Periodic Reporting for period 1 - LieLowerBounds (Lower bounds for partial differential operators on compact Lie groups)
Reporting period: 2019-06-01 to 2021-05-31
The problem we addressed in this project, which is of theoretical nature, concerns the validity of some fundamental lower bounds and other a priori estimates for pseudo-differential operators on Lie groups primarily of compact type.
Lower bounds are inequalities involving pseudo-differential operators with suitable properties. In the Euclidean setting such inequalities have been proved to be powerful tools to study a wide range of problems arising in mathematical analysis, especially problems related to partial differential equation.
Some problems to which the aforementioned bounds apply are, for instance, the study of hypoellipticity, solvability and well-posedness of initial value problems for evolution equations.
In the manifold setting not much is known about the validity of these inequalities for non-elliptic operators, therefore the understanding of the questions listed above is much limited with respect to the Euclidean case.
More in detail, we are interested in the application of a priori estimate in the resolution of problems concerning degenerate operators on compact Lie groups, in particular time-degenerate Schrödinger operators. These equations have recently attracted the attention of the mathematicians and physicists and they naturally appear in the study of Bose-Einstein condensations and nonlinear optics.
Among manifolds Lie groups are certainly of high importance, not only because they have nice geometric properties, but also because we encounter such structures in many physical situations.
Our objective here is to derive some unknown fundamental lower bounds and other inequalities for pseudo-differential operators of non-elliptic type in the Lie group setting. The final scope is to use these results to solve solvability and well-posedness problems for degenerate partial-differential operators.
Lower bounds are inequalities involving pseudo-differential operators with suitable properties. In the Euclidean setting such inequalities have been proved to be powerful tools to study a wide range of problems arising in mathematical analysis, especially problems related to partial differential equation.
Some problems to which the aforementioned bounds apply are, for instance, the study of hypoellipticity, solvability and well-posedness of initial value problems for evolution equations.
In the manifold setting not much is known about the validity of these inequalities for non-elliptic operators, therefore the understanding of the questions listed above is much limited with respect to the Euclidean case.
More in detail, we are interested in the application of a priori estimate in the resolution of problems concerning degenerate operators on compact Lie groups, in particular time-degenerate Schrödinger operators. These equations have recently attracted the attention of the mathematicians and physicists and they naturally appear in the study of Bose-Einstein condensations and nonlinear optics.
Among manifolds Lie groups are certainly of high importance, not only because they have nice geometric properties, but also because we encounter such structures in many physical situations.
Our objective here is to derive some unknown fundamental lower bounds and other inequalities for pseudo-differential operators of non-elliptic type in the Lie group setting. The final scope is to use these results to solve solvability and well-posedness problems for degenerate partial-differential operators.
Our work was divided into three main work packages (WP).
In WP1 we performed a detailed analysis of the geometric properties of pseudo-differential operators on Lie groups of compact type. A global Weyl quantisation for these operators is a far reaching result due to the geometry of the objects under study, therefore we focused our attention on alternative approaches to overcome the need of the Weyl calculus.
To start with, we considered the model case of the torus. We studied and defined some transformations, specifically the Bargmann transform, to approach in a different way the problem of the validity of lower bounds.
We derived properties similar to the Euclidean Bargmann transform, transform used in the Euclidean setting to derive, for instance, the Fefferman-Phong inequality.
We have considered and answered similar questions on the dual of the torus too, that is on the lattice, where a global pseudo-differential calculus is available.
In WP2 we analyzed Gårding’s inequality on compact Lie groups. Specifically, we obtained the general version of the sharp Gårding inequality on compact Lie groups. Moreover, some preliminary results about the validity of a subelliptic sharp Gårding inequality in the context of subelliptic pseudo-differential operators have been obtained.
Contemporarily, we analyzed other inequalities for degenerate operators in the Euclidean setting. Some results we achieved in this context are: we proved the validity of smoothing and Strichartz estimates for some classes of time-degenerate Schrödinger operators.
This was the first step for the analysis of similar problems on compact Lie groups.
Additionally, the validity of global Poincaré inequalities was established by the ER and collaborators on Lie groups of non compact type.
During this period the ER spent some research periods at Massachusetts Institute of Technology(MIT) and at the University of Bologna to collaborate with experts in the field. She has also co-organized 2 international conferences and participated as speaker in several scientific events.
WP3 was devoted to the study of the local well-posedness of the initial value problem associated with some time-degenerate Schrödinger operators.
Here we proved the local well-posedness of the nonlinear initial value problem by means of smoothing and Strichartz estimates. Additionally, local well-posedness results on compact Lie groups, specifically on the torus, were also obtained by the ER and collaborators.
Results. Some of the results obtained during the action have been submitted for publication in peer-reviewed journals, others will be submitted in the coming months. Preliminary versions are accessible on arXiv.org where the following results have been uploaded:
- S.Federico G. Staffilani, Sharp Strichartz estimates for some variable coefficient Schrödinger operators. Preprint. Arxiv https://arxiv.org/abs/2106.11940(opens in new window).
- M. Chatzakou, S. Federico, B. Zegarlinski, q-Poincaré inequalities on Carnot Groups, Preprint. Arxiv https://arxiv.org/abs/2007.04689(opens in new window).
- S. Federico, M. Ruzhansky, Smoothing and Strichartz estimates for degenerate Schrödinger-type equations. Preprint. Arxiv https://arxiv.org/abs/2005.01622(opens in new window) .
- S. Federico, G. Staffilani, Smoothing effect for time-degenerate Schrödinger operators (accepted for publication by the J. Diff. Eq). Arxiv https://arxiv.org/abs/2001.06708(opens in new window).
In WP1 we performed a detailed analysis of the geometric properties of pseudo-differential operators on Lie groups of compact type. A global Weyl quantisation for these operators is a far reaching result due to the geometry of the objects under study, therefore we focused our attention on alternative approaches to overcome the need of the Weyl calculus.
To start with, we considered the model case of the torus. We studied and defined some transformations, specifically the Bargmann transform, to approach in a different way the problem of the validity of lower bounds.
We derived properties similar to the Euclidean Bargmann transform, transform used in the Euclidean setting to derive, for instance, the Fefferman-Phong inequality.
We have considered and answered similar questions on the dual of the torus too, that is on the lattice, where a global pseudo-differential calculus is available.
In WP2 we analyzed Gårding’s inequality on compact Lie groups. Specifically, we obtained the general version of the sharp Gårding inequality on compact Lie groups. Moreover, some preliminary results about the validity of a subelliptic sharp Gårding inequality in the context of subelliptic pseudo-differential operators have been obtained.
Contemporarily, we analyzed other inequalities for degenerate operators in the Euclidean setting. Some results we achieved in this context are: we proved the validity of smoothing and Strichartz estimates for some classes of time-degenerate Schrödinger operators.
This was the first step for the analysis of similar problems on compact Lie groups.
Additionally, the validity of global Poincaré inequalities was established by the ER and collaborators on Lie groups of non compact type.
During this period the ER spent some research periods at Massachusetts Institute of Technology(MIT) and at the University of Bologna to collaborate with experts in the field. She has also co-organized 2 international conferences and participated as speaker in several scientific events.
WP3 was devoted to the study of the local well-posedness of the initial value problem associated with some time-degenerate Schrödinger operators.
Here we proved the local well-posedness of the nonlinear initial value problem by means of smoothing and Strichartz estimates. Additionally, local well-posedness results on compact Lie groups, specifically on the torus, were also obtained by the ER and collaborators.
Results. Some of the results obtained during the action have been submitted for publication in peer-reviewed journals, others will be submitted in the coming months. Preliminary versions are accessible on arXiv.org where the following results have been uploaded:
- S.Federico G. Staffilani, Sharp Strichartz estimates for some variable coefficient Schrödinger operators. Preprint. Arxiv https://arxiv.org/abs/2106.11940(opens in new window).
- M. Chatzakou, S. Federico, B. Zegarlinski, q-Poincaré inequalities on Carnot Groups, Preprint. Arxiv https://arxiv.org/abs/2007.04689(opens in new window).
- S. Federico, M. Ruzhansky, Smoothing and Strichartz estimates for degenerate Schrödinger-type equations. Preprint. Arxiv https://arxiv.org/abs/2005.01622(opens in new window) .
- S. Federico, G. Staffilani, Smoothing effect for time-degenerate Schrödinger operators (accepted for publication by the J. Diff. Eq). Arxiv https://arxiv.org/abs/2001.06708(opens in new window).
The validity of lower bounds and other inequalities in the Lie group setting have multiple consequences and applications in the resolution of problems related to partial differential equations. We established the general version of the sharp Gårding inequality on compact Lie groups and initiated the analysis of such problems in a subelliptic setting. In particular the subelliptic problem was not considered before and it can bring to the resolution of problems involving subelliptic pseudo-differential operators on compact Lie groups.
We also point put that the proof of other lower bounds is still an open problem. The importance of these inequalities on compact Lie groups have attracted the attention of the scientific community and signed the beginning of ongoing collaborations.
On the other hand, time-degenerate Schrödinger operators have interesting applications in the context of Bose-Einstein condensation and nonlinear optics. These kind of operators have not been intensively studied so far (no smoothing and Strichartz estimates were considered before), and the results we obtained in this MSCA opened the way to a more deep mathematical understanding of these equations and of the corresponding solutions. Through the dissemination of our results in several scientific events, these problems gained visibility and attracted the interest of many experts in the field. Moreover, the results obtained by the ER and G. Staffilani on the torus showed that sharp results can be obtained for some variable coefficients Schrödinger operators. This is a very interesting property that opened the investigation of other more general and difficult cases.
In conclusion, we can certainly affirm that the research conducted in this project opened up further perspective for investigations in the area.
We also point put that the proof of other lower bounds is still an open problem. The importance of these inequalities on compact Lie groups have attracted the attention of the scientific community and signed the beginning of ongoing collaborations.
On the other hand, time-degenerate Schrödinger operators have interesting applications in the context of Bose-Einstein condensation and nonlinear optics. These kind of operators have not been intensively studied so far (no smoothing and Strichartz estimates were considered before), and the results we obtained in this MSCA opened the way to a more deep mathematical understanding of these equations and of the corresponding solutions. Through the dissemination of our results in several scientific events, these problems gained visibility and attracted the interest of many experts in the field. Moreover, the results obtained by the ER and G. Staffilani on the torus showed that sharp results can be obtained for some variable coefficients Schrödinger operators. This is a very interesting property that opened the investigation of other more general and difficult cases.
In conclusion, we can certainly affirm that the research conducted in this project opened up further perspective for investigations in the area.