Weighted Nash inequalities

Nash or Sobolev inequalities are known to be equivalent to ultracontractive prop-erties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work, following works by F.-Y. Wang, we present a simple and extremely gen-eral 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 R naturally associated with the measure with density Ca exp(−|x|a), with 1 < a < 2, for which uniform bounds are known not to hold