# Questions tagged [pseudo-differential-operators]

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### Pseudo-differential operators and differetial operator

Hello I am totally new to Pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator. So, I want to show , using the definition of the symbol given ...
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### Derivative of a pseudo-differential operator [closed]

I want to show that $D^{\alpha}_x a(x,D)u(x) = (2n)^{-n}\langle \tilde{u},D^{\alpha}_x (e^{ix\xi}a(x,\xi))\rangle$, where $a(x,D)$ is a pseudo-differential operator, and to compute the derivative I'm ...
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### Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$

How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
108 views

### Principal symbol of a non-local operator and Atiyah–Singer index formula

I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
113 views

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### Conformal manifolds produce Fredholm modules-pseudodifferential operator

This question is a continuation of the discussion which can be found here. From the exterior derivative one constructs an operator $S$ with the property that the graph of $S$ is the (closure of) the ...
102 views

### Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far: 1. There exists a density defined ...
87 views

### Associating a pseudo-differential operator to the symbol in the SG setting

We all know that given a symbol $a(x,\xi) \in S^{\mu,\rho}(\mathbb{R}^n,\mathbb{R}^n)$, a pseudo-differential operator can be defined as \begin{equation} Op(a)u(x)=(2\pi)^{-n}\int \int e^{i(x-x')\cdot ...
This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator. Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...