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-Poincar\'e) inequalities. The
purpose of this note is to show how to check these weighted Nash inequalities
in concrete examples, 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
