Abstract
The Poincaré, logarithmic Sobolev and Sobolev inequalities capture different features of the associated semigroup or the invariant measure, in terms of convergence to equilibrium, estimates on the heat kernels or tail behaviors of the invariant measure. This chapter investigates intermediate or more general families of functional inequalities which are suited to a wide variety of regimes as well as to more precise, or different, features. The study is concerned with three main examples, entropy-energy, generalized Nash and weak Poincaré inequalities. The first part of the chapter describes the family of functional entropy–energy inequalities well-suited to heat kernel bounds by means of the method developed for hypercontractivity under logarithmic Sobolev inequalities. Off-diagonal heat kernel estimates may be achieved in the same way. Further sections in this chapter investigate generalized Nash inequalities on the basis of the example of the classical Nash inequality in Euclidean space, as well as weighted Nash inequalities, with a focus on non-uniform heat kernel bounds and tail inequalities. Weak Poincaré inequalities are studied as the main minimal tool to tighten families of standard functional inequalities. Weak Poincaré inequalities may be further studied in their own from the viewpoint of heat kernel bounds and tail estimates. Further related families of functional inequalities of interest and illustrative examples complete the exposition of this chapter.
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Bakry, D., Gentil, I., Ledoux, M. (2014). Generalized Functional Inequalities. In: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der mathematischen Wissenschaften, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-319-00227-9_7
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DOI: https://doi.org/10.1007/978-3-319-00227-9_7
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