# 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.

#### Pseudo-differential operators on compact Lie groups

Ruzhansky M., Turunen V., Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhauser, Basel, 2010. 724pp. Contents, description and samples, this book and time-frequency analysis, summary at Birkhauser review Bookmetrix

#### Pseudo-differential operators on graded Lie groups

Fischer V., Ruzhansky M., Quantization on nilpotent Lie groups, Progress in Mathematics, Vol. 314, Birkhauser, 2016. xiii+557pp. linkdownload (open access book, first by Imperial College London) Bookmetrix the winner of  Ferran Sunyer I Balaguer Prize 2014

#### Pseudo-differential operators on locally compact Type I groups

Mantoiu M., Ruzhansky M., Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups, Doc. Math., 22 (2017), 1539-1592. offprint (open access), link, arxiv

#### Pseudo-differential operators on nilpotent Lie groups with flat orbits

Mantoiu M., Ruzhansky M., Quantizations on nilpotent Lie groups and algebras having flat coadjoint orbits. J. Geom. Anal., 29 (2019), 2823-2861. arxiv, link

#### Nonharmonic analysis of pseudo-differential operators

1. Ruzhansky M., Tokmagambetov N., Nonharmonic analysis of boundary value problems, Int. Math. Res. Notices, (2016) 2016 (12), 3548-3615. offprint (open access), arxiv, link
2. Ruzhansky M., Tokmagambetov N., Nonharmonic analysis of boundary value problems without WZ condition, Math. Model. Nat. Phenom., 12 (2017), 115-140. arxiv, link
3. Delgado J., Ruzhansky M., Tokmagambetov N., Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary, J. Math. Pures Appl.,107 (2017), 758-783. offprint (open access), arxiv, link
4. Cardona D., Kumar V., Ruzhansky M., Tokmagambetov N., LpLq boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations. arxiv

#### Pseudo-differential operators on the lattice

Botchway L., Kibiti G., Ruzhansky M., Difference equations and pseudo-differential operators on Zn, J. Funct. Anal.,278 (2020), no. 11, 108473, 41pp. link (open access)arxiv

In the above paper one makes a consistent development of the calculus of pseudo-differential operators which can be called pseudo-difference operators in the context of the lattice ${\mathbb Z}^n$ Some interesting questions to answer:

• how to define Hörmander type or other symbol classes on ${\mathbb Z}^n$?
• what is the relation to the analysis on the torus ${\mathbb T}^n$?
• can it be used to derive new properties of operators on ${\mathbb Z}^n$?
• how to apply it to solving difference equations and to finding properties of their solutions?

#### Pseudo-differential operators on locally compact and quantum groups

1. Akylzhanov R., Ruzhansky M., Lp-Lq multipliers on locally compact groups, J. Funct. Anal., 278 (2020), no. 3, 108324, 49pp. link (open access)arxiv
2. Akylzhanov R., Majid S., Ruzhansky M., Smooth dense subalgebras and Fourier multipliers on compact quantum groups, Comm. Math. Phys., 362 (2018), 761-799.

#### Pseudo-differential operators with nonlinear quantizing functions

Esposito M., Ruzhansky M., Pseudo-differential operators with nonlinear quantizing  functions, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 103-130. arxiv, link