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.
166 questions
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Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
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5
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Why is symplectic geometry so important in modern PDE ?
First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds ...
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6
answers
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Square roots of the Laplace operator
In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...
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Applications of Atiyah-Singer using pseudodifferential operators
Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
- 2,998
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4
answers
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Green's operator of elliptic differential operator
Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
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14
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4
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Is every continuous microlocal operator a pseudo-differential operator?
Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave ...
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14
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2
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Applications of pseudodifferential operators to PDE
I am planning to build a PDE course centred around pseudodifferential operators. I know some important applications of pseudodifferential operators to PDEs, but I don't know enough to get the whole ...
13
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1
answer
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Atiyah-Singer for pseudodifferential operators via heat kernel?
The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
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1
answer
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applications of C$^*$-algebras in the field of PDEs
I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...
12
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1
answer
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Trace formula for PSDOs
In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian ...
- 9,853
12
votes
1
answer
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Does hypoellipticity imply the existence of a parametrix?
Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...
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2
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Why take 'complex powers' of pseudo-differential operators?
Given a pseudo-differential operator $P$ of order zero, Seeley showed that the holomorphic family of operators $\lbrace P^{z} : z\in \mathbb{C} \rbrace$ of all complex powers is contained in the ...
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4
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Is every non-negative test function the limit of a sequence of sums of squares of test functions?
Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-...
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How to define the square root of $1-\Delta $?
If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?
The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...
- 1,441
11
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1
answer
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Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
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2
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what's the motivation of Weyl calculus ?
In the pseudo-differential operator theory, we can define a pseudo-differential operator by $$a(x,D)u=(2\pi)^{-n}\int{a(x,\xi)e^{i\langle x-y,\xi \rangle}u(y)dyd\xi}$$ with $a(x,\xi)$ belong to some ...
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5
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What are the invariant Pseudo-differential operators on a Lie group?
It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group.
On the other ...
11
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1
answer
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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 ...
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3
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Characterization of inverse differential operators
If I have a partial differential operator $p(D)$, where $p$ is a polynomial with constant coefficients and $D$ is the derivative in Euclidean space. Its inverse is easily described in Fourier space: $\...
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10
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0
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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 ...
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3
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Functions of pseudodifferential operators
Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can ...
- 9,853
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Understanding the analytic index map of the Atiyah-Singer index theorem
Hi,
I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn.
I do not understand why the analytic index map ...
- 2,998
9
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2
answers
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Pseudodifferential operators on spaces with boundary
Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann ...
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3
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Characteristic Variety of the Principal Symbol solves PDE system?
In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, http://www.sciencedirect.com/science/...
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1
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Pseudo-differential operators with compactly supported symbols
If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO $...
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2
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when a pseudo-differential operators to be compact?
In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...
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8
votes
2
answers
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(sharp)Garding's inequality and inequality with lower bounds
The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part $$P_{2m}...
8
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1
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K-homology classes of Dirac operators on Hermitian manifolds
Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely
1) (d + d$^*,\Omega^{*})$
2) ($\partial$ + $\...
- 1,453
8
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0
answers
341
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Do pseudo-differential operators preserve smoothness without compact support assumption?
I've been reading Lawson's book, Spin Geometry recently. In this book, a pseudo-differential operator is defined as a linear map on Schwartz space $P\colon \mathcal{S} \longrightarrow \mathcal{S} $ by ...
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0
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Pseudodifferential operators on compact manifolds with boundary
I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
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6
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Fractional Leibniz formula
Let $T=(-\Delta)^{1/2}$.
Can we have estimates, similar to the one below
$$
\| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1}g\|_p,
$$
hold in $L^p$, where $...
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2
answers
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Do pseudo-differential operators form a sheaf of algebras?
Let $M$ be a smooth manifold.
I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on ...
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2
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When is a Pseudo-differential operator trace class or in Dixmier ideal?
Let's denote the set of all Pseudo-differential operators with symbol of “order” $d$ by $\Psi_d(M)$ and Sobolev space on $M$ by $H_s(M)$. It is known that
If $P\in\Psi_d(M)$ Then $P$ extends to a ...
7
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3
answers
627
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Criteria for Positivity of Pseudoddifferential Operators on Manifolds
Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L_0^2$ be the subspace which is orthogonal to the constant functions. When is ...
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1
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Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold
I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
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1
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Pseudo-differential operators which are independent of lower order perturbations
In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real ...
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0
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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 ...
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6
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2
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Differential structures and K-homology groups
What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...
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1
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What is a good reference for conormal distributions?
May I humblely ask what is a good reference for conormal distributions (for student with some rudimentary pseudo-differential operator background)? I heard from my advisor that it is useful in index ...
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6
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1
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The elliptic regularity theorem for differential operators with variable coefficients
I'm following the book "Introduction to the theory of distributions" by Friedlander and Joshi. There is the following result p. 109
Theorem (8.6.1). Let $X \subset \mathbb{R}^n$ be an open set, and ...
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0
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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
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1
answer
449
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Practical way to check whether a distribution is conormal
Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 \...
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5
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1
answer
355
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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 ...
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1
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$L^p$-estimates for elliptic pseudodifferential operators
Assume we have an pseudodifferential operator
$P:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S}(\mathbb{R}^n), Pf(x) = (2\pi)^{-n/2}\int\mathrm{d}\xi\; p(x,\xi)\,\hat{f}(\xi)e^{i\xi x}$
acting on ...
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1
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Is this a pseudodifferential operator?
Let $M$ be a non-compact manifold and $D$ a first-order self-adjoint elliptic differential operator on $M$. Then is the bounded operator
$$A:=\sqrt{(D^2+1)^{-1}}:L^2(M)\rightarrow H^1(M)$$
a ...
- 1,903
5
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1
answer
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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 ...
5
votes
1
answer
270
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Positivity of semiclassical pseudodifferential operators
Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis)
Let $m\in\mathbb{R}$, and $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\...
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5
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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 ...
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4
votes
2
answers
248
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second order operator with real coefficients and not locally solvable
This question is a follow up of the following answer I posted recently:
Counterexamples in PDE
Is there a second order partial differential operator with real coefficients which are not solvable in ...
- 2,239
4
votes
3
answers
644
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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:
$$
\...
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