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1
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0answers
31 views

Degenerate Carleman Estimate for Laplace Beltrami Operator

Suppose $(M,g)$ is a compact Riemannian manifold with boundary and that I have been able to prove a Carleman type estimate of the following form for an explicit phase function $\phi$ : $ \| e^ {\tau ...
6
votes
0answers
177 views

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 ...
1
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0answers
28 views

Equivalence of fractional power of second-order positive differential operator as pseudodifferential operator and a fractional definition

Let $A$ be a second-order differential operator on a closed manifold $M$ satisfying $$(Au,u) \geq 0$$ with $A=-\Delta$ the Laplace-Beltrami the model case. One can define for $s \in (0,1)$ the ...
2
votes
1answer
146 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 ...
1
vote
1answer
120 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 ...
6
votes
2answers
165 views

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 ...
1
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0answers
30 views

Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...
0
votes
0answers
50 views

regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies: \begin{equation*} b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...
2
votes
1answer
68 views

Compactness of Weyl pseudodifferential operators with integrable symbols

Given a tempered distribution $s \in \mathcal{S}'(\mathbb{R}^{2d})$, define the Weyl pseudodifferential operator of symbol $s$ as the mapping $\mathcal{S}(\mathbb{R}^{d}) \rightarrow ...
1
vote
1answer
259 views

Pseudo-differential evolution equation

I'm looking for results (or some ideas) on the following kind of pseudo-differential evolution equation: $$ \frac{\partial u(t,x)}{\partial t} = \int_{-\infty}^{t} B(t-s,x)\, A(x,D_{x})u(s,x)\,ds \; ...
4
votes
1answer
102 views

Abstract stationary phase

I have been reading Semi-classical analysis by Guillemin and Sternberg. At the end of Chapter 8, they gave an abstract version of the stationary phase method. I have a hard time figuring out what ...
0
votes
0answers
88 views

Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function. When I consider price of American ...
3
votes
1answer
171 views

Application of Egorov's Theorem for Pseudodifferential Operators

Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...
5
votes
1answer
110 views

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 ...
10
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1answer
278 views

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
votes
4answers
743 views

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 ...
0
votes
0answers
39 views

asking for some basic computation of pseudodifferential operator

i am reading Kashiwara's paper"Analyse microlocale du noyau de Bergman". In the page 9 he computed the following: suppose $h=\sum_{j=1}^{n}|z_j|^4$ and $\delta(f)$ is the dirac function $Y(f)$ is ...
2
votes
1answer
87 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 ...
2
votes
0answers
121 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 ...
3
votes
1answer
178 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. ( ...
8
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3answers
427 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 ...
3
votes
0answers
74 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 ...
2
votes
1answer
119 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 ...
1
vote
1answer
75 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 ...
4
votes
1answer
163 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 ...
3
votes
1answer
105 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 ...
20
votes
2answers
724 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
0answers
171 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} ...
10
votes
1answer
422 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. ...
5
votes
1answer
329 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 ...
2
votes
2answers
271 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 ...
4
votes
1answer
271 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 : $$ ...
11
votes
5answers
606 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 ...
12
votes
2answers
671 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 ...
3
votes
1answer
103 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 ...
4
votes
2answers
371 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 ...
9
votes
1answer
242 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 ...
3
votes
1answer
218 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 ...
3
votes
2answers
143 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 ...
1
vote
1answer
232 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
1answer
419 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 ...
8
votes
2answers
623 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 ...
1
vote
1answer
371 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 ...
7
votes
3answers
313 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 ...
25
votes
5answers
3k 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 ...
4
votes
2answers
183 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 ...
4
votes
2answers
771 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 ...
6
votes
1answer
666 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 ...
18
votes
6answers
3k 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 ...
7
votes
2answers
996 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 ...