Project: Hörmander-Weyl calculus in the framework of ultra distributions
This project concerns developing a calculus of Weyl operators, also feasible for objects which are more hard treated than usual. It is expected that the searched calculus is less sensitive for geometrical structures, compared to existing calculi.
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.