Wave Equation and Multiplier Estimates on Damek–Ricci Spaces

Abstract

Let S be a Damek–Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $${\rm {e}}^{it\sqrt{ L}}\psi\big(\sqrt{ L}/{\lambda}\big)$$ 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. This generalizes previous results in Muller and Thiele (Studia Math. 179:117–148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37–61, 2003) and Vallarino (J. Lie Theory 17:163–189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L.