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Wave equation and multiplier estimates on ax + b groups

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Müller D., Thiele C.

Studia Mathematica

2007

tom 179

nr 2

117-148

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.

Ograniczanie wyników

1 Studia Mathematica

1 Müller D.

1 Thiele C.

1 2007

Wyniki wyszukiwania

Wave equation and multiplier estimates on ax + b groups