Wave Equation and Multiplier

Let L be the distinguished Laplacian on certain semidirect products of R by R which are of ax+ b-type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e √ ψ( √ L/λ) for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [−2, 2] if λ ≤ 1 and supported in [1, 2] if λ ≥ 1. As corollary, we reprove a basic multiplier estimate from [5] for this particular class of groups, and derive Sobolev estimates for solutions to the wave equation associated to L. There appears no dispersive effect with respect to the L norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold .