Convolution operator and maximal function for the Dunkl transform

For a family of weight functionsh _K invariant under a finite reflection group onR ^d, analysis related to the Dunkl transform is carried out for the weightedL ^p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence. ST wishes to thank YX for the warm hospitality during his stay in Eugene. The work of YX was supported in part by the National Science Foundation under Grant DMS-0201669.     

Weighted Function Spaces and Dunkl Transform

We introduce first weighted function spaces on ${\mathbb{R}^d}$ using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on ${\mathbb{R}^d}$ weighted L ^p -estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L ¹ when this function belongs to a suitable Besov-Dunkl space.     

Linear and Bilinear Multiplier Operators for the Dunkl Transform

The main purpose of this article is to study the L _p -boundedness of linear and bilinear multiplier operators for the Dunkl transform in the one dimensional case.     

Relating transplantation and multipliers for Dunkl and Hankel transforms

By expressing the Dunkl transform of order α of a function f in terms of the Hankel transforms of orders α and α + 1 of even and odd parts of f, respectively, we show that a considerable part of harmonic analysis of the Dunkl transform on the real line may be reduced to known results for the Hankel transform. In particular, defining the modified Dunkl transform and then considering the Dunkl transplantation operator we transfer known multiplier results for the Hankel transform to the Dunkl transform setting. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Weyl transform and bounded operators on Lp(Rn)

A multiplier theorem for the Weyl transform is proved. This theorem is used to derive sufficient conditions for the boundedness of a general operator on L^p($\text{R}$ⁿ). An application to multipliers of the Hermite expansion is given.     

Un analogue d'un théorème de Hardy pour la transformation de DunklAn analogue of Hardy's theorem for the Dunkl transform

In this Note we give a generalization of Hardy's theorem for the Dunkl transform $\text{F}{}_{\mathrm{D}}$ on $\text{R}{}^{\mathrm{d}}$. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that and , where $\text{L}{}_{\mathrm{k}}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ are the L^p spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimèche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.     

Some problems of approximation theory in the spacesL p on the line with power weight

In the spaces L _p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.     

Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms

For all d?2 and p∈(1,max(2,(d+1)/2)], we prove sharp Lp to Lp(Lq) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((logt)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d?2 and p∈(1,2]. As corollaries, we prove Lp bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in Rd.     

Annihilating sets for the short time Fourier transform

We obtain a class of subsets of R2d such that the support of the short time Fourier transform (STFT) of a signal f∈L²(Rd) with respect to a window g∈L²(Rd) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L²-norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.     

Cardinal hermite interpolation with box splines

The study of cardinal interpolation (CIP) by the span of the lattice translates of a box spline has met with limited success. Only the case of interpolation with the box spline determined by the three directionsd ₁=(1, 0),d ₂=(0, 1), andd ₃=(1, 1) inR ² has been treated in full generality [2]. In the case ofR ^d,d ≥ 3, the directions that define the box spline must satisfy a certain determinant condition [6], [9]. If the directions occur with even multiplicities, then this condition is also sufficient. For Hermite interpolation (CHIP) both even multiplicities and the determinant condition for the directions does not prevent the linear dependence of the basis functions. This leads to singularities in the characteristic multiplier when using the standard Fourier transform method. In the case of derivatives in one direction, these singularities can be removed and a set of fundamental splines can be given. This gives the existence of a solution to CHIP inL ^p (R ^d) for data inl ^p (Z ^d), 1≤p≤2.     

An analogue of the Riesz-Haviland theorem for the truncated moment problem

Let β≡β(2n)={βi}_|i|?2n denote a d-dimensional real multisequence, let K denote a closed subset of Rd, and let . Corresponding to β, the Riesz functionalL≡Lβ:P2n→R is defined by L(∑aixi):=∑aiβi. We say that L is K-positive if whenever p∈P2n and pK_|?0, then L(p)?0. We prove that β admits a K-representing measure if and only if Lβ admits a K-positive linear extension . This provides a generalization (from the full moment problem to the truncated moment problem) of the Riesz-Haviland theorem. We also show that a semialgebraic set solves the truncated moment problem in terms of natural “degree-bounded” positivity conditions if and only if each polynomial strictly positive on that set admits a degree-bounded weighted sum-of-squares representation.     

Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(Xd) to Lp(Xd) for all d/(d+α)<p?∞ and, consequently, is of weak type (1,1), where 0<α?1 is depending only on θ and X=R or X=T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f∈L₁(Xd) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(Xd) and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.     

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.     

Mean convergence of Fourier-Dunkl series

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the Ap classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions.     

Separable anisotropic differential operators in weighted abstract spaces and applications

In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev-Lions type spaces are discussed when E₀, E are two Banach spaces and E₀ is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E₀. These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established.     

L-Fourier multipliers for the Dunkl operator on the real line

We consider Fourier multipliers for L^p associated with the Dunkl operator on and establish a version of Hörmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials.     

Bounds for the weighted Lp discrepancy and tractability of integration

Quite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted L^p discrepancy of points in the d-dimensional unit cube for a sequence γ=(γ₁,γ₂,…) of weights. In this paper we prove a nice formula for the weighted L^p discrepancy for even p. We use this formula to derive an upper bound for the average weighted L^p discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0,1)^d for which the weighted L^p discrepancy is uniformly bounded in d and goes to zero polynomially in N⁻¹.Finally we use these facts to generalize some results from Sloan and Woźniakowski (1998) on (strong) QMC-tractability of integration in weighted Sobolev spaces.     

Weighted Oscillation and Variation Inequalities for Singular Integrals and Commutators Satisfying Hrmander Type Conditions

This paper is devoted to investigating the weighted L~p-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type(p, p) estimates for 1 p ∞ and the weighted weak type(1,1) estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain Hormander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini's conditions as model examples. Meanwhile, we also obtain the weighted L~p-boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO(R~d)-functions.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.