Quantization on Nilpotent Lie Groups

Abstract

The purpose of this monograph is to give an exposition of the global quantization of operators on nilpotent homogeneous Lie groups. We also present the background analysis on homogeneous and graded nilpotent Lie groups. The analysis on homo- geneous nilpotent Lie groups drew a considerable attention from the 70’s onwards. Research went in several directions, most notably in harmonic analysis and in the study of hypoellipticity and solvability of partial differential equations. Over the decades the subject has been developing on different levels with advances in the analysis on the Heisenberg group, stratified Lie groups, graded Lie groups, and general homogeneous Lie groups. In the last years analysis on homogeneous Lie groups and also on other types of Lie groups has received another boost with newly found applications and further advances in many topics. Examples of this boost are subelliptic estimates, multi- plier theorems, index formulae, nonlinear problems, potential theory, and symbolic calculi tracing full symbols of operators. In particular, the latter has produced fur- ther applications in the study of linear and nonlinear partial differential equations, requiring the knowledge of lower order terms of the operators.