Dunkl spectral multipliers with values in UMD lattices

摘要

We show a Hormander spectral multiplier theorem for A = A(0) circle times Id(Y) acting on the Bochner space L-p (R-d, h(k)(2); Y), where A(0) is the Dunkl Laplacian, h(k)(2) 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 (R-d, dx) by Girardi-Weis, Hytonen and others before. We apply our main result to maximal regularity for Cauchy problems involving A. (C) 2016 Elsevier Inc. All rights reserved.