Wave equation and multiplier estimates on Damek-Ricci spaces (Q967574)

Let \(S\) be a Damek-Ricci space and \(L\) a distinguished left invariant Laplacian on \(S\). The authors of the article under review prove point-wise estimates for the convolution kernels of spectrally localized multiplier operators of the form \(e^{it\sqrt{L}}\psi(\sqrt{L}/\lambda)\) for arbitrary time \(t\) and \(\lambda>0\), where \(\psi\) is a bump function supported in \([-2,2]\) if \(\lambda\leq 1\), and in \([1,2]\) if \(\lambda>1\). Such estimates were first proved by \textit{D. Müller} and \textit{C. Thiele} [Stud. Math. 179, No. 2, 117--148 (2007; Zbl 1112.43002)] in the case of \(ax+b\) groups. Point estimates of the gradient of these kernels were also given. As applications, a new proof of basic multiplier estimates, which is entirely based on the wave equation, from \textit{W. Hebisch} and \textit{T. Steger} [Math. Z. 245, No. 1, 37--61 (2003; Zbl 1035.43001)] on the exponential growth groups and from \textit{M. Vallarino} [J. Lie Theory 17, No. 1, 163--189 (2007; Zbl 1124.22002)] on harmonic extensions of H-type groups is given. Moreover, the authors obtain \(L^p\)-estimates for Fourier multiplier operators of the form \(m(\sqrt{L})\cos(t\sqrt{L})\) and \(m(\sqrt{L})\frac {\sin(t\sqrt{L})}{\sqrt{L}}\), where \(m\) is a suitable symbol, and derive Sobolev estimates for solutions of the wave equation associated with \(L\) on Damek-Ricci spaces.