Super-Poincaré and Nash-type inequalities for subordinated semigroups
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We prove that if a super-Poincaré inequality is satisfied by an infinitesimal generator \(-A\) of a symmetric contraction semigroup on \(L^2\) and that is contracting on \(L^1\), then it implies a corresponding super-Poincaré inequality for \(-g(A)\) for any Bernstein function \(g\). We also study the converse of this statement. We prove similar results for Nash-type inequalities. We apply our results to Euclidean, Riemannian, hypoelliptic and Ornstein–Uhlenbeck settings.
KeywordsSuper-Poincaré inequality Nash-type inequality Symmetric semigroup Subordination in the sense of Bochner Bernstein function Super-Poincaré profile
The authors thank the anonymous referee for helpful comments on this paper and her/his patience during the submission of this paper. This research was supported in part by the ANR Project EVOL. The second author thanks the CNRS for a period of delegation during which this paper has been completed.
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