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• # Artykuł - szczegóły

## Studia Mathematica

2007 | 179 | 2 | 117-148

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

EN

### Abstrakty

EN
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 $e^{it√L} ψ(√L/λ)$ 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 $L^{∞}$-norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.

117-148

wydano
2007

### Twórcy

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