  Nash or Sobolev inequalities are known to be equivalent to ultracontractive
properties of Markov semigroups, hence to uniform bounds on their kernel
densities. In this work we present a simple and extremely general method, based
on weighted Nash inequalities, to obtain non-uniform bounds on the kernel
densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm
of the heat kernels. We illustrate the method on the heat kernel on $\dR$
naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $1
