# Questions tagged [pseudo-differential-operators]

This tag is for questions regarding to pseudo-differential operators, which are generalizations of differential operators and Fourier multipliers. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.

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### On class of Hörmander symbols. [duplicate]

In An Introduction to Pseudo-Differential Operators by Wong, the Hörmander's class is defined as every function $\sigma(x,\xi)\in\mathcal{C}^{\infty}(\mathbb{R}^n\times \mathbb{R}^n)$ such that for ...
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### Lawson's Spin Geometry book - pieces that do not fit

This is from Lawson's book "Spin Geometry", there is a problem here, there are pieces that do not fit. In (4.3) and (4.4) shows how to construct the operator Q from P and its symbol. For ...
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### Reference needed: Symbol sequence for pseudodifferential operators

In Higsons's book Analytic K-Homology there is a section (subsection (b) in 2.8 "Geometric Examples of Extensions", starting from page 46) which discusses the following exact sequence called ...
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### Good books and lecture notes to learn pseudo-differential operators and spectral theory

I am looking for a list of good books and lecture notes to learn pseudo-differential operators and spectral theory (for infinite dimensions.) I am familiar with introductory functional analysis, ...
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### Extending symbol of pseudodifferential operator defined outside of a ball

While reading Wong's book "An introduction to pseudo-differential operators" (3rd edition), I came across the following statement in the exercises : Let $\sigma \in C^\infty(R^n\times R^n)$...
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### Pseudodifferential operator properly supported

Be $P$ a pseudodifferential operator in $\Omega$, it have properly supported when both $P$ and $P^*$ have the for each compact $K \subset \Omega$ there is a compact $K^′ \subset \Omega$ such that ...
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### What is a Symbol (as in "Symbol Calculus")?

I have been reading through several papers on Deformation Quantization and the terms "symbol" and "symbol calculus" keep cropping up. I am somewhat well acquainted with most of the ...
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### Product with a smooth function, Sobolev regularity and pseudo-differential operator

I am working on microlocal Sobolev spaces using Hörmander's books, and while trying to write down the details of a proof, I have faced the following problem. It looks easy, but I can't find a correct ...
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### Comparison of quantization procedures

Let $M$ be a compact manifold and let $U\subset M$ be an open set with a coordinate chart $\Phi:U\to V$ with $V\subset \mathbb{R}^d$. Suppose that $w\in C_c^\infty(U)$ is a smooth function supported ...
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### Distribution Theory

I am using http://www.math.mcgill.ca/gantumur/math581w12/downloads/pseudodiff.pdf as a self study on Pseudo-Differential Operators and Distribution Theory. In example 3 on page 6 the following ...
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Suppose I have a pseudo-differential operator on $\mathbb R$ whose symbol is $a(x,\xi) = e^{-\xi^2/2}$ (notice, no dependence on $x$) Question 1: Are the following statements correct? $a$ is an ...