Growth properties of the Dunkl transform on the real line

Growth Properties of the Dunkl Transform on the Real Line

Mohamed El Hamma

Faculty of Sciences Ain Ckock, Casablanca, Morocco m elhamma@yahoo.fr

Radouan Daher

Faculty of Sciences Ain Ckock, Casablanca, Morocco r daher@yahoo.fr

Abstract

Using a generalized Dunkl translation, we obtain an analog of Titch- marsh’s Theorem for the Dunkl transform on the real line.

Mathematics Subject Classification: 44A15; 44A35; 33C52

Keywords: Dunkl operator, Dunkl transform, generalized Dunkl transla- tion

1 Introduction and preliminaries

Dunkl operators are differential-difference operators introduced in 1989, by Dunkl [2]. On the real line, these operators, which are denoted by D_α, depend on a real parameter α >−¹₂.

Forα >−¹₂, Dunkl kerneleαis defined as the unique solution of a differential- difference equation related to D_α and satisfying eα(0) = 0. This kernel is used to defined Dunkl transform which was introduced by Dunkl in [3].

Titchmarsh [5, Theorem 84] characterized of functions in L^p(R), (1 < p≤ 2) satisfying on estimate growth of the norm of their Fourier transform, namely we have

Theorem 1.1 Let f(x)belong to L^p(R) (1< p≤2), and let

_∞

−∞|f(x+h)−f(x−h)|^pdx=O(h^αp); 0< α≤1 as h−→0.

Then F(x) belongs to L^β for p

p+αp−1 < β≤ p p−1 where F stands for the Fourier transform of f.

The main result is the proof an analog of Theorem 1.1, using the generalized Dunkl translations.

The Dunkl operator is a differential-difference operator D_α D_αf(x) = df(x)

dx + (α+1

2)f(x)−f(−x)

x , α >−1 2, where f ∈L^p(R,|α|^2α+1dx), 1< p≤2).

Letjα(x) is a normalized Bessel function of the first kind, i.e., jα(x) = 2^αΓ(α+ 1)Jα(x)

x^α ,

where Jα(x) is a Bessel function of the first kind ([1]).

The function jα(x) is infinitely differentiable and is defined also by jα(z) = Γ(α+ 1)

∞

n=0

(−1)ⁿ(^z₂)²ⁿ

n!Γ(n+α+ 1), z ∈C (1) Moreover, from (1) we see that

z−→0lim

jα(z)−1 z² = 0 by consequence, there exist c > 0 andη >0 satisfying

|z| ≤η=⇒ |j_α(z)−1| ≥c|z|². (2) Let kernel Dunkl function is defined by the formula

eα(x) = jα(x) +i(2α+ 2)⁻¹xjα+1(x). (3) The function y = eα(x) satisfies the equation D_αy = iy with the initial conditiony(0) = 1. In the limit case withα=−¹₂ the kernel function coinsides with the usual exponential function e^ix.

We use the formula (3), then

|1−jα(hx)| ≤ |1−eα(hx)| (4) The Dunkl transform is defined by

f(λ) =

_+∞

−∞ f(x)eα(λx)|x|^2α+1dx, λ∈R The inverse Dunkl transform is defined by the formula

f(x) = 1

(2^α+1Γ(α+ 1)²

_+∞

−∞

f(λ)eα(−λx)|λ|^2α+1dλ.

Plancherel’s theorem and the Marcinkiewicz interpolation theorem (see [5]) we get for f ∈L^p(R,|x|^2α+1dx) with 1< p≤2 and q such that ¹_p +¹_q = 1,

fq ≤C1fp, (5)

where C1 is a positive constant and fp =

_∞

−∞|f(x)|^p|x|^2α+1dx _1/p

.

K. Trim´eche has introduced in [6] the Dunkl translation operators τh, h∈ R, in [4], we have

τhf(x) =eα(xh)f(x). (6)

2 Main Result

Theorem 2.1 Let f(x)∈L^p(R,|x|^2α+1dx), and let

_∞

−∞|τhf(x)−f(x)|^p|x|^2α+1dx=O(h^γp); as, h−→0 Then f∈L^β(R,|x|^2α+1dx) for

2pα+ 2p

2p+ 2α(p−1) +γp−2 < β ≤ p p−1

Proof: From formula (6) the Dunkl transform ofτhf(x)−f(x) is (eα(hx)− 1)f(x).

By formulas (5) and (6), we obtain

_∞

−∞|1−eα(hx)|^q|f(x)|^q|x|^2α+1dx ≤ C1

_∞

−∞|τ_hf(x)−f(x)|^p|x|^2α+1dx _1/p−1

≤ C1h^γq. From formulas (2) and (4), we have

_η/h

0 |hx|^2q|f(x)|^q|x|^2α+1dx≤Kh^γq, where K is a positive constant.

Therefore

_η/h

0 x^2q|f(x)|^q|x|^2α+1dx=O(h^(γ−2)q). Let

φ(t) =

_t

1 |x²f(x)|^βx^(2α+1)q/βdx.

Then, if β < q, by H¨older inequality we have

φ(t) ≤

_t

1 |x²f(x)|^qx^2α+1dx

_{β/q t}

1 dx _1−β/q

= O(t^(2−γ)qβqt^1−β/q)

= O(t2β−γβ+1−β/q)

= O(t^{1−γβ+β(p+1p )}). Hence

_t

1 |f(x)|^βx^2α+1dx =

_t

1 x^{−2β−(2α+1)βq}φ(x)x^2α+1dx

= t^{−2β−(2α+1)βq}t^2α+1φ(t) + (2β+ (2α+ 1)β

q −(2α+ 1))

_t

1 x^{−2β−(2α+1)βq+2α}φ(x)dx

= O(t^{−2β−(2α+1)βq}+2α+1+1−γβ+β(^p+1_p )) +O(

_t

1 x^{−2β−(2α+1)βq+2α}x^{1−γβ+β(p+1p} dx)

= O(t^{−2β−(2α+1)βq}+2α+2−γβ+β(^p+1_p )).

and this bounded as t −→ ∞if −2β−(2α+ 1)^β_q + 2α+ 2−γβ+β(^p+1_p )<0, i.e. if

β > 2pα+ 2p

2p+ 2α(p−1) +γp−2.

Similarly for the integral over (−t,−1). This proves the theorem.

References

[1] H. Bateman and A.Erd´elyi, Higher Transcendental Functions, Vol II(Mc Graw-Hill, New York, 1953; Nauka, Moscow, 1974).

[2] C.F. Dunkl, Differential-difference operators associated to reflexion groups, Trans. Amer. Math. Soc., 311 (1) (1989), 167-183.

[3] C.F. Dunkl, Hankel transforms associated to finite reflection groups inProceedings of Special Session on Hypergeometric Functions on Domains of Positivity, Jack Polynomials and Applications, Proceedings, Tampa 1991, Contemp. Math. 138(1992) 123-138.

[4] F. Soltani, L^p Fourier multiplies for the Dunkl operator on the real line, J. Funct. Anal. 209 (2004) 16-35.

[5] E.C.Titchmarsh, Introduction to the theory of Fourier Integrals , Oxford University Press, Amen House, London, E.C.4, 1948.

[6] K. Trim´eche, Pley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), 17-38.

Received: January , 2012