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
42
votes
5
answers
6k
views
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 ...
- 2,239
3
votes
2
answers
536
views
pseutotensor category
In the paper B. Bakalov, A.D'Andrea and V.G.Kac, Theory of finite pseudoalgebras, section 3, one finds the following definition of pseudotensor category:
A pseudotensor category is a class of ...
- 3,357
7
votes
1
answer
870
views
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 ...
CommunityBot
- 1
3
votes
1
answer
480
views
about smoothing pseudodifferential operators
Hi,
I have a question which involves pdo.
Let us consider a pseudodifferential operator $A:S(\mathbb{R}^d)\rightarrow S(\mathbb{R}^d) $ whose symbol $a(x,\xi)$ lives in the $S_{0,0}^0$ class :
$$ \...
- 16.2k
11
votes
5
answers
1k
views
What are the invariant Pseudo-differential operators on a Lie group?
It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group.
On the other ...
CommunityBot
- 1
11
votes
2
answers
1k
views
what's the motivation of Weyl calculus ?
In the pseudo-differential operator theory, we can define a pseudo-differential operator by $$a(x,D)u=(2\pi)^{-n}\int{a(x,\xi)e^{i\langle x-y,\xi \rangle}u(y)dyd\xi}$$ with $a(x,\xi)$ belong to some ...
CommunityBot
- 1
3
votes
1
answer
166
views
Analogue of the integral Fourier operator with angle in some cone
Let $\Phi(x,y,\theta)$ be a phase function defined on $X \times X \times (\mathbb R^n-0)$ where $X$ is some domain in $\mathbb R^n$, let $A(x,y,\theta)$ be an amplitude function. As usually an ...
- 16.2k
12
votes
1
answer
687
views
Trace formula for PSDOs
In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian ...
CommunityBot
- 1
3
votes
2
answers
221
views
a question about first-order hyperbolic equations
Performing certain manipulations on pseudo-differential equations I have come across the following first order equation:
$$
D_{t}u-\lambda(t,x,D_{t},D_{x})u=0, \ \ (*)
$$
where $\lambda$ is a scalar ...
- 27.5k
1
vote
1
answer
338
views
On the generalization of the Mittag-Leffler function and fractional derivative
The Mittag-Leffler function $E_{\alpha}(x)$ has an important property:
$$
\frac{\partial^{\alpha}}{\partial x^{\alpha}} E_{\alpha}(x^{\alpha}) = E_{\alpha}(x^{\alpha}).
$$
I tried to find an ...
4
votes
2
answers
248
views
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 ...
CommunityBot
- 1
3
votes
1
answer
678
views
Is this kernel space of finite dimension ?
Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
- 16.4k
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 ...
- 16.2k
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\|_{...
- 2,935
10
votes
0
answers
409
views
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
9
votes
2
answers
1k
views
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 ...
- 29.2k