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7
votes
3answers
323 views

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
vote
0answers
103 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 ...
1
vote
0answers
82 views

Extension of pseudodifferential operators

I'm very sorry if this is the wrong place to ask this question, but I've asked it on StackExchange and received no answers. ( ...
2
votes
0answers
47 views

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 ...
1
vote
0answers
62 views

Elliptic Fourier integral operators

I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point. But ...
0
votes
1answer
55 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 ...
2
votes
1answer
113 views

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 ...
2
votes
1answer
80 views

Support-preserving pseudodifferential operators

Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the ...
18
votes
2answers
560 views

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
votes
6answers
593 views

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 ...
0
votes
0answers
24 views

Vanishing of non commutative ( Wodzicki) residue on pseudo differential projections

Its a known fact that the non-commutative (Wodzicki) residue of a pseudo-differential projection is always zero. My question is: Is it possible to get this result by looking at structure of the ...
1
vote
0answers
89 views

Schwartz kernel of a pseudodifferential operator with singular symbols

If we have a smooth symbol $r(x,\xi)$ of order $d$ the corresponding pseudodifferential operator $P$ is integral operator with kernel given by $$K(x,y)=\int_{\mathbb{R}^n}e^{i(x-y)\cdot \xi} ...
9
votes
1answer
330 views

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. ...
7
votes
2answers
515 views

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 ...
4
votes
1answer
280 views

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 ...
6
votes
2answers
715 views

(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 ...
25
votes
8answers
3k views

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 ...
5
votes
3answers
281 views

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 ...
10
votes
2answers
480 views

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 ...
23
votes
5answers
2k 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
votes
2answers
250 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 ...
5
votes
1answer
606 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 ...
3
votes
1answer
201 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 : $$ ...
10
votes
5answers
473 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 ...
8
votes
2answers
662 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 ...
2
votes
1answer
87 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 ...
3
votes
2answers
291 views

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 ...
6
votes
1answer
205 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 ...
2
votes
0answers
130 views

Exponential decay for the gradient of a solution

Dear all, I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$: $$ \sqrt{-\Delta+m^2} u +u= f(u), $$ where I can assume that $m \neq ...
2
votes
2answers
126 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 ...
0
votes
1answer
204 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 ...
3
votes
2answers
174 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 ...
12
votes
6answers
2k views

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 ...
2
votes
1answer
360 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 ...
0
votes
1answer
279 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 ...
5
votes
2answers
1k views

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: ...
3
votes
2answers
635 views

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 ...
7
votes
3answers
539 views

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 ...
1
vote
1answer
363 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)} + ...
4
votes
0answers
258 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 ...
4
votes
2answers
647 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 ...