Research Proposal: The project is devoted to advancing aspects of non-commutaive analysis and its applications to PDEs fuelled by recent advances in the global quantization theories and functional analysis on nilpotent Lie groups.
We will mainly work in the setting of (homogeneous) stratified groups (or Carnot groups). These are the nilpotent Lie groups allowing for a collection of vector fields whose iterated commutators span the whole of the Lie algebra. Such groups play a central role in the sub-Riemannian analysis, allowing for the use of techniques from analysis, geometry, algebra, and the theory of partial differential equations.
One of the aims of this project is the development of the non-commutative Beals-Fefferman theory. This theory is one of the major breakthroughs of the classical theory of pseudo-differential operators, but very little is known about its non-commutative version. In the Euclidean setting, Fefferman and Phong in 1978 improved the well-known sharp Gårding inequality with the use of the (Euclidean) Beals-Fefferman calculus. This further improvement is now called the Fefferman-Phong inequality. Thus, by the development of the non-commutative analogue of the theory, and with the recent developments on the Weyl quantization in the setting of graded groups, we hope to make a succeful attempt and resolve the long-standing problem of establishing the Fefferman-Phong inequality on stratified groups.