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 paper, following work of F.-Y. Wang, we present a simple and extremely general method, based on weighted Nash inequalities, for obtaining non-uniform bounds on kernel densities. Such bounds imply control of the trace or the Hilbert–Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on \mathbb{R} naturally associated with the measure with density C_a\exp(-|x|^a) , with 1< a < 2 , for which uniform bounds are known not to hold.