## FEFFERMAN-STEIN INEQUALITIES FOR THE Z d 2 DUNKL MAXIMAL OPERATOR (2008)

### BibTeX

@MISC{Deleaval08fefferman-steininequalities,

author = {Luc Deleaval},

title = {FEFFERMAN-STEIN INEQUALITIES FOR THE Z d 2 DUNKL MAXIMAL OPERATOR},

year = {2008}

}

### OpenURL

### Abstract

In this article, we establish the Fefferman-Stein inequalities for the Dunkl maximal operator associated with a finite reflection group generated by the sign changes. Similar results are also given for a large class of operators related to Dunkl’s analysis.

### Citations

1379 |
Singular Integrals and Differentiability Properties of Functions
- Stein
- 1970
(Show Context)
Citation Context ...R κ and the covering lemma) a suitable sequence of disjoint sets Rn such that µκ(E+) � C ∑ n µκ(Rn), where C depends only on κ1, . . .,κd. We can then follow the standard techniques (see for instance =-=[13]-=-) in order to prove that µκ(E+) � C λ ‖f‖κ,1. Finally, the basic but crucial observation (3.14) M R κ f(x) = MR κ f(ε1x1, . . .,εdxd), with εj = ±1, allows us to deduce the weak-type inequality, that ... |

732 |
Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals
- Stein
- 1993
(Show Context)
Citation Context ... inequality is easy to prove. Indeed, we can show the key inequality ˜µκ(K) � C ∫ |f(y)|M λ R κ W(y)dµκ(y) R d for any compact set K in E+ just as in the proof for the classical maximal operator (see =-=[15]-=-). Therefore ˜µκ(E+) � C ∫ |f(y)|M λ R κ W(y)dµκ(y) R d and we then deduce (3.15) on account of (3.14). To conclude, we claim that we can combine the maximal theorem and the weighted inequality for MR... |

695 |
A treatise on the theory of Bessel functions
- Watson
- 1995
(Show Context)
Citation Context ... known. More precisely, it is given for both x and y in C by j=1 xy Eκ(x, y) = j 1 κ− (ixy) + 2 2κ + 1 j κ+ 1 2 (ixy), where jκ is the normalized Bessel function of the first kind and of order κ (see =-=[19]-=-). Moreover, we have a crucial one-dimensional product formula for this kernel. Before formulating it, let us introduce some notations. Notations. as well as (1) For x, y, z ∈ R, we put { 1 2xy σx,y,z... |

128 |
Differential-difference operators associated to reflection groups
- Dunkl
- 1989
(Show Context)
Citation Context ..., . . .,d}, so G is isomorphic to Zd 2. Let κ1, κ2, . . . , κd be nonnegative real numbers. Associated with these objects are the Dunkl operators Dk (for k = 1, . . .,d) which have been introduced in =-=[4]-=- by C. F. Dunkl. They are given for x ∈ Rd by d∑ f(x) − f Dkf(x) = ∂kf(x) + κj ( σj(x) ) f(x) − f 〈ek, ej〉 = ∂kf(x) + κk 〈x, ej〉 ( σk(x) ) , j=1 where ∂k denotes the usual partial derivative. A fundam... |

119 |
Topics in Harmonic Analysis Related to the Littlewood-Paley Theory
- Stein
- 1970
(Show Context)
Citation Context ...together with the maximal result for M R κ (Theorem 3.1), implies a maximal theorem for Mκ (proved in [16]) without using the Hopf-Dunford-Schwartz ergodic theorem (which is a general method given in =-=[14]-=-). 4. Application Since the Fefferman-Stein inequalities are an important tool in Harmonic analysis, we would like to define a large class of operators such that each operator of this class satisfies ... |

96 |
Some maximal inequalities
- Fefferman, Stein
- 1971
(Show Context)
Citation Context ...ts are also given for a large class of operators related to Dunkl’s analysis. hal-00333258, version 1 - 22 Oct 2008 1. Introduction In the early seventies, C. Fefferman and E. M. Stein have proved in =-=[6]-=- the following extension of the Hardy-Littlewood maximal theorem. Theorem 1.1. Let (fn)n�1 be a sequence of measurable functions defined on Rd and let M be the well-known maximal operator given by ∫ 1... |

75 |
transforms associated to finite reflection groups, Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications
- Hankel
- 1992
(Show Context)
Citation Context ...tion and the Dunkl convolution. We restrict the statement from Dunkl’s analysis to the special case considered in this article. For a large survey about this theory, the reader may especially consult =-=[3, 5, 10, 11, 16, 18]-=-. Let e1, . . . , ed be the standard basis of R d . We denote by σj (for each j from 1 to d) the reflection with respect to the hyperplane perpendicular to ej, that is to say for every x = (x1, . . .,... |

44 | Generalized Hermite polynomials and the heat equation for Dunkl operators
- Rösler
- 1998
(Show Context)
Citation Context ...ion 1 - 22 Oct 2008 Moreover, we know that τκ x (qt κ )(y) > 0 for x and y in Rd and that ∫ (3.2) τ κ x (qt κ )(y)dµκ(y) = 1 . R d For all these results (and for more details), the reader may consult =-=[8]-=- or [10]. We now turn to the proof of Lemma 3.1. Proof. One begins with the proof of the following one-dimensional equality (3.3) τ κ ∫ x (χ )(y) = χ (z)dν [−r,r] [−r,r] κ,+ x,y (z), x, y ∈ R \ {0}. R... |

44 | Positivity of Dunkl’s intertwining operator
- Rösler
- 1999
(Show Context)
Citation Context ... Pn, Vκ(1) = 1, DkVκ = Vκ∂k for k = 1, . . .,d, with Pn the subspace of homogeneous polynomials of degree n in d variables. Even if the positivity of the intertwining operator has been established in =-=[9]-=- by M. Rösler, an explicit formula of Vκ is not known in general. However, in our setting, the operator Vκ is given according to [20] by the following integral representation ∫ d∏ Vκf(x) = f(x1t1, . .... |

31 | Markov processes related with Dunkl operators
- Rösler, Voit
- 1998
(Show Context)
Citation Context ... t > 0 we have φt(x) = , with aκ = cκ d γκ+ 2 aκ t (t2 + ‖x‖2 d+1 ) γκ+ 2 = P t κ(x), 2 Γ ( γκ + d+1 2 which is the Dunkl-Poisson kernel (for more details about this kernel, the reader is referred to =-=[12]-=- and [16]). Thus, in this case, Mφ κ is the maximal function associated with the Dunkl-Poisson semigroup. We now state the Fefferman-Stein inequalities for Mφ κ (for φ, ˜ φ and φt as above). Theorem 4... |

29 |
Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transform
- Trimèche
(Show Context)
Citation Context ...tion and the Dunkl convolution. We restrict the statement from Dunkl’s analysis to the special case considered in this article. For a large survey about this theory, the reader may especially consult =-=[3, 5, 10, 11, 16, 18]-=-. Let e1, . . . , ed be the standard basis of R d . We denote by σj (for each j from 1 to d) the reflection with respect to the hyperplane perpendicular to ej, that is to say for every x = (x1, . . .,... |

25 | A positive radial product formula for the Dunkl kernel
- Rösler
(Show Context)
Citation Context ...tion and the Dunkl convolution. We restrict the statement from Dunkl’s analysis to the special case considered in this article. For a large survey about this theory, the reader may especially consult =-=[3, 5, 10, 11, 16, 18]-=-. Let e1, . . . , ed be the standard basis of R d . We denote by σj (for each j from 1 to d) the reflection with respect to the hyperplane perpendicular to ej, that is to say for every x = (x1, . . .,... |

25 | Orthogonal polynomials for a family of product weight functions on the spheres
- XU
- 1997
(Show Context)
Citation Context ...the positivity of the intertwining operator has been established in [9] by M. Rösler, an explicit formula of Vκ is not known in general. However, in our setting, the operator Vκ is given according to =-=[20]-=- by the following integral representation ∫ d∏ Vκf(x) = f(x1t1, . . .,xdtd) Mκj(1 + tj)(1 − t 2 j )κj−1 dt, 1 Γ(κj+ 2 ) [−1,1] d with Mκj = Γ(κj)Γ( 1 (where Γ is the well-known Gamma function). 2 ) In... |

23 | Convolution operator and maximal function for Dunkl transform, arXiv:math.CA/0403049, version 3
- Thangavelu, Xu
(Show Context)
Citation Context ...very λ > 0 n=1 ∥ p }) > λ � C ( ∞ ∑ 1 r λ ∥ |fn(·)| r) ∥ , One would like to extend this result to the case of the Dunkl maximal operator Mκ which is defined according to S. Thangavelu and Y. Xu (see =-=[16]-=-) by Mκf(x) = sup r>0 1 ∣ (f ∗κ χ )(x) Br µκ(Br) ∣ d , x ∈ R , ∥ 1 2000 Mathematics Subject Classification. 42B10, 42B25. Key words and phrases. Dunkl maximal operator, Dunkl transform, Fefferman-Stei... |

21 | Dunkl operators: Theory and Applications
- Rösler
- 2003
(Show Context)
Citation Context |

20 |
de Jeu, The Dunkl transform
- E
- 1993
(Show Context)
Citation Context |

16 |
Bessel-type signed hypergroups on R
- Rösler
- 1995
(Show Context)
Citation Context ... z) denotes the area of the triangle (perhaps degenerated) with sides x, y, z. With these notations in mind, we can now state the product formula for the Dunkl kernel (this formula has been proved in =-=[7]-=- in the more general setting of signed hypergroups).Fefferman-Stein inequalities for the Zd 2 Dunkl maximal operator 5 hal-00333258, version 1 - 22 Oct 2008 Proposition 2.1. Let x, y ∈ R. (1) For eve... |

9 |
Trigonometric series. 2nd ed. Vol. I
- Zygmund
- 1959
(Show Context)
Citation Context ...ator is defined for both f and g in L2 (µκ) by ∫ (f ∗κ g)(x) = cκ f(y)τ κ x (g)(−y)dµκ(y), x ∈ R d . R d Thanks to Proposition 2.3, the usual Young’s inequality holds (for the proof, see for instance =-=[21]-=-). Proposition 2.5. Assume that p −1 + q −1 = 1 + r −1 with p, q, r ∈ [1, +∞]. Then, the map (f, g) ↦→ f ∗κ g defined on L 2 (µκ) × L 2 (µκ) extends to a continuous map from L p (µκ) × L q (µκ) to L r... |

4 |
The Hardy-Littlewood maximal function for Chebli-Trimeche hypergroups. Applications of hypergroups and related measure algebras
- Bloom, Xu
- 1993
(Show Context)
Citation Context ...pendent of x and f. The last result we mention about the generalized translation is the following onedimensional inequality which has been recently proved by C. Abdelkefi and M. Sifi in [1] (see also =-=[2]-=-). Proposition 2.4. There exists a positive constant C such that for x, y ∈ R and for every r > 0 we have ∣ κ τx (χ )(y) [−r,r] ∣ ( ) µκ ] − r, r[ � C ( ) , I(x, r) where we denote by I(x, r) the foll... |

3 | Riesz transform and Riesz potentials for Dunkl transforms - Thangavelu, Xu - 412 |

1 |
Dunkl translation and uncentered maximal operator on the real line
- Abdelkefi, Sifi
(Show Context)
Citation Context ...here C is independent of x and f. The last result we mention about the generalized translation is the following onedimensional inequality which has been recently proved by C. Abdelkefi and M. Sifi in =-=[1]-=- (see also [2]). Proposition 2.4. There exists a positive constant C such that for x, y ∈ R and for every r > 0 we have ∣ κ τx (χ )(y) [−r,r] ∣ ( ) µκ ] − r, r[ � C ( ) , I(x, r) where we denote by I(... |