Title: Global Hypoellipticity and Solvability on Product Manifolds (in Dutch: Globale hypoellipticiteit en oplosbaarheid op productvariëteiten)

Brief description: We propose to investigate global properties of certain systems of partial differential equations of geometric importance on spaces known as tube structures or the product manifolds. They are constituted by systems of vector fields with symmetries that can be studied via Fourier analysis. Our aim is to determine necessary and/or sufficient conditions for their solvability and for the regularity of their solutions when the ambient space is a so-called Lie group, which encodes extra symmetries of the equations and their solutions. Certain second-order operators associated with such systems(known as sub-Laplacians, or sums-of-squares of vector fields) will also be investigated from this point of view since their properties are related and also connect the former systems with applications. Both the general theory and concrete special cases will be studied. We will be addressing questions of fundamental importance for these systems, such as the hypoellipticity and solvability of the corresponding system of partial differential equations. 

The project will combine in a unique way the expertise of the Brazilian team on hypoellipticity and solvability of partial differential operators with the expertise of the Belgian team on different aspects of the noncommutative Fourier analysis and the theory of pseudo-differential operators on Lie groups.

The research program proposed here splits into three interacting work packages:

WP1: Global hypoellipticity of first order partial differential operators on tube structure built on compact Lie groups.

WP2: Global hypoellipticity of sums of squares of Hörmander’s vector fields on tube structure built on compact Lie groups.

WP3: Global solvability of partial differential equations given by tube structure built on compact Lie groups.

PI: Prof. Michael Ruzhansky   and Co-PI: Dr. Vishvesh Kumar