Elliptic operators and Lie groups. (English) Zbl 0747.47030

Oxford Science Publications. Oxford etc.: Clarendon Press. xi, 558 p. (1991).
This beautiful book develops the basic theory of elliptic operators on a Lie group and extends the theory of parabolic evolution equations to the non-commutative context. The starting point of the theory is R. P. Langlands’ largely unpublished thesis on elliptic operators associated with continuous representations of a Lie group [Proc. Nat. Acad. Sci. 46, 361-363 (1960; Zbl 0095.105)]. Much of the book is concerned with strongly elliptic operators with constant coefficients, but the general theory for operators with smooth bounded coefficients is also developed. In addition, the principal results of the second order theory are extended to subelliptic operators and to strongly elliptic operators with variable coefficients satisfying minimal smoothness hypothesis.
A main problem of the book consists of establishing that a closed elliptic differential operator is a semigroup generator. The semigroups generated by elliptic operators are given by integral kernels. Their basic properties are obtained by Lie group versions of the Sobolev inequalities in a form due to Nash in combination with the Gårding inequalities.
This book is devoted to students (mathematics, theoretical physics) who want to learn the theory of elliptic operators on Lie groups, and it leads directly to the current research. Therefore, it is also of interest for the research worker. The book contains five chapters (1. Elliptic operators, 2. Analytic elements, 3. Semigroup kernels, 4. Second order operators, 5. Elliptic operators with variable coefficients) and four appendices (on \(L^{p,q}\)-spaces, interpolation theory, inequalities on Lie groups etc.).
Reviewer: J.Marschall


47F05 General theory of partial differential operators
35J30 Higher-order elliptic equations
35K25 Higher-order parabolic equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
22E30 Analysis on real and complex Lie groups


Zbl 0095.105