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Aspects of Sobolev-type inequalities. (English) Zbl 0991.35002
London Mathematical Society Lecture Note Series. 289. Cambridge: Cambridge University Press. x, 190 p. (2002).
The book provides a very nice illustration of various aspects of Sobolev type inequalities when considered over complete (non-compact) Riemannian manifolds that simply do not emerge in \(\mathbb{R}^n\).
In fact, as a start the author begins with a clear presentation of the familiar Sobolev inequalities in \(\mathbb{R}^n\), in a spirit that already ancitipates their (possible) extension to manifolds. This point of view is extended to the subsequent discussion on the role of Sobolev inequalities in Moser’s iteration scheme when deriving Harnack’s inequality for (positive) solutions of elliptic operators over \(\mathbb{R}^n\).
The author then returns to discuss Sobolev type inequalities over manifolds, their weak and/or equivalent forms and how they reveal geometrical properties to be satisfied by the underlying Riemannian manifold. Such a discussion motivates the introduction of ‘local’ versions for such inequalities as to relax the requirements on the class of manifolds to be considered.
Beside interesting other applications, the author concludes by discussing the impact of Sobolev inequalities on parabolic equations on manifolds. There, the validity of certain inequalities of Sobolev type is strictly related to the growth properties of the corresponding heath-kernel and (via Moser’s scheme) on the validity of Harnack type inequalities.
The book is very well written and organized. It contains so many comments and explanations that both experts and non-experts on the subject may enjoy reading it.
Reviewer: G.Tarantello

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58J05 Elliptic equations on manifolds, general theory
58J35 Heat and other parabolic equation methods for PDEs on manifolds