Sobolev-type inequalities play a fundamental role in many areas of function space theory and in partial differential equations. The purpose of this volume is to extend results on Sobolev-type inequalities from the classical version (domains in Euclidean space) to the framework of Riemannian manifolds.

The first chapter presents fundamental Sobolev inequalities for embedding into Lebesgue spaces, together with two classical proofs using alternatively Gagliardo inequality and Riesz potentials. Related results (embeddings into Hölder spaces, best embedding constants and the Poincaré inequality) are also discussed. Possibilities of introduced techniques are illustrated in Chapter 2 in a discussion of Moser’s famous proof of Harnack’s inequality for weak positive solutions of elliptic equations in divergence form. Chapter 3 concentrates on Sobolev-type inequalities on Riemannian manifolds. After explaining basic facts about Riemannian manifolds, relations between the Sobolev and Poincaré inequalities and the volume growth of the manifold are established. Strong and weak forms of the Sobolev inequality are proved, together with an interesting form of pseudo-Poincaré inequality. The fourth chapter is devoted to applications: ultracontractivity of the heat equation, Gaussian heat kernel estimates, the Rozenblum-Lieb–Cwikel inequality and the Birman-Schwinger principle. The final chapter studies a parabolic version of the Harnack inequality and characterises manifolds that satisfy a scale invariant parabolic Harnack principle.

This book is a well-written and self-contained account of the topic. It is accessible to advanced graduate students and to researchers.

Reviewer:

jsta