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 naturally associated with the measure with density , with , for which uniform bounds are known not to hold.
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Dominique Bakry, François Bolley, Ivan Gentil, Patrick Maheux, Weighted Nash inequalities. Rev. Mat. Iberoam. 28 (2012), no. 3, pp. 879–906