Analysis and geometry on groups.
Paperback reprint of the 1992 original.

*(English)*Zbl 1179.22009
Cambridge Tracts in Mathematics 100. Cambridge: Cambridge University Press (ISBN 978-0-521-08801-5/pbk). xii, 156 p. (2008).

[For a review of the 1992 original that appeared under the title “Analysis on Lie groups” see Zbl 0813.22003.]

The present research monograph is an exposition of mathematical works done by the authors during the 1980s and involves the following basic material. The real analysis background such as Sobolev, Harnack and Dirichlet inequalities; the existing semigroup theory, especially the Beurling-Deny theory; the theory of second order subelliptic differential operators and especially the “sum of squares operators”.

The exposé is organized as follows. In Chapter II the semigroup machinery is built necessary to link the Sobolev inequalities with the behavior of the heat kernel. These functional analytic results are of independent interest. In Chapter III some basic properties of the sums of squares of vector fields are described. A given set of vector fields \(X_1,\dots,X_{k+1}\) satisfies the Hörmander condition if the fields \(X_1,\dots,X_{k+1}\) together with their brackets of every order span the tangent space at each point. Under this condition, a genuine distance can be defined by considering the “minimal length” of absolutely continuous paths tangent to the fields \(X_1,\dots, X_{k+1}.\) Moreover, the operator \(\sum_{i=1}^k X_i^2+X_{k+1}\) is hypoelliptic (Hörmander’s theorem) and a local Harnack inequality holds. Chapter IV focuses on the study of the sublaplacian associated with a Hörmander system of left invariant vector fields on a nilpotent Lie group, where the analysis is based on Harnack’s principle. Indeed, any connected nilpotent Lie group can be covered by another nilpotent Lie group that admits a dilation structure. This dilation structure, together with the local Harnack principle, yields the scaled Harnack principle (Section 3. Chapter III) and from this, a two-sided Gaussian bound for the heat kernel follows. The volume growth of nilpotent Lie groups is also studied. This shows the existence of a local dimension \(d\) that governs the behavior of the volume of small balls, and of a dimension at infinity \(D\) that governs the volume of large balls. Finally, heat kernel and volume estimates, together with Chapter II, yield optimal Sobolev inequalities. Chapter IV also serves as a model for a general study of Hörmander systems of vector fields. In Chapter V it is shown how Harnack’s principle and a local scaling technique yield satisfactory local results for the heat equation associated with sublaplacians on groups and manifolds. Chapter VI introduces in the simple setting of discrete groups the main ideas leading to the analytic and geometric study of groups at infinity. Chapter VII develops the various tools needed to extend and refine the results of Chapter VI. The main result establishes the sharp relationship between volume growth and the decay of convolution powers in the setting of locally compact, compactly generated groups. Chapter VIII considers unimodular connected Lie groups. Here two-sided Gaussian estimates for the heat kernel, optimal Sobolev inequalities, and Harnack’s principle are obtained. Chapter IX concentrates on Sobolev inequalities for non-unimodular Lie groups. Finally Chapter X contains various geometric applications of the above theory.

The book does not aim at being self-contained.

The present research monograph is an exposition of mathematical works done by the authors during the 1980s and involves the following basic material. The real analysis background such as Sobolev, Harnack and Dirichlet inequalities; the existing semigroup theory, especially the Beurling-Deny theory; the theory of second order subelliptic differential operators and especially the “sum of squares operators”.

The exposé is organized as follows. In Chapter II the semigroup machinery is built necessary to link the Sobolev inequalities with the behavior of the heat kernel. These functional analytic results are of independent interest. In Chapter III some basic properties of the sums of squares of vector fields are described. A given set of vector fields \(X_1,\dots,X_{k+1}\) satisfies the Hörmander condition if the fields \(X_1,\dots,X_{k+1}\) together with their brackets of every order span the tangent space at each point. Under this condition, a genuine distance can be defined by considering the “minimal length” of absolutely continuous paths tangent to the fields \(X_1,\dots, X_{k+1}.\) Moreover, the operator \(\sum_{i=1}^k X_i^2+X_{k+1}\) is hypoelliptic (Hörmander’s theorem) and a local Harnack inequality holds. Chapter IV focuses on the study of the sublaplacian associated with a Hörmander system of left invariant vector fields on a nilpotent Lie group, where the analysis is based on Harnack’s principle. Indeed, any connected nilpotent Lie group can be covered by another nilpotent Lie group that admits a dilation structure. This dilation structure, together with the local Harnack principle, yields the scaled Harnack principle (Section 3. Chapter III) and from this, a two-sided Gaussian bound for the heat kernel follows. The volume growth of nilpotent Lie groups is also studied. This shows the existence of a local dimension \(d\) that governs the behavior of the volume of small balls, and of a dimension at infinity \(D\) that governs the volume of large balls. Finally, heat kernel and volume estimates, together with Chapter II, yield optimal Sobolev inequalities. Chapter IV also serves as a model for a general study of Hörmander systems of vector fields. In Chapter V it is shown how Harnack’s principle and a local scaling technique yield satisfactory local results for the heat equation associated with sublaplacians on groups and manifolds. Chapter VI introduces in the simple setting of discrete groups the main ideas leading to the analytic and geometric study of groups at infinity. Chapter VII develops the various tools needed to extend and refine the results of Chapter VI. The main result establishes the sharp relationship between volume growth and the decay of convolution powers in the setting of locally compact, compactly generated groups. Chapter VIII considers unimodular connected Lie groups. Here two-sided Gaussian estimates for the heat kernel, optimal Sobolev inequalities, and Harnack’s principle are obtained. Chapter IX concentrates on Sobolev inequalities for non-unimodular Lie groups. Finally Chapter X contains various geometric applications of the above theory.

The book does not aim at being self-contained.

Reviewer: Lubomira Softova (Aversa)

##### MSC:

22E30 | Analysis on real and complex Lie groups |

43A80 | Analysis on other specific Lie groups |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

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

22E40 | Discrete subgroups of Lie groups |

35K05 | Heat equation |

20F05 | Generators, relations, and presentations of groups |

20F65 | Geometric group theory |

##### MathOverflow Questions:

Embedding theorem for anisotropic Sobolev spacesGeneralization of Gagliardo-Nirenberg Inequality