Stratified Lie Groups and Potential Theory for Their Sub-Laplacians [Paperback]

This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra or differential geometry.

Elements of Analysis of Stratified Groups.- Stratified Groups and Sub-Laplacians.- Abstract Lie Groups and Carnot Groups.- Carnot Groups of Step Two.- Examples of Carnot Groups.- The Fundamental Solution for a Sub-Laplacian and Applications.- Elements of Potential Theory for Sub-Laplacians.- Abstract Harmonic Spaces.- The ?-harmonic Space.- ?-subharmonic Functions.- Representation Theorems.- Maximum Principle on Unbounded Domains.- ?-capacity, ?-polar Sets and Applications.- ?-thinness and ?-fine Topology.- d-Hausdorff Measure and ?-capacity.- Further Topics on Carnot Groups.- Some Remarks on Free Lie Algebras.- More on the CampbellHausdorff Formula.- Families of Diffeomorphic Sub-Laplacians.- Lifting of Carnot Groups.- Groups of Heisenberg Type.- The Carath?odoryChowRashevsky Theorem.- Taylor Formula on Homogeneous Carnot Groups.

From the reviews:

The book is about sub-Laplacians on stratified Lie groups. The authors present the material using an elementary approach. They achieve the level of current research starting from the basic notions of differential geometry and Lie group theory. The book is full of extensive examples which illustrate the general problems and results. Exercises are included at the end of each chapter. & The book is clearly and carefully written. It will be useful for both the graduate student and researchers in different areas. (Roman Urban, Zentralblatt MATH, Vol. 1128 (6), 2008)

The monograph under review is a comprehensive treatment of many interesting results regarding subelliptic partilS¶