Lp harmonic analysis for differential-reflection operators

Saïd, Salem Ben; Boussen, Asma; Sifi, Mohamed

doi:10.1016/j.crma.2016.01.020

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2016

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“…Our first step is Theorem 3.3, where we prove that Ψ A,ε (λ, ·) > 0 whenever λ ∈ iR. In [4,5], these estimates are crucial tools for developing L p -Fourier analysis associated with the (A, ε)-operators (1.1). We note that Ψ A,ε reduces to the Dunkl kernel in the (A α , ε)-case [11,27]; to the Heckman kernel in the (A α,β , 0)-case [2,9,20]; and to the Cherednik kernel (or Opdam's kernel) in the (A α,β , 1)-case [1,15,26].…”

Section: Introductionmentioning

confidence: 99%

Intertwining Operators Associated to a Family of Differential-Reflection Operators

2016

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We introduce a family of differential-reflection operators ΛA,ε acting on smooth functions defined on R. Here, A is a Sturm-Liouville function with additional hypotheses and −1 ≤ ε ≤ 1. For special pairs (A, ε), we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators ΛA,ε is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of ΛA,ε. As the operators ΛA,ε are a mixture of d/dx and reflection operators, we prove the existence of an intertwining operator VA,ε between ΛA,ε and the usual derivative. The positivity of VA,ε is also established.Mathematics Subject Classification. 34K99, 34B25, 33E30.

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