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Three results in Dunkl analysis

Béchir Amri, Jean-Philippe Anker, Mohamed Sifi (2010)

Colloquium Mathematicae

We first establish a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the L p L p norm of Dunkl translations in dimension 1. Finally, we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

Wave equation and multiplier estimates on ax + b groups

Detlef Müller, Christoph Thiele (2007)

Studia Mathematica

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 i t 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...

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