Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 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 -norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-2-2,
author = {Detlef M\"uller and Christoph Thiele},
title = {Wave equation and multiplier estimates on ax + b groups},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {117-148},
zbl = {1112.43002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-2-2}
}
Detlef Müller; Christoph Thiele. Wave equation and multiplier estimates on ax + b groups. Studia Mathematica, Tome 178 (2007) pp. 117-148. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-2-2/