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Journal Articles Journal of Functional Analysis Year : 2017

Dunkl spectral multipliers with values in UMD lattices


We show a H\"ormander spectral multiplier theorem for $A = A_0 \otimes \mathrm{Id}_Y$ acting on the Bochner space $L^p(\mathbb{R}^d , h^2_\kappa; Y)$, where $A_0$ is the Dunkl Laplacian, $h^2_\kappa$ a weight function invariant under the action of a reflection group and $Y$ is a UMD Banach lattice. We follow hereby a transference method developed by Bonami-Clerc and Dai-Xu, passing through a Marcinkiewicz multiplier theorem on the sphere. We hereby generalize works for $A_0 = − \Delta$ acting on $L^p(\mathbb{R}^d,dx)$ by Girardi-Weis, Hyt\"onen and others before. We apply our main result to maximal regularity for Cauchy problems involving $A$.
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hal-01900687 , version 1 (22-10-2018)



Luc Deleaval, Christoph Kriegler. Dunkl spectral multipliers with values in UMD lattices. Journal of Functional Analysis, 2017, 272 (5), pp.2132 - 2175. ⟨10.1016/j.jfa.2016.12.013⟩. ⟨hal-01900687⟩
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