Dunkl transform of (β γ)-Dunkl Lipschitz functions

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Dunkl transform of ð; Þ-Dunkl Lipschitz functions

By Radouan DAHER,^ÞMustapha BOUJEDDAINE^Þand Mohamed El HAMMA^*^Þ (Communicated by Kenji FUKAYA,M.J.A., Oct. 14, 2014)

Abstract: In this paper, we obtain an analog of Younis’s Theorem 5.2 in [7] for the Dunkl transform on the real line for functions satisfying the ð; Þ-Dunkl Lipschitz condition in the spaceL^pðR;jxj^2þ1dxÞ, where ¹₂.

Key words: Dunkl operator; Dunkl transform; generalized translation operator.

1. Introduction and preliminaries. Dunkl operators provide an essential tool to extend Four- ier analysis on Euclidean spaces. These operators have been introduced in 1989, by C. Dunkl in [2], the Dunkl kernel e is used to deﬁne the Dunkl transform F which was introduced by C. Dunkl in [3]. The Dunkl transform on the real line, which enjoys properties similar to those of Fourier transform, is generalization of the Fourier transform.

Younis ([7], Theorem 5.2) characterized the set of functions in L²ðRÞ satisfying the Dini-Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely we have

Theorem 1.1 ([7], Theorem 5.2). Let f 2 L²ðRÞ. Then the following are equivalents:

1. kfð:þhÞ fð:Þk_L²_ðRÞ¼O _h

ðlog¹_hÞ

as h!0, 0< <1, >0,

2. R

jjrjFðfÞðÞj²d¼Oðr²ðlogrÞ²Þas r! þ1,

whereF stands for the Fourier transform of f.

In this paper, we obtain an analog of Theorem 1.1 for the Dunkl transform on the real line. For this purpose, we use a generalized translation operator.

Let Lp;¼L^pðR;jxj^2þ1dxÞ, where ¹₂, denote the L^p space of functions f deﬁned on R endowed with the following ﬁnite norm

kfk_p;¼ 1 2^þ1ðþ1Þ

Z

R

jfðxÞj^pjxj^2þ1dx

1=p

; where1< p2.

LetjðzÞdenote the normalized Bessel function of the ﬁrst kind of order given by

jðzÞ ¼ðþ1ÞX¹

n¼0

ð1Þⁿ n!ðnþþ1Þ

z 2

2n

; z2C;

where C denotes the complex plane. Deﬁne the Dunkl kernel e by

eðxÞ ¼jðxÞ þicxj_þ1ðxÞ; wherec¼ ð2þ2Þ¹,i¼ ffiffiffiffiffiffiffi

p1 .

We define a differential-difference Dunkl operator

DfðxÞ ¼df

dxðxÞ þ þ1 2

fðxÞ fðxÞ

x ;

f 2C¹ðRÞ:

So the functiony¼eðxÞsatisﬁes the equation Dyiy¼0with the initial conditionyð0Þ ¼1and it is the unique solution (see [4]). In the limit case with ¼ ¹₂ the Dunkl kernel coincides with the usual exponential function e^ix.

Lemma 1.2. For x2R the following in- equalities are fulﬁlled

1. jeðxÞj 1, and the equality is attained only withx¼0,

2. j1eðxÞj 2jxj,

3. j1eðxÞj c, with jxj 1, where c is a certain constant which depends only on.

Proof. See Lemma 2.9 in [1].

The Dunkl transform of order for f2Lp; is deﬁned by

FðfÞðÞ ¼ 1 2^þ1ðþ1Þ

Z

R

fðxÞeðxÞjxj^2þ1dx:

For ¼ ¹₂, F coincides with the classical Fourier transform.

The inverse Dunkl transform is deﬁned by the formula

doi: 10.3792/pjaa.90.135

#2014 The Japan Academy

2000 Mathematics Subject Classiﬁcation. Primary 46E30;

41A25; 41A17.

Þ Department of Mathematics, Faculty of Sciences A€n Chock, University of Hassan II, BP 5366 Maarif, Casablanca, Morocco.

Þ Department of Mathematics and Computer Sciences, Faculty of Sciences, Equipe d’Analyse Harmonique et Proba- bilite´s, Universite´ Moulay Isma€l, BP 11201 Zitoune, Mekne`s, Morocco.

No. 9] Proc. Japan Acad.,90, Ser. A (2014) 135

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fðxÞ ¼ 1 2^þ1ðþ1Þ

Z

R

FðfÞðÞeðxÞjj^2þ1d:

By Plancherel’s theorem and Marcinkiewics interpolation theorem (see [5]) we get for f 2Lp;

with1< p2and qsuch that ¹_pþ¹_q¼1, kFðfÞk_q;C₁kfk_p;; ð1Þ

whereC₁ is a positive constant.

K. Trime`che introduced in [6] the generalized translation operator Th, on Lp;. For f 2Lp; we have

FðT_hfÞðÞ ¼eðhÞFðfÞðÞ: ð2Þ

2. Main results. In this section we give the main results of this paper. We need ﬁrst to deﬁne ð; Þ-Dunkl Lipschitz class.

Deﬁnition 2.1. A function f2L_p; is said to be in theð; Þ-Dunkl Lipschitz class, denoted by DLipðp; ; Þ; if

kThfð:Þ fð:Þk_p;

¼O h ðlog¹_hÞ

!

ash!0; ; >0:

Theorem 2.2. Let fðxÞ belong to DLipðp; ; Þ. Then

Z

jjr

jFðfÞðÞj^qjj^2þ1d

¼Oðr^qðlogrÞ^qÞasr! þ1;

where ¹_pþ¹_q¼1.

Proof. Letf2DLipðp; ; Þ. Then kT_hfð:Þ fð:Þk_p;¼O h

ðlog_h¹Þ

!

ash!0:

If jj 2 ½¹_h;²_h then jhj 1 and (3) of Lemma 1.2 implies that

1 1

c^qj1eðhÞj^q: Therefore

Z

1=hjj2=h

1 c^q

Z

1=hjj2=h

j1eðhÞj^qjFðfÞðÞj^qjj^2þ1d 1

c^q Z

R

j1eðhÞj^qjFðfÞðÞj^qjj^2þ1d:

From formulas (1) and (2) we have

Z

R

j1eðhÞj^qjFðfÞðÞj^qjj^2þ1d C^q₁kThfð:Þ fð:Þk^q_p;:

Then there exists a positive constant K such that

Z

1=hjj2=h

jFðfÞðÞj^qjj^2þ1dK h^q ðlog_h¹Þ^q: We obtain

Z

rjj2r

jFðfÞðÞj^qjj^2þ1dK r^q ðlogrÞ^q: So that

Z

jjr

¼ Z

rjj2r

þ Z

2rjj4r

þ Z

4rjj8r

þ

" #

jFðfÞðÞj^qjj^2þ1d K r^q

ðlogrÞ^q þK ð2rÞ^q

ðlog 2rÞ^qþK ð4rÞ^q ðlog 4rÞ^q þ K r^q

ðlogrÞ^q þK ð2rÞ^q

ðlogrÞ^q þK ð4rÞ^q ðlogrÞ^qþ K r^q

ðlogrÞ^q½1þ2^qþ ð2^qÞ²þ ð2^qÞ³þ CK r^q

ðlogrÞ^q; whereC¼ ð12^qÞ¹.

Finally, we get Z

jjr

¼O r^q ðlogrÞ^q

as r! þ1:

Thus, the proof is ﬁnished.

Deﬁnition 2.3. A function f 2L_p; is said to be in theðp; ; Þ-Dini Lipschitz Dunkl, denoted by DLipðp; ; Þ; if

kThfð:Þ fð:Þk_p;¼O ðhÞ ðlog_h¹Þ

!

ash!0; >0;

where

1. ðtÞ a continuous increasing function on

½0;1Þ, 2. ð0Þ ¼0,

3. ðtsÞ ¼ ðtÞ ðsÞfor allt; s2 ½0;1Þ.

136 R. DAHER, M. BOUJEDDAINEand M. El HAMMA [Vol. 90(A),

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Theorem 2.4. Let fðxÞ belong to DLipðp; ; Þ. Then

Z

jjr

¼Oð ðr^qÞðlogrÞ^qÞasr! þ1; where ¹_pþ¹_q¼1.

Proof. Letf2DLipðp; ; Þ. Then we have kThfð:Þ fð:Þk_p;¼O ðhÞ

ðlog¹_hÞ

!

as h!0:

If jj 2 ½_h¹;_h² then jhj 1 and from (3) of Lemma 1.2, we obtain

1 1

c^qj1eðhÞj^q: Z Then

1=hjj2=h

1 c^q

Z

1=hjj2=h

j1eðhÞj^qjFðfÞðÞj^qjj^2þ1d 1

c^q Z

R

j1eðhÞj^qjFðfÞðÞj^qjj^2þ1d KkThfð:Þ fð:Þk^q_p;

¼O ð ðhÞÞ^q ðlog¹_hÞ^q

!

¼O ðh^qÞ ðlog¹_hÞ^q

!

;

whereKis a positive constant.

We obtain Z

rjj2r

jFðfÞðÞj^qjj^2þ1d¼O ðr^qÞ ðlogrÞ^q

: Then there exists a positive constant C such that

Z

rjj2r

jFðfÞðÞj^qjj^2þ1dC ðr^qÞ ðlogrÞ^q: So that

Z

jjr

¼ Z

rjj2r

þ Z

2rjj4r

þ Z

4rjj8r

þ

" #

jFðfÞðÞj^qjj^2þ1d C ðr^qÞ

ðlogrÞ^q þC ðð2rÞ^qÞ

ðlog 2rÞ^q þC ðð4rÞ^qÞ ðlog 4rÞ^q þ C ðr^qÞ

ðlogrÞ^q þC ðð2rÞ^qÞ

ðlogrÞ^q þC ðð4rÞ^qÞ ðlogrÞ^q þ C ðr^qÞ

ðlogrÞ^q½1þ ð2^qÞ þ ²ð2^qÞ þ ³ð2^qÞ þ : Therefore

Z

jjr

jFðfÞðÞj^qjj^2þ1dC ðr^qÞ ðlogrÞ^q; whereC ¼Cð1 ð2^qÞÞ¹ since ð2^qÞ<1.

Finally, we get Z

jjr

¼O ðr^qÞ ðlogrÞ^q

as r! þ1:

and this ends the proof.

References

[ 1 ] E. S. Belkina and S. S. Platonov, Equivalence of K-functionals and moduli of smoothness con- structed by generalized Dunkl translations, Izv.

Vyssh. Uchebn. Zaved. Mat.2008, no. 8, 3–15;

translation in Russian Math. (Iz. VUZ) 52 (2008), no. 8, 1–11.

[ 2 ] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer.

Math. Soc.311(1989), no. 1, 167–183.

[ 3 ] C. F. Dunkl, Hankel transforms associated to ﬁnite reﬂection groups, in Hypergeometric functions on domains of positivity, Jack poly- nomials, and applications (Tampa, FL,1991), 123–138, Contemp. Math., 138, Amer. Math.

Soc., Providence, RI, 1992.

[ 4 ] C. F. Dunkl, Integral kernels with reﬂection group invariance, Canad. J. Math.43 (1991), no. 6, 1213–1227.

[ 5 ] E. C. Titchmarsh, Introduction to the theory of Fourier Integrals, Clarendon Press, Oxford, 1937.

[ 6 ] K. Trime`che, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), no. 1, 17–38.

[ 7 ] M. S. Younis, Fourier transforms of Dini-Lipschitz functions, Internat. J. Math. Math. Sci. 9 (1986), no. 2, 301–312.

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