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Heat kernels and spectral theory. (English) Zbl 0699.35006
Cambridge Tracts in Mathematics, 92. Cambridge etc.: Cambridge University Press. ix, 197 p. £27.50; $ 49.50 (1989).
This advanced monograph investigates through a study of heat kernels the spectral properties of linear, selfadjoint elliptic differential operators and obtains pointwise bounds for associated eigenfunctions. The book is a first account of the new techniques which can be used as a result of the recent improvements in understanding of the heat kernels obtained by means of quadratic form technique and logarithmic Sobolev inequalities. In contrast to older theories the new method enables pointwise upper and lower bounds to be obtained in terms of constants which are computable and, in many cases, sharp. The adopted approach, which is entirely analytical, enables operators in divergence form with measurable second order coefficients to be dealt with in a straightforward way. Previous works in the area relied on the Moser’s Harnack inequality, here the results of E. B. Fabes and D. W. Stroock [Arch. Ration. Mech. Anal. 96, 327-338 (1986; Zbl 0652.35052)] are used instead. The theory for operators with variable coefficients is developed in a region of Euclidean space whilst heat kernels of Laplace- Beltrami operators are studied on Riemannian manifolds.
The first chapter summarises the important concepts and results which are used extensively in the rest of the book. Although the longest chapter in the book it is concisely and precisely written and forms a valuable feature of the book. Chapter two introduces the concepts of contractivity and ultracontractivity and extends the theory of logarithmic Sobolev inequalities, developed by Gross, to yield ultracontractive bounds. With future applications in mind the development is given an abstract and quite general setting. In the third chapter ultracontractive bounds are used to obtain Gaussian upper bounds on heat kernels. It is then shown that these upper bounds, which offer considerable improvement on previous results, can be used to provide lower bounds. In chapter four the behaviour of heat kernels and eigenfunctions is studied near the boundary of the region on which the elliptic operator acts. Examples are given to illustrate the complicated nature of this topic even under relatively favourable conditions. It is shown that the behaviour of the heat kernel and Green’s function near the boundary is controlled by the ground state. Corresponding results are obtained for the Schrödinger operator and harmonic oscillator. In the final chapter Laplace-Beltrami operators are studied on Riemannian manifolds whose Ricci curvature is bounded below. The parabolic Harnack inequality of Li and Yau is presented together with heat kernel bounds for manifolds of non-negative Ricci curvature.
Examples are used liberally throughout the text either to motivate or to illustrate the theory and its development. A particularly attractive feature of the book is the set of notes at the end of each chapter which although mainly of an historical nature also contain additional information. The material which at times can be quite demanding, is presented in a persuasive and decidedly readable manner. The book will be welcomed by those working in the area.
Reviewer: G.F.Roach

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P05 General topics in linear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
35J10 Schrödinger operator, Schrödinger equation
35K05 Heat equation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58J10 Differential complexes