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13
votes
0answers
186 views

What is the significance of Hochschild homology of pseudo-differential operators?

I want to ask what is the motivation for people to compute the Hochschild homology for the algebra of pseudo-differential operators on a given manifold. While surveying the literature, it seems much ...
9
votes
0answers
292 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 ...
7
votes
0answers
201 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 ...
5
votes
0answers
55 views

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 ...
3
votes
0answers
80 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
0answers
55 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 ...
2
votes
0answers
123 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 ...
2
votes
0answers
204 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} ...
1
vote
0answers
79 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 ...
1
vote
0answers
44 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 ...
1
vote
0answers
39 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 $m$: ...