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Spectral multipliers for wave operators

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, Schrödinger operators and sublaplacians on Lie groups. However, for the wave equation functions f (s) = (1 + s)^{-α} e^{its} a better estimate is available, in the standard case (works of Miyachi and Peral) and on Heisenberg Lie groups (Müller 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.