# Tagged Questions

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### 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 ...

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83 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. ( ...

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115 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 ...

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569 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 ...

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334 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. ...

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282 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 ...

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296 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 ...

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282 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 ...

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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 ...

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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 ...