# American Institute of Mathematical Sciences

March  2015, 35(3): 857-870. doi: 10.3934/dcds.2015.35.857

## Non ultracontractive heat kernel bounds by Lyapunov conditions

 1 Ceremade, Umr Cnrs 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris cedex 16, France 2 Institut Universitaire de France and Laboratoire de Mathématiques, Umr Cnrs 6620, Université Blaise Pascal, Avenue des Landais, F-63177 Aubière cedex, France 3 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received  November 2013 Revised  July 2014 Published  October 2014

Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be derived by means of weighted Nash (or super-Poincaré) inequalities. The purpose of this note is to show how to check these weighted Nash inequalities in concrete examples of reversible diffusion Markov semigroups in $\mathbb{R}^d$, in a very simple and general manner. We also deduce off-diagonal bounds for the Markov kernels of the semigroups, refining E. B. Davies' original argument.
Citation: François Bolley, Arnaud Guillin, Xinyu Wang. Non ultracontractive heat kernel bounds by Lyapunov conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 857-870. doi: 10.3934/dcds.2015.35.857
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