Wave equation and multiplier estimates on $ax+b$ groups

Detlef Müller ¹   ; Christoph Thiele ²

¹ Mathematisches Seminar C.A.-Universität Kiel Ludewig-Meyn-Strasse 4 D-24098 Kiel, Germany
² Department of Mathematics UCLA Los Angeles, CA 90095-1555, U.S.A.

Studia Mathematica, Tome 179 (2007) no. 2, pp. 117-148

Résumé

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.

Détail

DOI : 10.4064/sm179-2-2

Keywords: distinguished laplacian certain semidirect products mathbb mathbb which type prove pointwise estimates convolution kernels spectrally localized wave operators form sqrt psi sqrt lambda arbitrary time arbitrary lambda where psi smooth bump function supported lambda lambda corollary reprove basic multiplier estimate hebisch steger math particular class groups derive sobolev estimates solutions wave equation associated there appears dispersive effect respect infty norms large times estimates seems unlikely non trivial strichartz type estimates