# Pseudo-differential operators

Pseudo-differential operators on ${\mathbb R}^n$ are operators of the form

$Tf(x)=\int_{{\mathbb R}^n} e^{2\pi i x\cdot\xi} a(x,\xi) \widehat{f}(\xi) d\xi, \ \ \ \ \ \ \ (1)$

where $\widehat{f}(\xi)=\int_{{\mathbb R}^n} e^{-2\pi i x\cdot\xi} f(x) dx$ is the Fourier transform of $f$ and $a(x,\xi)$ is called the symbol of $T.$
It may look complicated but in fact, roughly,

every linear continuous operator taking functions to functions is a pseudo-differential operator!

This is very easy to see, even rigorously, in the case of the torus ${\mathbb T}^n={\mathbb R}^n/{\mathbb Z}^n.$ For a smooth function $f\in C^\infty({\mathbb T}^n)$ its Fourier coefficients are defined by

$\widehat{f}(k):=\int_{{\mathbb T}^n} e^{-2\pi i x\cdot\xi} f(x) dx, \ \ \ k\in{\mathbb Z}^n. \ \ \ \ (2)$

The Fourier inversion formula then gives

$f(x)=\sum_{k\in{\mathbb Z}^n} e^{2\pi i x\cdot k} \widehat{f}(k). \ \ \ \ \ \ \ (3)$

Let now $T:C^\infty({\mathbb T}^n)\to C^\infty({\mathbb T}^n)$ be a linear continuous operator, and let us define its so-called toroidal symbol by

$a(x,k):=e^{-2\pi i x\cdot k}T(e^{2\pi i x\cdot k}). \ \ \ \ (4)$

Then using (3) and the fact that $T$ is linear and continuous we have

$Tf(x)=T(\sum_{k\in{\mathbb Z}^n} e^{2\pi i x\cdot k} \widehat{f}(k)) \ \ \ \$

$\ \ \ \ \ \ \ \ \ =\sum_{k\in{\mathbb Z}^n} T(e^{2\pi i x\cdot k}) \widehat{f}(k) \ \ \ \$

$\ \ \ \ \ \ \ \ \ =\sum_{k\in{\mathbb Z}^n} e^{2\pi i x\cdot k} a(x,k) \widehat{f}(k), \ \ \ \$

where we used (4) in the last line. Thus, we have shown an analogue of (1) on the torus:

$Tf(x)=\sum_{k\in{\mathbb Z}^n} e^{2\pi i x\cdot k} a(x,k) \widehat{f}(k), \ \ \ \ \ \ \ (5)$

which is called the toroidal quantization on the torus ${\mathbb T}^n.$

A more or less comprehensive analysis of this toroidal quantization, its properties and relations to (1)  can be read here:

Ruzhansky M., Turunen V., Quantization of pseudo-differential operators on the torus, J. Fourier Anal. Appl., 16 (2010), 943-982download from arxiv, link

You can also read a bit more on pseudo-differential operators at nLab or in Wikipedia.