Super-Poincar\'e and Nash-type inequalities for Subordinated Semigroups

We prove that if a super-Poincar'e inequality is satisfied by an infinitesimal generator $-A$ of a symmetric contracting semigroup then it implies a corresponding super-Poincar'e inequality for $-g(A)$ with any Bernstein function $g$. We also study the converse statement. We deduce similar results for the Nash-type inequality. Our results applied to fractional powers of $A$ and to $\log (I+A)$ and thus generalize some results of Biroli and Maheux, and Wang 2007. We provide several examples.