Projektdetails
Abstract
Wider research context
This project deals with questions of ultradifferentiable (u.d.) regularity of PDEs, using mainly microlocal tools. In 1971 Hörmander defined the u.d. wavefront set with respect to Denjoy-Carleman classes, that is u.d. classes given by weight sequences, which has applications regarding ud regularity of PDEs by different authors since then, generalizing results from the smooth and analytic category to the u.d. setting. The microlocal theory of Hörmander has also been extended to other u.d. classes, like classes given by weight functions. However, if we consider more advanced tools of microlocal analysis like pseudodifferential operators or Fourier integral operators (FIOs), then we see that there are some works on u.d. pseudodifferential operators but only for Gevrey classes there is a comprehensive theory which is analogous to the theory in the smooth or analytic category. In particular there is no u.d. pseudodifferential calculus which is geometric, i.e. u.d. pseudodifferential operators are well-defined on u.d. manifolds of the same class.
Objectives
The main goal of the project is to develop a geometric theory of pseudodifferential operators and FIOs with respect to large classes of u.d. classes including quasianalytic classes. Microlocal means that the u.d. pseudodifferential operators decrease the u.d. wavefront set. Subsequently we apply these and other tools of microlocal analysis to a variety of regularity problems in PDE theory.
These problems include the problem of iterates and u.d. hypoellipticity of PDOs. Another problem considered is the regularity problem of CR maps. Here we generalize the smooth regularity problem for CR maps given by Lamel-Mir to the u.d. category.
Regarding u.d. hypoellipticity of PDOs our main application is to extend the characterization of smooth (resp. analytic) hypoellipticity of operators of principal type given by Treves to the u.d. setting.
Methods
We use the usual techniques and methods from microlocal and harmonic analysis, like Fourier transform, cutoff functions etc. additonal to the tools like pseudodifferential operators and FIOs in the u.d. category which we will develop during the project. The most important tool are almost analytic functions, which have been used in microlocal analysis before, both in the smooth and u.d. category. Moreover, the theory of u.d. functions, which has been developed in last few years, will be applied throughout the project. More precisely we work in the setting of u.d. functions given by weight matrices, i.e. families of weight sequences. In particular, these classes include Denjoy-Carleman classes and classes given by weight functions.
Innovation
This is the first time that u.d. pseudodifferential operators and u.d. FIOs with respect to general quasianalytic classes are considered.
Prinicipal Researcher involved
Stefan Fürdös
This project deals with questions of ultradifferentiable (u.d.) regularity of PDEs, using mainly microlocal tools. In 1971 Hörmander defined the u.d. wavefront set with respect to Denjoy-Carleman classes, that is u.d. classes given by weight sequences, which has applications regarding ud regularity of PDEs by different authors since then, generalizing results from the smooth and analytic category to the u.d. setting. The microlocal theory of Hörmander has also been extended to other u.d. classes, like classes given by weight functions. However, if we consider more advanced tools of microlocal analysis like pseudodifferential operators or Fourier integral operators (FIOs), then we see that there are some works on u.d. pseudodifferential operators but only for Gevrey classes there is a comprehensive theory which is analogous to the theory in the smooth or analytic category. In particular there is no u.d. pseudodifferential calculus which is geometric, i.e. u.d. pseudodifferential operators are well-defined on u.d. manifolds of the same class.
Objectives
The main goal of the project is to develop a geometric theory of pseudodifferential operators and FIOs with respect to large classes of u.d. classes including quasianalytic classes. Microlocal means that the u.d. pseudodifferential operators decrease the u.d. wavefront set. Subsequently we apply these and other tools of microlocal analysis to a variety of regularity problems in PDE theory.
These problems include the problem of iterates and u.d. hypoellipticity of PDOs. Another problem considered is the regularity problem of CR maps. Here we generalize the smooth regularity problem for CR maps given by Lamel-Mir to the u.d. category.
Regarding u.d. hypoellipticity of PDOs our main application is to extend the characterization of smooth (resp. analytic) hypoellipticity of operators of principal type given by Treves to the u.d. setting.
Methods
We use the usual techniques and methods from microlocal and harmonic analysis, like Fourier transform, cutoff functions etc. additonal to the tools like pseudodifferential operators and FIOs in the u.d. category which we will develop during the project. The most important tool are almost analytic functions, which have been used in microlocal analysis before, both in the smooth and u.d. category. Moreover, the theory of u.d. functions, which has been developed in last few years, will be applied throughout the project. More precisely we work in the setting of u.d. functions given by weight matrices, i.e. families of weight sequences. In particular, these classes include Denjoy-Carleman classes and classes given by weight functions.
Innovation
This is the first time that u.d. pseudodifferential operators and u.d. FIOs with respect to general quasianalytic classes are considered.
Prinicipal Researcher involved
Stefan Fürdös
| Kurztitel | Ultradifferenzierb. Regularität von PDEs |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/12/24 → 30/11/28 |