Marie Curie Project H2020-MSCA-IF-2018 LieLowerBounds: Lower bounds for partial differential operators on compact Lie groups (2019-2021) |

**Name of Researcher: Dr Serena Federico****Scientist in Charge: ****Prof Michael Ruzh**an**sky****Host: Ghent University**

**PROJECT ABSTRACT**

The goal of this project is to investigate the validity of some fundamental lower bounds for partial differential operators on compact Lie groups.

The motivations moving the interest for this problem are explained by the fact that the validity of such inequalities will yield the development of several results for PDEs on compact Lie groups, as, for instance, in the problems related to solvability, hypoellipticity, and well-posedness of the (weakly-hyperbolic) Cauchy problem. The theory of pseudo-differential operators and the global quantization on compact Lie group introduced by Ruzhansky and Turunen, the latter given in terms of the irreducible representations of the group, represent the key tools for the development of the project. Of course, in order to obtain lower bounds for partial differential operators on compact Lie groups, some geometric quantities attached to the operators which play a crucial role in the analysis of the argument will be studied. Our final goal is to use these fundamental estimates to treat the problem of solvability of partial differential operators on compact Lie groups.

Historically, the theory of partial differential operators is one of the most important branches of mathematics with several consequences in many other mathematical fields and with applications in other sciences. This project intends to investigate the validity of some fundamental lower bounds for partial differential operators, and in general for pseudo-differential operators, on compact Lie groups. The validity of such a priori estimates will lead to the development of results about interesting problems in the theory of partial differential operators, such as, for instance, in the solvability problem of partial differential operators on compact Lie groups. We will combine *microlocal analysis techniques *and* noncommutative methods* to define a new *global approach* to be applied to the resolution of problems in the setting of compact Lie groups.

**Publications:**

- S. Federico, M. Ruzhansky,
*Smoothing and Strichartz estimates for degenerate Schrödinger-type equations*. arxiv - S. Federico, G. Staffilani,
*Smoothing effect for time-degenerate Schrödinger operators*. arxiv - S. Federico, A. Parmeggiani,
*On the Solvability of a Class of Second-Order Degenerate Operators.**P. Boggiatto et al. (eds.),*Advances in Microlocal and Time-Frequency Analysis, Springer Nature Switzerland AG 2020, pp. 207-226. - S. Federico,
*Sufficient conditions for local solvability of some degenerate PDO with complex subprincipal symbol*, J. Pseudo-Differ. Oper. Appl. (10) (2019) 929–940. https://doi.org/10.1007/s11868-018-0264-x - S. Federico, A. Parmeggiani,
*On the local solvability of a class of degenerate second order operators with complex coefficients*, Comm. Partial Differential Equations 43 (10) (2018) 1485-1501. - S. Federico,
*Local solvability of a class of degenerate second order operators,*Bruno Pini Mathematical Analysis Seminar, Vol. 8 (2017) 185-203. - S. Federico,
*A model of solvable second order PDE with non smooth coefficients*, J. Math. Anal. Appl. 440 (2016) 661-676. - S. Federico, A. Parmeggiani,
*Local solvability of a class of degenerate second order operators*, Comm. Partial Differential Equations 41 (03) (2016) 484-514.