric. One can define a family of operators $A(z):H^1(\mathbb{S}^n)\to H^1(\mathbb{S}^n)$ as the following $$A(z)=\Delta+z$$ ...
Slm2004's user avatar
  • 633
1 vote
1 answer
178 views

Expressing a matrix operator as a sum of an identity and a compact operator

My problem concerns with the unique solvability of a linear system of integral equations. In my problem, as I was able to write the system in matrix form: $$ M \begin{align} \begin{bmatrix} ...
Julienne Franz's user avatar
1 vote
0 answers
97 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
  • 1,171
1 vote
0 answers
104 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
1 vote
0 answers
323 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
67 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
0 answers
74 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
  • 4,135
1 vote
0 answers
312 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
55 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
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
1 vote
0 answers
59 views

Bound of analytic torsion for a line bundle

Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...
Bombyx mori's user avatar
  • 6,249
1 vote
0 answers
260 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
  • 377
1 vote
0 answers
180 views

Implicit function theorem for operators

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
gipom's user avatar
  • 115
1 vote
0 answers
61 views

Convergence of PDE/PsiDE - expansion of pseudo-differential operators

I have am working with a nonlinear pseudo-differential evolution equation of the form $$u_t + \mathcal{N}(u) + \mathcal{D}_{\epsilon} u = 0$$ where $\mathcal{N}$ is a nonlinear operator and $\mathcal{...
Frubiclé's user avatar
  • 155
1 vote
0 answers
108 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
  • 481