Global hypoellipticity and solvability on product manifolds

• Grant number: 24/08416-6
• Support Opportunities: Regular Research Grants
• Start date: January 01, 2025
• End date: December 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Paulo Leandro Dattori da Silva
• Grantee: Paulo Leandro Dattori da Silva
• Principal researcher abroad: Michael Ruzhansky
• Institution abroad: Ghent University (UGent), Belgium
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated researchers: Gabriel Cueva Candido Soares de Araújo

Abstract

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 areconstitued 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 encode 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 as well as 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, with the expertise of the Belgian team on different aspects of the Fourier analysis and the theory of pseudo-differential operators on Lie groups. (AU)