## Project information

**Project manager**

Joachim Toft

**Other project members**

Albin Petersson

**Financier**

The Swedish Research Council (Vetenskapsrådet)

**Timetable**

1 Jan 2020–31 Dec 2023

**Subject**

Mathematics (Department of Mathematics, Faculty of Technology)

## More about the project

The purpose of the project is to develop the Hörmander-Weyl Calculus (HWC) in the theory of pseudo-differential operators into a Gevrey and Gelfand-Shilov framework, called Gevrey-HWC. The involved weights, metrics and symbols in Gevrey-HWC are allowed to grow or decay faster than polynomially (i.e. sub-exponentially, exponentially or super-exponentially), but must obey stronger regularity (of Gevrey type), than the classical situation where growth and decay is restricted to be polynomial.

Especially we intend to:

- Extend the notion of metric confinement of symbols to Gevrey-HWC.
- Deduce integral localization in Gevrey-HWC.
- Apply integral localization on operators in Gevrey-HWC and use confinement of symbols to deduce continuity and algebraic properties in the calculus.
- Solve symbol-valued evolution equations to deduce existence of inverse pairs in the calculus. This can be used to simplify the weights in several problems.
- Define Hilbert spaces of functions and ultra-distributions adapted to the calculus, which corresponds to Sobolev spaces in the classical pseudo-differential calculus.

It is expected that the obtained results will give new tools for the study of pseudo-differential operators. The heat equation has a Gaussian fundamental solution. Questions on propagation of singularities and well-posedness is therefore naturally discussed in the framework of Gevrey and Gelfand-Shilov regularity which concerns growth and decay beyond polynomial.