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We show that certain functional inequalities, e.g. Nash-type and Poincaré-type inequalities, for infinitesimal generators of C0 semigroups are preserved under subordination in the sense of Bochner. Our result improves earlier results by Bendikov and Maheux (Trans Am Math Soc 359:3085–3097, 2007, Theorem 1.3) for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.

Functional inequalities and subordination: stability of Nash and Poincaré inequalities