6. Sobolev Inequalities

After the Poincaré and logarithmic Sobolev inequalities, this chapter is devoted to the investigation of Sobolev inequalities. Sobolev inequalities play a central role in analysis, providing in particular compact embeddings and tight connections with heat kernels bounds. They are also deeply linked with the geometric structure of the underlying state space through conformal invariance. The study here focuses on the aspects of Sobolev inequalities in the context of Markov diffusion operators and semigroups. The chapter starts with a brief exposition of the classical Sobolev inequalities on the model spaces, namely the Euclidean, spherical and hyperbolic spaces. Next, various definitions of Sobolev-type inequalities in the Markov Triple context are emphasized, in particular (logarithmic) entropy-energy and Nash-type inequalities. The basic equivalence between Sobolev inequalities and (uniform) heat kernel bounds (ultracontractivity) is studied in the further sections. Local inequalities under the semigroup are investigated next, providing in particular a heat flow approach to the celebrated Li–Yau parabolic inequality. The sharp Sobolev inequality under curvature-dimension condition, covering the example of the standard sphere, is established in the framework of this monograph. Further sections describe the conformal invariance properties of Sobolev inequalities, and, as a consequence, the sharp Sobolev inequalities in the Euclidean and hyperbolic spaces on the basis of the one on the sphere, Gagliardo–Nirenberg inequalities and non-linear porous medium and fast diffusion equations and their geometric counterparts, with in particular a fast diffusion approach to several Sobolev inequalities of interest.