Wave equation and multiplier estimates on Damek-Ricci spaces

Details Wave equation and multiplier estimates on Damek-Ricci spaces Wave equation and multiplier estimates on Damek-Ricci spaces

2008-06-18

arxiv.org/abs/0806.2921v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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 associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L.

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