Stratified Lie groups and potential theory for their sub-Laplacians.

*(English)*Zbl 1128.43001
Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-3-540-71896-3/hbk). xxvi, 800 p. (2007).

Let us start with some basic definitions. Let \(\mathbb{R}^N\) be equipped with a Lie group structure by the multiplication law \(\circ,\) and with a family \(\{\delta_\lambda\}_{\lambda>0}\) of automorphisms (dilations) of \((\mathbb{R}^N,\circ)\) of the form \(\delta_\lambda(x^{(1)},x^{(2)},\ldots,x^{(r)})=(\lambda x^{(1)},\lambda^2x^{(2)},\ldots,\lambda^rx^{(r)}),\) where \(x^{(i)}\in\mathbb{R}^{N_i},\) \(1\leq i\leq r\) and \(N_1+\ldots+N_r=N.\) For \(1\leq i\leq N_1,\) let \(Z_i\) be the vector field from \(\mathfrak{g}\) (the Lie algebra of the group \((\mathbb{R}^N,\circ)\)) satisfying \(Z_i(0)=\frac{\partial}{\partial x_i}.\) The group \(G=(\mathbb{R}^N,\circ)\) with dilations \(\delta_\lambda\) is called homogeneous Carnot group. It is not difficult to see (Proposition 2.2.17) that a homogeneous Carnot group is a stratified group. Recall that a stratified group (or Carnot group) is a connected and simply connected Lie group \(H\) whose Lie algebra \(\mathfrak{h}\) admits a stratification \(\mathfrak{h}=V_1\oplus V_2\oplus\ldots\oplus V_r\) such that \([V_1,V_{i-1}]=V_i\) for \(2\leq i\leq r,\) and \([V_1,V_r]=\{0\}.\) It turns out (Theorem 2.2.18) that if \(H\) is a stratified group then there exists a homogeneous Carnot group \(H^*\) which is isomorphic to \(H.\) A sub-Laplacian \(\mathcal{L}\) on a stratified group \(H\) is a second order differential operator \(\mathcal{L}=\sum_{j=1}^mX_j^2,\) where \(X_1,\ldots,X_m\) is a linear basis of \(V_1.\)

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 divided into three parts.

Part I of the book: Elements of analysis of stratified groups is a self-contained introduction to stratified Lie groups in \(\mathbb{R}^N.\) A lot of examples of explicit stratified groups, such as the Heisenberg-Kaplan groups, Kolmogorov-type groups, Bony-type groups, are given. The main topic of Part I is the analysis of the fundamental solution for the sub-Laplacian \(\mathcal{L}.\) Chapter 5 is devoted to this and related problems. In particular, it is proved that for a given \(\mathcal{L}\) there exists an \(\mathcal{L}\)-gauge, i.e., a homogeneous norm \(d\) such that \(d^{Q-2}\) is \(\mathcal{L}\)-harmonic outside \(\{0\}\) (here \(Q\) is the homogeneous dimension of the group). Moreover, Liouville-type theorems, Harnack-type inequalities and the Sobolev-Stein embedding theorem are proved in Chapter 5.

In Part II of the book Elements of potential theory for sub-Laplacians the authors use an abstract harmonic space theory (presented in Chapter 6) to develop potential theory for the sub-Laplacians on stratified groups. This potential theory turns out to be analogous to the potential theory of the classical Laplacian. Part II, which is the core of the book, contains the following chapters: Abstract harmonic spaces; The \(\mathcal{L}\)-harmonic space; \(\mathcal{L}\)-subharmonic functions; Representation theorems; Maximum principle on unbounded domains; \(\mathcal{L}\)-capacity, \(\mathcal{L}\)-polar sets and applications; \(\mathcal{L}\)-thinness and \(\mathcal{L}\)-fine topology; \(d\)-Hausdorff measure and \(\mathcal{L}\)-capacity.

Finally, Part III of the book Further topics on Carnot groups contains some more specialized material. It includes, among others, the following topics treated in Chapters 14–20: the study of free Lie algebras, the Campbell-Hausdorff formula, the equivalence of the sub-Laplacian under diffeomorphisms, the Rothschild-Stein lifting theorem, the structure of Heisenberg-Kaplan groups, the Carathéodory-Chow-Rashevsky theorem on the connectedness of \(\mathbb{R}^N\) with respect to a family of Hörmander vector fields, the Lagrange mean value theorem, and several versions of the Taylor formula for smooth functions on homogeneous Carnot groups.

The book is clearly and carefully written. It will be useful for both the graduate student and researchers in different areas.

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 divided into three parts.

Part I of the book: Elements of analysis of stratified groups is a self-contained introduction to stratified Lie groups in \(\mathbb{R}^N.\) A lot of examples of explicit stratified groups, such as the Heisenberg-Kaplan groups, Kolmogorov-type groups, Bony-type groups, are given. The main topic of Part I is the analysis of the fundamental solution for the sub-Laplacian \(\mathcal{L}.\) Chapter 5 is devoted to this and related problems. In particular, it is proved that for a given \(\mathcal{L}\) there exists an \(\mathcal{L}\)-gauge, i.e., a homogeneous norm \(d\) such that \(d^{Q-2}\) is \(\mathcal{L}\)-harmonic outside \(\{0\}\) (here \(Q\) is the homogeneous dimension of the group). Moreover, Liouville-type theorems, Harnack-type inequalities and the Sobolev-Stein embedding theorem are proved in Chapter 5.

In Part II of the book Elements of potential theory for sub-Laplacians the authors use an abstract harmonic space theory (presented in Chapter 6) to develop potential theory for the sub-Laplacians on stratified groups. This potential theory turns out to be analogous to the potential theory of the classical Laplacian. Part II, which is the core of the book, contains the following chapters: Abstract harmonic spaces; The \(\mathcal{L}\)-harmonic space; \(\mathcal{L}\)-subharmonic functions; Representation theorems; Maximum principle on unbounded domains; \(\mathcal{L}\)-capacity, \(\mathcal{L}\)-polar sets and applications; \(\mathcal{L}\)-thinness and \(\mathcal{L}\)-fine topology; \(d\)-Hausdorff measure and \(\mathcal{L}\)-capacity.

Finally, Part III of the book Further topics on Carnot groups contains some more specialized material. It includes, among others, the following topics treated in Chapters 14–20: the study of free Lie algebras, the Campbell-Hausdorff formula, the equivalence of the sub-Laplacian under diffeomorphisms, the Rothschild-Stein lifting theorem, the structure of Heisenberg-Kaplan groups, the Carathéodory-Chow-Rashevsky theorem on the connectedness of \(\mathbb{R}^N\) with respect to a family of Hörmander vector fields, the Lagrange mean value theorem, and several versions of the Taylor formula for smooth functions on homogeneous Carnot groups.

The book is clearly and carefully written. It will be useful for both the graduate student and researchers in different areas.

Reviewer: Roman Urban (Wrocław)

##### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

43A80 | Analysis on other specific Lie groups |

35J70 | Degenerate elliptic equations |

35H20 | Subelliptic equations |

35A08 | Fundamental solutions to PDEs |

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

31C15 | Potentials and capacities on other spaces |

35B50 | Maximum principles in context of PDEs |

22E60 | Lie algebras of Lie groups |