Corrigendum to “Lp -Fourier multipliers for the Dunkl operator on the real line” [J. Funct. Anal. 209 (2004) 16–35]

Journal of Functional Analysis 242 (2007) 672–673 www.elsevier.com/locate/jfa

Corrigendum

Corrigendum to “Lp -Fourier multipliers for the Dunkl operator on the real line” [J. Funct. Anal. 209 (2004) 16–35] Fethi Soltani Department of Mathematics, Faculty of Sciences of Tunis, Campus University, 1060 Tunis, Tunisia Available online 25 September 2006

Example 3 (Partial sum operator). Let  = [0, ∞[ or  = ]−∞, 0]. We define the partial sum operator S by Fα (S f ) := χ Fα (f ), where χ is the characteristic function of . Corollary 3. For all f ∈ Lp (μα ), p ∈ ]1, ∞[, there exists a constant Cp > 0 independent of f , so that S f p,α  Cp f p,α . Proof. Let 1 = [0, ∞[ and 2 = ]−∞, 0]. Then 1 S1 = (I − iHα ) 2

1 and S2 = (I + i Hα ), 2

where I is the identity operator and Hα is the transform given by Fα (Hα f )(λ) = i sgn(λ)Fα (f )(λ), DOI of original article: 10.1016/j.jfa.2003.11.009. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2006.08.007

λ ∈ R.

F. Soltani / Journal of Functional Analysis 242 (2007) 672–673

673

We put m(λ) = i sgn(λ). Now, we consider a dyadic decomposition m(λ) = with ∞    φ 2j λ , m0 (λ) = i sgn(λ)

mq (λ) = i sgn(λ)φ(λ),

∞

q=0 mq (λ/2

q ),

q  1,

j =0

where φ is the function given by (15) and is assumed to be even. Then mq ∈ S(R). Hence there exists a constant c independent of q, for which  s   1 + λ2 0 F(mq )(λ)  c, λ ∈ R and s0 > (s + 1)/2. This gives  mq H s  c

 s−2s0 1 + λ2 dλ

1/2 ,

s0 > (s + 1)/2.

R s,−1/2

Thus we deduce that m ∈ H∞

, s > α + 1. This finishes the proof.

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