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.
57 questions with no upvoted or accepted answers
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409
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Between Being a Connection and Being an Elliptic Operator
Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
- 505
8
votes
0
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341
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Do pseudo-differential operators preserve smoothness without compact support assumption?
I've been reading Lawson's book, Spin Geometry recently. In this book, a pseudo-differential operator is defined as a linear map on Schwartz space $P\colon \mathcal{S} \longrightarrow \mathcal{S} $ by ...
- 525
8
votes
0
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278
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Pseudodifferential operators on compact manifolds with boundary
I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
- 81
7
votes
0
answers
80
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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 ...
- 808
6
votes
0
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162
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Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
5
votes
0
answers
143
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Extension of elliptic complex to an exact sequence
This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.
Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...
- 5,824
4
votes
1
answer
311
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?...
4
votes
0
answers
310
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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 ...
4
votes
0
answers
171
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 ...
- 737
4
votes
0
answers
256
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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 ...
- 873
4
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0
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46
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Spectrum of the hypoelliptic transverse signature operator
Let $D$ be the transverse signature operator constructed by Connes and Moscovici in the paper "Local index formula in Noncommutative Geometry":this is first order hypoelliptic pseudodifferential ...
- 9,330
4
votes
0
answers
2k
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Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
4
votes
0
answers
237
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Contact manifolds and pseudodifferential operators
By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...
- 41
3
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0
answers
82
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$,...
- 2,095
3
votes
0
answers
129
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 ...
- 1,903
3
votes
0
answers
320
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 ...
- 395
3
votes
0
answers
98
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 ...
- 445
3
votes
0
answers
91
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\...
- 1,903
3
votes
0
answers
81
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 ...
- 9,330
3
votes
0
answers
128
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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 ...
- 9,330
3
votes
0
answers
170
views
Examples of unbounded pseudo-differential operators in $L^{\infty}$
During my graduate studies I've been told that pseudo-differential operators with symbols in $S^0=S^0_{1,0}$ (the simplest class) are bounded $L^2 \to L^2$, and also $L^p \to L^p$ for all $p \in (1, \...
- 115
3
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0
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114
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Parametrix of external product of elliptic operators
Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
- 1,903
3
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0
answers
588
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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 ...
- 509
3
votes
0
answers
68
views
Pseudodifferential calculus for the Diffeomorphism Invariant Geometry
In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup ...
- 9,330
3
votes
0
answers
170
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Closure of pseudodifferential operators of order 0 on compact manifolds
Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...
- 2,998
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 ...
- 205
2
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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 ...
- 350
2
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0
answers
85
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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 ...
- 1,171
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 ...
- 371
2
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0
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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 ...
- 41
2
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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 ...
- 509
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 ...
- 503
2
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0
answers
240
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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 ...
- 53
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 ...
- 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 $\...
- 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 ...
- 63
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}\...
- 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 ...
- 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, ...
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{...
- 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)-\...
- 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...
- 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\...
- 938
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 ...
- 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 ...
- 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, ...
- 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 ...
- 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{...
- 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 $C^\infty$ ...
- 481
1
vote
0
answers
70
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 ...
- 4,135