Using the Davies-Helffer-Sjöstrand functional calculus based on almost analytic extensions, we address the following problem: Given a self-adjoint operator S in H, and functions f in an appropriate class, for instance, f ∈ C ∞ 0 (R), how to control the norm f (S) B(H) in terms of the norm of the resolvent of S , (S − z 0 I H) −1 B(H) , for some z 0 ∈ C\R. We are particularly interested in the case where B(H) is replaced by a trace ideal, B p (H), p ∈ [1, ∞).