科研之友
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摘要
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 C-a exp(-vertical bar x vertical bar(a)), with 1 %26lt; a %26lt; 2, for which uniform bounds are known not to hold.
- 出版日期2012
