Wave equation and multiplier estimates on ax + b groups

• Autorzy: Detlef Müller , Christoph Thiele
• Języki publikacji: 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.

Kategorie tematyczne

• 42B15: Multipliers
• 43A15: L p -spaces and other function spaces on groups, semigroups, etc.
• 22E30: Analysis on real and complex Lie groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 179
• Numer: 2
• Strony: 117-148

Daty

wydano

2007

Twórcy

autor

Detlef Müller

• Mathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Strasse 4, D-24098 Kiel, Germany

autor

Christoph Thiele

• Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A.

Identyfikatory

DOI

10.4064/sm179-2-2