Ultracontractivity and embedding into L ^∞

Given a self-adjoint semigroup e^-tA satisfying an ultracontractivity bound of the type ∥e^-tA∥2→∞ ≤ e^m(t), we find conditions on the sequence ∥A ⁿf∥₂^1/n that imply that f is a bounded function. Sobolev's classical embedding theorem says that, when A is the Laplace operator on ℝ^d, ∥A^kf∥2 < ∞ for some k>d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence ∥Aⁿf∥ ₂^1/n and depends on the behavior of the ultracontractivity function m.