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arXiv · 1004.3456

Weighted Nash Inequalities

Dominique Bakry·François Bolley·Ivan Gentil·Patrick Maheux
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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. 2010-06-24. Weighted Nash Inequalities. https://arxiv.org/abs/1004.3456

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