Dunkl Translation and Uncentered Maximal Operator on the Real Line

(1)

Volume 2007, Article ID 87808,9pages doi:10.1155/2007/87808

Research Article

Dunkl Translation and Uncentered Maximal Operator on the Real Line

Chokri Abdelkefi and Mohamed Sifi

Received 22 November 2006; Accepted 5 July 2007 Recommended by Ahmed Zayed

We establish estimates of the Dunkl translation of the characteristic functionχ_[₋_ε,ε],ε >0, and we prove that the uncentered maximal operator associated with the Dunkl operator is of weak type (1, 1). As a consequence, we obtain theL^p-boundedness of this operator for 1< p≤+∞.

Copyright © 2007 C. Abdelkefi and M. Sifi. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl [1] and are denoted byΛα, whereα is a real parameter>−1/2. These operators are associated with the reflection groupZ2onR. The Dunkl kernelEαis used to define the Dunkl transformᏲαwhich was introduced by Dunkl in [2]. Rösler in [3]

shows that the Dunkl kernels verify a product formula. This allows us to define the Dunkl translationτx,x∈R. As a result, we have the Dunkl convolution.

The Hardy-Littlewood maximal function was first introduced by Hardy and Little- wood in 1930 for functions defined on the circle (see [4]). Later it was extended to various Lie groups, symmetric spaces, some weighted measure spaces (see [5–10]), and diﬀerent hypergroups (see [11–14]).

In this paper, we establish an estimate of the Dunkl translation of the characteristic functionτx(χ[−ε,ε])(y), x,y∈R,x=0, based on the inversion formula which extends some results of [11] to the Dunkl operator onR, and we prove the weak type (1, 1) of the uncentered maximal operatorMdefined for each integrable function f on (R,dμ_α) by

M(f)(x)= sup

ε>0,|z|∈B(x,ε)

1 μα(]−ε,ε[)

_ε

−ετz(f)(−y)dμα(y), x∈R, (1.1)

(2)

whereB(x,ε) is the interval [max{0,|x| −ε},|x|+ε[ andμαis a weighted Lebesgue measure onR(seeSection 2). Finally, we obtain for 1< p≤+∞theL^p-boundedness ofM. In the casez=x, these results are already proved onR^din [9] by using the maximal function associated to the Poisson semigroup.

The contents of this paper are as follows.

In Section 2, we collect some basic definitions and results about harmonic analysis associated with Dunkl operator.

In Section 3, we establish estimates of τx(χ[₋ε,ε])(y),x,y∈R, x=0, and we prove the weak type (1, 1) of the uncentered maximal operatorMand theL^p-boundedness for 1< p≤+∞ofM.

In the sequel,crepresents a suitable positive constant which is not necessarily the same in each occurrence. Furthermore, we denote by

(i)Ᏹ(R) the space ofC^∞-functions onR,

(ii)D_∗(R) the space of even functions inᏱ(R) with compact support, (iii)S_∗(R) the space of even functions inᏱ(R) decreasing rapidly.

2. Preliminaries

For a real parameterα >−1/2, we consider the diﬀerential-diﬀerence operator defined by Λα(f)(x)=df

dx(x) +2α+ 1 x

f(x)−f(−x) 2

, f ∈Ᏹ(R), (2.1) called Dunkl operator.

Forλ∈C, the initial problem

Λα(f)(x)=λ f(x), f(0)=1,x∈R, (2.2) has a unique solutionEα(λ) called Dunkl kernel and given by

E_α(λx)=j_α(iλx) + λx

2(α+ 1)j_α+1(iλx), x∈R, (2.3) wherejαis the normalized Bessel function of the first kind and orderα, defined by

jα(λx)=

⎧⎪

⎨

⎪⎩

2^αΓ(α+ 1)Jα(λx)

(λx)^α ifλx=0,

1 ifλx=0,

(2.4) whereJ_αis the Bessel function of first kind and orderα(see [15]).

We have for allx∈Rthat

the functionλ−→j_α(λx) is even onR,

E_α(−iλx)≤1. (2.5)

LetAαbe the function defined onRby A_α(x)= |x_|^2α+1

2^α+1Γ(α+ 1), x∈R, (2.6)

(3)

and letμαbe the weighted Lebesgue measure onRgiven by

dμα(x)=Aα(x)dx. (2.7)

For every 1≤p≤+∞, we denote byL^p(μ_α) the spaceL^p(R,dμ_α) and we use·p,αas a shorthand for · L^p(μα).

The Dunkl transformᏲαwhich was introduced by Dunkl in [2] is defined for f ∈ L¹(μ_α) by

Ᏺα(f)(x)=

REα(−ixy)f(y)dμα(y), x∈R. (2.8) According to [16], we have the following results:

(i) for all f ∈L¹(μ_α), we haveᏲα(f)∞,α≤ f1,α;

(ii) for all f ∈L¹(μα) such thatᏲα(f)∈L¹(μα), we have the inversion formula f(x)=

RE_α(iλx)Ᏺα(f)(λ)dμ_α(λ), a.ex∈R; (2.9) (iii) for every f ∈L²(μ_α), we have

Ᏺα(f)_2,α= f2,α. (2.10)

In the sequel, we consider the signed measureγx,yonRgiven by

dγ_x,y(z)₌

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

W_α(x,y,z)dμ_α(z) ifx,y∈R\{0}, dδx(z) ify=0, dδy(z) ifx=0,

(2.11)

whereWα(see [3]) is an even function satisfying the following properties:

W_α(x,y,z)=W_α(y,x,z)=W_α(−x,z,y)=W_α(−z,y,−x),

R

W_α(x,y,z)dμ_α(z)≤4. (2.12)

We have

suppγ_x,y=S_x,y∪

−S_x,y withS_x,y=|x| − |y|,|x|+|y|

. (2.13)

Forx,y∈Rand f a continuous function onR, the Dunkl translation operatorτxgiven by

τ_x(f)(y)=

Rf(z)dγ_x,y(z) (2.14)

(4)

satisfies the following properties (see [17]):

(i)τxis a continuous linear operator fromᏱ(R) into itself;

(ii) for all f ∈Ᏹ(R), we have

τx(f)(y)=τy(f)(x), τ0(f)(x)=f(x). (2.15) The Dunkl convolution f ∗αg, of two continuous functions f andgonRwith compact support, is defined by

f ∗αg(x)=

Rτx(f)(−y)g(y)dμα(y), x∈R. (2.16) The convolution∗αis associative and commutative (see [3]). The following results are shown in [18].

(i) For all x∈R, the operatorτx extends to L^p(μα), p≥1, and we have for f ∈ L^p(μ_α) that

τx(f)_p,α≤4fp,α. (2.17) (ii) For allx,λ∈Rand f ∈L¹(μ_α), we have

Ᏺα

τx(f)(λ)=Eα(iλx)Ᏺα(f)(λ). (2.18) (iii) Assume that p,q,r∈[1, +∞[ satisfyies 1/ p+ 1/q=1 + 1/r (the Young condi- tion). Then, the map (f,g)→ f ∗αg defined onCc(R)×Cc(R) extends to a continuous map fromL^p(μα)×L^q(μα) toL^r(μα), and we have

f∗αg_r,α≤4fp,αgq,α. (2.19) (iv) For all f ∈L¹(μα) andg∈L²(μα), we have

Ᏺα

f ∗αg=Ᏺα(f)Ᏺα(g). (2.20) 3. Estimates for Dunkl translation and weak type (1, 1) of

the uncentered maximal operator

In this section, we establish estimates ofτx(χ[₋ε,ε])(y),x,y∈R,x=0, whereχ[₋ε,ε]is the characteristic function of the interval [−ε,ε], and we prove the weak-type (1, 1) of the uncentered maximal operatorMand theL^p-boundedness for 1< p≤+∞ofM.

We observe that forx,y∈R\{0}andε >0, τ_xχ[−ε,ε]

(y)≤cτ˘_|_x_|χ_[0,ε]|y|

, (3.1)

where for a continuous function f on [0, +∞[ andr,s >0, ˘τr denotes the translation of the Bessel hypergroup given by

˘

τ_r(f)(s)=2²⁻^αΓ(α+ 1)²

√πΓ(α+ 1/2) +∞

0 f(z)Δα(r,s,t)dμ_α(t) (3.2)

(5)

with

Δα(r,s,t)=

⎧⎪

⎪⎨

⎪⎪

⎩

(r+s)²−t²t²−(r−s)²^α⁻^1/2

(rst)^2α if|r−s|< t < r+s,

0 otherwise.

(3.3) On the other hand, we have from (2.3), (2.5), and (2.8) that

Ᏺα

χ[₋ε,ε]

(λ)≤ ε

α+ 1Aα(ε) forε >0,λ∈R, (3.4) and by (2.4),

Ᏺα

χ[−ε,ε]

(λ)≤cε^α+1/2λ⁻^α⁻^3/2 forλ∈[ε⁻¹, +∞[. (3.5) Then, using (3.4), (3.5), and the fact that|E_α(iλx)| ≤c(A_α(x))⁻^1/2|λ|⁻^α⁻^1/2, for|x|>2ε, λ∈R\{0}, the next lemma follows closely the argumentations of [11, Proposition 4.6 and Lemma 5.1].

Lemma 3.1. There exists a positive constantcsuch that for anyx,y∈R,x=0, andε >0, one has

τx χ[₋ε,ε]

(y)≤cAα ε

Aα(x). (3.6)

Notation 3.2. For x∈Randε >0, we denote byB(x,ε) the interval [max{0,|x| −ε},

|x|+ε[.

Lemma 3.3. There exists a positive constantcsuch that for anyx,y∈Randε >0, one has τx

χ[₋ε,ε]

(y)≤cμ_α]−ε,ε[

μα

B(x,ε). (3.7)

Proof. On the one hand, we have for|x| ≤εthat μα

B(x,ε)=

B(x,ε)dμα(y)= _|_x_|_+ε

0 dμα(y)≤cμα

]−ε,ε[, (3.8)

since

1 4τx

χ[₋ε,ε]

(−y)≤1, x,y∈R, (3.9)

then we obtain (3.7) for|x| ≤ε.

On the other hand, we have for|x|> ε, μα

B(x,ε)= _|_x_|_+ε

|x|−εdμα(y)≤c|x|+ε^2α+1 _|_x_|_+ε

|x|−εd y

≤cμα

]−ε,ε[A_α(x) Aα(ε).

(3.10)

Then by (3.6), we obtain (3.7) for|x|> ε, which proves the result.

(6)

According to [7, Lemma 1.6] (see also [11, Lemma 4.21]), we have the following Vitali covering lemma.

Lemma 3.4. LetEbe a measurable subset ofR+(with respect toμ_α) which is covered by the union of a family of bounded intervals{B_j}, whereB_j=B(x_j,r_j). Then from this family, one can select a disjoints subsequence,B1,B2,. . .,Bh,. . ., (which may be finite) such that

h

μ_αB_h≥cμ_α(E). (3.11)

Theorem 3.5. The uncentered maximal operatorMis of weak type (1, 1).

Proof. Forε >0,x∈R,|z| ∈B(x,ε), and f ∈L¹(μα), we have _ε

−ετz(f)(−y)dμα(y)=

f∗αχ[₋ε,ε]

(z)=

Rf(y)τz

χ[₋ε,ε]

(−y)dμα(y), (3.12)

then using (2.13), (2.14), and (3.7), we obtain

_ε

−ετz(f)(−y)dμα(y)| ≤

|y|∈B(z,ε)

τz

χ[−ε,ε]

(−y)f(y)dμα(y)

≤c

|y|∈B(z,ε)

f(y)dμα(y)

μ_α]−ε,ε[

μα

B(z,ε),

(3.13)

hence we deduce that

M(f)(x)_≤cM( f)(x), (3.14)

whereM(f) is defined by M( f)(x)= sup

ε>0,|z|∈B(x,ε)

1 μα

B(z,ε)

|y|∈B(z,ε)

f(y)dμα(y). (3.15)

Observe that we have

M( f)(−x)=M( f)(x), x∈R. (3.16) Forλ >0, put

Eλ=

x∈R;M( f)(x)> λ, E⁺_λ =

x∈R+;M( f)(x)> λ, E⁻_λ =

x∈R^∗₋;M( f)(x)> λ.

(3.17)

By (3.16) we obtain

μα E⁺_λ=μα E⁻_λ, μα Eλ

=2μα E⁺_λ. (3.18)

(7)

Now, for eachx∈E_λ⁺, there existε >0 andz∈Rsuch that

|z| ∈B(x,ε),

|y|∈B(z,ε)

f(y)dμ_α(y)> λμ_αB(z,ε). (3.19)

Furthermore, note thatx∈B(z,ε), then whenxruns through the setE⁺_λ, the union of the correspondingB(z,ε) coversE⁺_λ. Thus, usingLemma 3.4, we can select a disjoint subse- quenceB(z1,ε1),. . .,B(zh,εh),. . ., (which may be finite) such that

h

μ_αBz_h,ε_h≥cμ_α E⁺_λ. (3.20)

We have

|y|∈

hB(zh,εh)

f(y)dμα(y)≥

h

|y|∈B(zh,εh)

f(y)dμα(y). (3.21)

Applying (3.19) and (3.20) to each of the mutually disjoint intervals, we get

|y|∈

hB(zh,εh)

f(y)dμ_α(y)> λ

h

μ_αBz_h,ε_h≥λcμ_α E_λ⁺. (3.22)

But since the first member of this inequality is majorized byf1,α, we obtain μ_α E⁺_λ≤cf1,α

λ , (3.23)

and by (3.18), we deduce that

μ_α E_λ≤cf1,α

λ , (3.24)

which gives thatMis of weak type (1, 1), and hence from (3.14), the same is true forM.

As consequence ofTheorem 3.5, we obtain the following corollary.

Corollary 3.6. If 1< p≤+∞and f ∈L^p(μα), then one has M(f)∈L^pμα

, M(f)_p,α≤cfp,α. (3.25)

Proof. Using theTheorem 3.5, [15, Corollary 21.72], and proceeding in the same manner as in the proof on [2, 1.3.Theorem 1], we obtain the desired results.

(8)

Acknowledgments

The authors thank the referees for their remarks and suggestions. The authors are sup- ported by the DGRST research project 04/UR/15-02.

References

[1] C. F. Dunkl, “Diﬀerential-diﬀerence operators associated to reflection groups,” Transactions of the American Mathematical Society, vol. 311, no. 1, pp. 167–183, 1989.

[2] C. F. Dunkl, “Hankel transforms associated to finite reflection groups,” in Hypergeometric Func- tions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, Fla, 1991), vol. 138 of Contemporary Mathematics, pp. 123–138, American Mathematical Society, Providence, RI, USA, 1992.

[3] M. R¨osler, “Bessel-type signed hypergroups onR,” in Probability Measures on Groups and Re- lated Structures, XI (Oberwolfach, 1994), H. Heyer and A. Mukherjea, Eds., pp. 292–304, World Scientific, River Edge, NJ, USA, 1995.

[4] G. H. Hardy and J. E. Littlewood, “A maximal theorem with function-theoretic applications,”

Acta Mathematica, vol. 54, no. 1, pp. 81–116, 1930.

[5] J. L. Clerc and E. M. Stein, “L^p-multipliers for noncompact symmetric spaces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 71, no. 10, pp. 3911–3912, 1974.

[6] G. Gaudry, S. Giulini, A. Hulanicki, and A. M. Mantero, “Hardy-Littlewood maximal functions on some solvable Lie groups,” Journal of the Australian Mathematical Society. Series A, vol. 45, no. 1, pp. 78–82, 1988.

[7] E. M. Stein, Singular Integrals and Diﬀerentiability Properties of Functions, Princeton Mathemat- ical Series, no. 30, Princeton University Press, Princeton, NJ, USA, 1970.

[8] J.-O. Str¨omberg, “Weak typeL¹ estimates for maximal functions on noncompact symmetric spaces,” Annals of Mathematics, vol. 114, no. 1, pp. 115–126, 1981.

[9] S. Thangavelu and Y. Xu, “Convolution operator and maximal function for the Dunkl trans- form,” Journal d’Analyse Math´ematique, vol. 97, pp. 25–55, 2005.

[10] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123 of Pure and Applied Math- ematics, Academic Press, Orlando, Fla, USA, 1986.

[11] W. R. Bloom and Z. Fu Xu, “The Hardy-Littlewood maximal function for Ch´ebli-Trim`eche hy- pergroups,” in Applications of Hypergroups and Related Measure Algebras (Seattle, Wash, 1993), vol. 183 of Contemporary Mathematics, pp. 45–70, American Mathematical Society, Providence, RI, USA, 1995.

[12] W. C. Connett and A. L. Schwartz, “The Littlewood-Paley theory for Jacobi expansions,” Trans- actions of the American Mathematical Society, vol. 251, pp. 219–234, 1979.

[13] W. C. Connett and A. L. Schwartz, “A Hardy-Littlewood maximal inequality for Jacobi type hypergroups,” Proceedings of the American Mathematical Society, vol. 107, no. 1, pp. 137–143, 1989.

[14] K. Stempak, “La théorie de Littlewood-Paley pour la transformation de Fourier-Bessel,” Comptes Rendus des Séances de l’Académie des Sciences, vol. 303, no. 1, pp. 15–18, 1986.

[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cam- bridge, UK, 1966.

[16] M. F. E. de Jeu, “The Dunkl transform,” Inventiones Mathematicae, vol. 113, no. 1, pp. 147–162, 1993.

[17] M. Ali Mourou, “Transmutation operators associated with a Dunkl type diﬀerential-diﬀerence operator on the real line and certain of their applications,” Integral Transforms and Special Func- tions, vol. 12, no. 1, pp. 77–88, 2001.

(9)

[18] F. Soltani, “L^p-Fourier multipliers for the Dunkl operator on the real line,” Journal of Functional Analysis, vol. 209, no. 1, pp. 16–35, 2004.

[19] E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer, NewYork, NY, USA, 1965.

Chokri Abdelkefi: Department of Mathematics, Preparatory Institute for Engineering Studies of Tunis, Monfleury, Tunis 1089, Tunisia

Email address:[email protected]

Mohamed Sifi: Department of Mathematics, Faculty of Sciences of Tunis, Tunis-El Manar University, Tunis 1060, Tunisia

Email address:[email protected]