Hörmander's multiplier theorem for the Dunkl transform

2018/07/07 by Dziubański, Jacek, Hejna, Agnieszka · 1 citation
#42B15 #42B20 #42B25 #42B30 #42B35 #47D03 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1807.02640

Abstract

For a normalized root system R in \mathbb R^N and a multiplicity function k≥ 0 let \mathbf N=N+∑_{α∈ R} k(α). Denote by dw(\mathbf x)=∏_{α∈ R}|⟨ \mathbf x,α⟩|^k(α) d\mathbf x the associated measure in \mathbb R^N. Let \mathcal F stands for the Dunkl transform. Given a bounded function m on \mathbb R^N, we prove that if there is s>\mathbf N such that m satisfies the classical Hörmander condition with the smoothness s, then the multiplier operator \mathcal T_mf=\mathcal F^-1(m\mathcal Ff) is of weak type (1,1), strong type (p,p) for 1

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