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In this work, new characterizations of Herz and Herz-type Hardy spaces associated with the Dunkl operator on the real line are introduced.

In this paper, we study a class of generalized Fock spaces for the complex Dunkl operator associated with the cyclic group, generalizing the Bargmann representations for Lie algebra su(1,1)-generators. We present an explicit realization of Cr-extended oscillator in terms of this operator. Furthermore, we construct the wave functions and we determine the coherent states. We also determine an explicit realization of a polynomial deformation sud(1,1) of the classical Lie algebra su(1,1). Finally, we give a deforming functional which maps sud(1,1) to su(1,1).

In this paper, we obtain several versions of the real Paley-Wiener theorems for a generalized Fourier transform associated to a Dunkl-type differential-difference operator on the real line, which is related to Lions’ second-order singular differential operator, and includes Dunkl operator associated with the reflection group $${\mathbb {Z}}_2$$ on $${\mathbb {R}}$$ as a special case.

For $$p\ge p_0:=2\lambda /(2\lambda +1)$$ with $$\lambda >0$$ , the Hardy space $$H_\lambda ^p({\mathbb {R}}^2_+)$$ associated with the Dunkl transform $${\mathscr {F}}_{\lambda }$$ and the Dunkl operator D on the line $${\mathbb {R}}$$ , where $$(D_xf)(x)=f'(x)+\frac{\lambda }{x} (f(x)-f(-x))$$ , is the set of functions $$F=u+iv$$ on the half plane $${\mathbb {R}}^2_+=\{(x,y):\,x\in {\mathbb {R}}, y>0\}$$ , satisfying the generalized Cauchy–Riemann equations $$D_xu-\partial _yv=0$$ , $$\partial _yu+D_xv=0$$ , and $$\sup _{y>0}\int _{{\mathbb {R}}}| F(x,y)|^p|x|^{2\lambda }dx

In the paper we study the Schrodinger oscillatory integrals $$T^t_{\lambda ,a}f(x)$$ ( $$\lambda \ge 0$$ , $$a>1$$ ) associated with the one-dimensional Dunkl transform $${\mathscr {F}}_{\lambda }$$ . If $$a=2$$ , the function $$u(x,t):=T^t_{\lambda ,2}f(x)$$ solves the free Schrodinger equation associated to the Dunkl operator, with f as the initial data. It is proved that, if f is in the Sobolev spaces $$H^s_{\lambda }({\mathbb {R}})$$ associated with the Dunkl transform, with the exponents s not less than 1 / 4, then $$T^t_{\lambda ,a}f$$ converges almost everywhere to f as $$t\rightarrow 0$$ . A counterexample is constructed to show that 1 / 4 can not be improved for $$a=2$$ , and when $$1/4\le s\le 1/2$$ , the Hausdorff dimension of the divergence set of $$T^t_{\lambda ,a}f$$ for $$f\in H_{\lambda }^s({\mathbb {R}})$$ is proved to be $$1-2s$$ at most.

In this paper the notion of a Dunkl two-wavelet is introduced. The resolution of the identity formula for the Dunkl continuous wavelet transform is then formulated and proved. Calderon’s type reproducing formula in the context of the Dunkl two-wavelet theory is proved. The two-wavelet localization operators in the setting of the Dunkl theory are then defined. The Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. It is proved that under suitable conditions on the symbols and two Dunkl wavelets, the boundedness and compactness of these localization operators on \(L^{p}_{k}(\mathbb {R}^{d})\), \(1 \le p \le \infty \). Finally typical examples of localization operators are presented.

In this paper, we establish new spectral theorems for the Dunkl transform on \(\mathbb {R}^{d}\).

The purpose of this paper is to present some recent results for maximal functions and to study the $$L^p$$ -boundedness of the spherical maximal function associated to the Dunkl operators.

We characterize the Hardy space \(H^1\) in the rational Dunkl setting associated with the reflection group \(\mathbb {Z}_2^n\) by means of special Riesz transforms. As a corollary we obtain Riesz transforms characterization of \(H^1\) for product of Bessel operators in \((0,\infty )^n\).

Using a generalized dual translation operator, we obtain an analog of Titchmarsh’s theorem for the generalized Dunkl transform for functions satisfying the Q-Dunkl Lipschitz condition in the space \(\mathrm {L}_{Q}^{2}\).

In the present paper, we investigate in Dunkl analysis, the action of some fundamental operators on the atomic Hardy space \(H^1_k\).

In Dunkl theory on Rd which generalizes classical Fourier analysis, we study first the behavior at infinity of the Riesz potential of a non compactly supported function. Second, we give for 1, Comment: 22 pages. arXiv admin note: text overlap with arXiv:1301.6267, arXiv:1305.4394, arXiv:1208.5034

We establish that imaginary powers of the Dunkl harmonic oscillator are bounded from L p /(ℝ d , dμ K ) to itself, 1 < p < ∞, and from L 1(ℝ d , dμ K ) to L 1,∞(ℝ d , dμ K ).

This paper is perhaps the first attempt at a study of the Hardy space $$H^1$$ in the rational Dunkl setting. Following Uchiyama’s approach, we characterize $$H^1$$ atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for $$H^1$$ . These results are proved here in the one-dimensional case and in the product case.

Using a generalized spherical mean operator, we define generalized modu-lus of smoothness in the space \(\mathrm {L}_{k}^{2}(\mathbb {R}^{d})\). Based on the Dunkl operator we define Sobolev-type space and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness for the Dunkl transform on ℝd.

In this work we consider the Dunkl operator on the real line, defined by $$ {\cal D}_kf(x):=f'(x)+k\dfrac{f(x)-f(-x)}{x},\,\,k\geq0. $$ We define and study Dunkl–Sobolev spaces \(L^p_{n,k}(\mathbb{R})\), Dunkl–Sobolev spaces \({\cal L}^p_{\alpha,k}(\mathbb{R})\) of positive fractional order and generalized Dunkl–Lipschitz spaces \(\wedge^k_{\alpha,p,q}(\mathbb{R})\). We provide characterizations of these spaces and we give some connection between them.

In this paper, we first review the theory of Dunkl operators for complex reflection groups and then the theory of hyper-Bessel functions, which are a particular case of Meijer’s G-function and satisfy a higher order differential equation. Then we show that there exists a close relation between both theories. In fact, the components of the eigenfunctions of a Dunkl operator for a complex reflection group in the rank one case can be expressed in terms of hyper-Bessel functions.

In this paper, we study the Jacobi–Dunkl convolution operators on some distribution spaces. We characterize the Jacobi–Dunkl convolution operators as those ones that commute with the Jacobi–Dunkl translations and with the Jacobi–Dunkl operators. Also we prove that the Jacobi–Dunkl convolution operators are hypercyclic and chaotic on the spaces under consideration and we give a universality property for the generalized heat equation associated with them.

In Dunkl theory on \(\mathbb R ^d\) which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the \(d\)-dimensional Hardy inequalities whose constants are sharp and independent of \(d\). Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.

This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy–Riemann equations D x u−∂ y v=0,∂ y u+D x v=0, where D x is the Dunkl operator (D x f)(x)=f′(x)+(λ/x)(f(x)−f(−x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform , a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces $H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})$ on the half-plane ${\mathbb{R}}^{2}_{+}$ for p≥p 0=2λ/(2λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform is shown to be a bounded mapping from $H_{\lambda}^{1}({\mathbb{R}})$ to $L^{1}_{\lambda}({\mathbb{R}})$ ; and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for $f\in H^{1}_{\lambda}({\mathbb{R}})$ .

From the Dunkl analogue of Gegenbauer’s expansion of the plane wave, we derive an explicit closed formula for the spectrum of a right inverse of the Dunkl operator. This is done by stating the problem in such a way it is possible to use the technique due to Ismail and Zhang.

We introduce moduli of continuity on the basis of the generalized translation generated by the Dunkl operators $$ \Lambda f(x) = f^{\prime}(x) + \frac{{2\alpha + 1}}{x}\frac{{f(x) - f\left( { - x} \right)}}{2} $$ and estimate the functionals in terms of these moduli. The constants in these inequalities are explicitly expressed in terms of the moments of functionals, i.e. their upper bounds on generalized Sobolev classes. Jackson-type inequalities in the Lebesgue spaces on the real line with the weight $ {A_\alpha }(x) = {\left| x \right|^{2\alpha + 1}} $ , $ \alpha > - \frac{1}{2} $ , are particular cases of the inequalities established. Bibliography: 27 titles.

In this paper we present an approach to obtain harmonic function in the higher dimensional spaces in Dunkl-Clifford analysis. A very important application of this method is to establish some counterparts of Fueter’s Theorem in Dunkl case. To this end the corresponding Vekua-type systems for poly-axial Dunkl-monogenic functions are considered and studied.

In this article, we present a survey of some new results obtained in [2] , [8] . First, we give a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension. We also investigate the L p → L q boundedness properties of the Riesz potentials I α κ and the related fractional maximal function M κ , α associated to the Dunkl transform. Finally we prove the L p -boundedness, (1 p

In this note, Kelvin transform is introduced in the framework of Dunkl-Clifford analysis. It is shown that this transform preserves the class of Dunkl monogenic functions. As an application, we use it to generate Dunkl monogenic polynomials by a classical process due to Maxwell.

In the framework of superspace in Clifford analysis for the Dunkl version, the Fischer decomposition is established for solutions of the Dunkl super Dirac operators. The result is general without restrictions on multiplicity functions or on super dimensions. The Fischer decomposition provides a module for the Howe dual pair G × osp(1|2) on the space of spinor valued polynomials with G the Coxeter group, while the generators of the Lie superspace reveal the naturality of the Fischer decomposition.

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

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal–Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Said and Orsted and independently by Soltani) and the Dunkl heat kernel, due to Rosler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal–Bargmann analysis on Euclidean space. Hall’s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in μ-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal–Bargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall’s Version C generalized Segal–Bargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal–Bargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal–Bargmann kernels for Versions A and C.

Let R be a root system, for example, the root system associated to a semisimple Lie algebra. In [1], V. A. Golubeva and V. P. Lexin constructed two algebraic manifolds (Bethe and Dunkl manifolds) using the "universal" Dunkl operators. These manifolds were defined as subsets of the complex space ? N of dimension equal to the number of roots of the root system under consideration. The first manifold (Bethe manifold) is characterized by the following property: the Laplace operator constructed by means of Dunkl operators coincides with the "universal" Hamiltonian of the Calogero model. The second one (Dunkl manifold) is characterized by the property: the "universal" Dunkl operators commute. In this paper, the manifolds associated with the irreducible root system of Coxeter type are considered. We give their construction supposing that these manifolds are embedded in ? N/2. A theoremon the coincidence of Bethe and Dunkl manifolds is proved.

This is an overview of the use of reflection groups in analysis, applications in algebra, mathematical physics, and probability.

In this paper we consider the Dunkl operators Tj, j = 1, . . . , d, on \(\mathbb {R}^d\) and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an \(L^2_{k}\) inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimeche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.

We define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator., 22 pages

In this paper we give a necessary and sufficient condition in terms of the Jacobi–Dunkl transform in order that a Jacobi–Dunkl convolution of distributions is hypoelliptic.

We prove some potential theoretical properties of harmonic functions associated to Dunkl operators. We solve the corresponding Dirichlet problem and establish the related Harnack principle and normality criteria.

In this paper, a product formula for the eigenfunction of the Jacobi-Dunkl differential-difference operator is derived. It leads to a uniformly bounded convolution of point measures and a signed hypergroup on IR.

This paper is devoted to the heat equation associated with the Jacobi–Dunkl operator on the real line. In particular we show that the heat semigroup has a strictly positive kernel and a finite Green operator. As a direct application, we solve the Poisson equation and we introduce a new family of one-dimensional Markov processes.

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on ℝN. We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space Open image in new window of holomorphic functions on Open image in new window with reproducing kernel equal to the Dunkl-kernel. The definition and properties of Open image in new window extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrodinger model and in the Fock model. Further, we prove a branching decomposition of Open image in new window as a unitary Open image in new window-module and a general version of Hecke's formula for the Dunkl transform.

For a family of weight functions $h_k$ invariant under a finite reflection group on $R^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ 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.

Let Ω be a G-invariant convex domain in ℝN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)nf = 0 for some integer n. Here333-01is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, $$\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,$$ where Kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form $$f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,$$ where fj are Dunkl harmonic functions, i.e. Δhfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

We construct the shift operators and the intertwining operators for the spherical parts of generalized Calogero operators related to classical Coxeter systems.

We consider a generalized convolution \(\), \(\), on the real line generated by the Dunkl operator $$$$

The purpose of this Letter is to demonstrate a close connection between the problem of describing supercomplete commutative rings of partial differential operators (in the sense of Krichever–Veselov) and the theory of lacunae for hyperbolic Riesz kernels. As an application, we give a simple and explicit construction of additional quantum integrals for the generalized Calogero–Moser problem associated with a finite reflection group W and W-invariant integer root multiplicities mα. These quantum integrals are not W-invariant and cannot be constructed directly using the standard technique of Dunkl operators.

The interrelations between Calogero quantum problem and Knizhnik-Zamolodchikov equation are described following Matsuo, Cherednik, Felder, and the author. As the basic tool of the considerations the Dunkl operator is used. The generalizations related to an arbitrary Coxeter group and the applications to the Hadamard problem about the hyperbolic equations with the Huygens principle are discussed.

We obtain estimates for the eigenfunctions of Dunkl's operators and solve the spectral problem for these operators.

Some generalizations of the Lusztig-Lascoux-Schutzenberger operators for affine Hecke algebras are considered. As corollaries we obtain Lusztig's isomorphisms from affine Hecke algebras to their degenerate versions, a “natural” interpretation of the Dunkl operators and a new class of differential-difference operators generalizing Dunkl's ones and the Knizhnik-Zamolodchikov operators from the two dimensional conformal field theory.