  A classical theorem of Mihlin yields Lp estimates for spectral multipliers
Lp(R^d) -> Lp(R^d); g -> F^{-1}[f(| |^2) Fg] in terms of L^\infty bounds of the
multiplier function f and its weighted derivatives up to an order > d/2. This
theorem, which is a functional calculus for the standard Laplace operator, has
generalisations in several contexts such as elliptic operators on domains and
manifolds, Schr\"odinger operators and sublaplacians on Lie groups. However,
for the wave equation functions f (s) = (1 + s)^{-\alpha} e^{its} a better
estimate is available, in the standard case (works of Miyachi and Peral) and on
Heisenberg Lie groups (M\"uller and Stein). By a transference method for
polynomially bounded regularized groups, we obtain a new class of spectral
multipliers for operators that have these better wave spectral multipliers and
that admit a spectral decomposition of Paley-Littlewood type.
