Nash type inequalities for fractional powers of non-negative self-adjoint operators

Abstract

Assuming that a Nash type inequality is satisfied by a non-negative self-adjoint operator A A , we prove a Nash type inequality for the fractional powers A α A^{\alpha } of A A . Under some assumptions, we give ultracontractivity bounds for the semigroup ( T t , α ) (T_{t,{\alpha }}) generated by − A α -A^{\alpha } .