Wave equation and multiplier estimates on $ax+b$ groups
Volume 179 / 2007
Abstract
Let $L$ be the distinguished Laplacian on certain semidirect products of $\mathbb R$ by $\mathbb R^n$ which are of $ax+b$ type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $e^{it\sqrt{L}} \psi(\sqrt{L}/\lambda)$ for arbitrary time $t$ and arbitrary $\lambda>0$, where $\psi$ is a smooth bump function supported in $[-2,2]$ if $\lambda\le 1$ and in $[1,2]$ if $\lambda\ge 1$. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] 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 $L^\infty$-norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.
