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Weighted Nash Inequalities

Abstract : 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
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Dominique Bakry, François Bolley, Ivan Gentil, Patrick Maheux. Weighted Nash Inequalities. Revista Matemática Iberoamericana, European Mathematical Society, 2012, 28 (3), pp.879-906. ⟨10.4171/RMI/695⟩. ⟨hal-00474543v2⟩

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