Dunkl Operators Research Articles

Some Characterization of Herz and Herz-Type Hardy Spaces for the Dunkl Operator

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

Infinite Dimensional Systems of Particles with Interactions Given by Dunkl Operators

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl Laplacian with drift terms. For this semigroup we prove gradient bounds involving a symmetrised carre-du-champ operator, and we show that for small coefficients this semigroup has a unique invariant measure which satisfies ergodicity properties. We then extend this analysis to an infinite dimensional model on $(\mathbb {R}^{N})^{\mathbb {Z}^{d}}$ , consisting of interacting finite dimensional models. We construct an associated Markov semigroup for this model using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.

Orthogonal polynomial projection error in Dunkl–Sobolev norms in the ball

We study approximation properties of weighted L2-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form (1−‖x‖2)α∏i=1d|xi|γi, α,γ1,…,γd>−1. Said properties are measured in Dunkl–Sobolev-type norms in which the same weighted L2 norm is used to control all the involved differential–difference Dunkl operators, such as those appearing in the Sturm–Liouville characterization of similarly weighted L2-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms.

Characterization of fractional maximal operator and its commutators on Orlicz spaces in the Dunkl setting

On the $${\mathbb {R}}^{d}$$ the Dunkl operators $$\big \{D_{k,j}\big \}_{j=1}^{d}$$ are the differential-difference operators associated with the reflection group $${\mathbb {Z}}_2^d$$ on $${\mathbb {R}}^{d}$$ . In this paper, in the setting $${\mathbb {R}}^{d}$$ we find necessary and sufficient conditions for the boundedness of the fractional maximal operator $$M_{\alpha ,k}$$ on Orlicz spaces $$L_{\varPhi ,k}({\mathbb {R}}^{d})$$ . As an application of this result we show that $$b \in {\textit{BMO}}_{k}({\mathbb {R}}^{d})$$ if and only if the maximal commutator $$M_{b,k}$$ is bounded on Orlicz spaces $$L_{\varPhi ,k}({\mathbb {R}}^{d})$$ .

Matrix elements of vertex operators and fermionic limit of spin Calogero-Sutherland system

We present a construction of an integrable model as a projective type limit of spin Calogero-Sutherland model with N fermionic particles, where N tends to infinity. It is implemented in the multicomponent fermionic Fock space. Explicit formulas for limits of Dunkl operators and the Yangian generators are presented by means of fermionic fields.

Translation operator and maximal function for the (k,1)-generalized Fourier transform

In this paper we study a translation operator associated with the n-dimensional (k,1)-generalized Fourier transform, where k is a multiplicity function for the Dunkl operators. In particular, we prove that the translation is a positivity-preserving operator acting on a suitable space of radial functions on Rn. We then use it to define a Hardy-Littlewood type maximal operator, where weak-type (1,1) and strong-type (p,p) estimates for the maximal operator are established with a precise behavior in n and k.

The Dunkl-Hausdorff operators and the Dunkl continuous wavelets transform

In the present paper, we prove the boundedness of Dunkl-Hausdorff operators in space $$L^p_{\alpha }({\mathbb {R}})$$ and in the Hardy space $$H^{1}_{\alpha }({\mathbb {R}})$$ associated with the Dunkl operators, investigate continuous Dunkl wavelet transformation, and obtain some useful results. The relation between Dunkl wavelet transformation and Dunkl-Hausdorff operator is also established.

Fock Spaces for the Complex Dunkl Operator and Deformed (1, 1) Algebra

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).

Sobolev-type inequalities for Dunkl operators

In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case p=2, as well as an extension to the coefficient p=1 that was previously unknown. We also find estimates of the sharp constants for the Sobolev inequality for Dunkl gradient. Related inequalities and some improvements are also considered (Nash inequality, Besov space embeddings).

Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

In this paper, we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, [Formula: see text] inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli–Kohn–Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalized root systems are proved, which in the particular case of root systems [Formula: see text] improve some well-known results.

Semigroup and Riesz transform for the Dunkl–Schrödinger operators

Let $$L_k=-\Delta _k+V$$ be the Dunkl–Schrodinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ is the Dunkl Laplace operator associated to the Dunkl operators $$T_j$$ on $$\mathbb {R}^d$$ and V is a nonnegative potential function. In the first part of this paper we introduce the Riesz transform $$R_j= T_j L_k^{-1/2}$$ as a $$L^2$$ -bounded operator and we first prove that it is of weak type (1, 1) and then that it is bounded on $$L^p(\mathbb {R}^d,d\mu _k(x))$$ for $$1<p\le 2$$ . The second part is devoted to the $$L^p$$ smoothing of the semigroup generated by $$L_k$$ , when V belongs to the standard Kato class.

A Class of Quantum Briot–Bouquet Differential Equations with Complex Coefficients

Quantum inequalities (QI) are local restraints on the magnitude and range of formulas. Quantum inequalities have been established to have a different range of applications. In this paper, we aim to introduce a study of QI in a complex domain. The idea basically, comes from employing the notion of subordination. We shall formulate a new q-differential operator (generalized of Dunkl operator of the first type) and employ it to define the classes of QI. Moreover, we employ the q-Dunkl operator to extend the class of Briot–Bouquet differential equations. We investigate the upper solution and exam the oscillation solution under some analytic functions.

Real Paley–Wiener theorems for the (k,a)‐generalized Fourier transform

In this paper, we obtain several versions of the real Paley–Wiener theorems for the (k,a)‐generalized Fourier transform recently investigated at length by Ben Saïd, Kobayashi, and Ørsted. This generalized Fourier transform can be regarded as a two‐parameter generalization of Howe's description of classical Fourier transform, where k is a multiplicity function for the Dunkl operators on and a&gt;0 arises from the interpolation of the two Lie algebra actions on the Weil representation of and the minimal unitary representation of the O(d+1,2).

Singular Nonsymmetric Jack Polynomials for Some Rectangular Tableaux

In the intersection of the theories of nonsymmetric Jack polynomials in N variables and representations of the symmetric groups S N one finds the singular polynomials. For certain values of the parameter κ there are Jack polynomials which span an irreducible S N -module and are annihilated by the Dunkl operators. The S N -module is labeled by a partition of N, called the isotype of the polynomials. In this paper the Jack polynomials are of the vector-valued type, i.e., elements of the tensor product of the scalar polynomials with the span of reverse standard Young tableaux of the shape of a fixed partition of N. In particular, this partition is of shape m , m , … , m with 2 k components and the constructed singular polynomials are of isotype m k , m k for the parameter κ = 1 / m + 2 . This paper contains the necessary background on nonsymmetric Jack polynomials and representation theory and explains the role of Jucys–Murphy elements in the construction. The main ingredient is the proof of uniqueness of certain spectral vectors, namely the list of eigenvalues of the Jack polynomials for the Cherednik–Dunkl operators, when specialized to κ = 1 / m + 2 . The paper finishes with a discussion of associated maps of modules of the rational Cherednik algebra and an example illustrating the difficulty of finding singular polynomials for arbitrary partitions.

Boundedness of Multidimensional Dunkl-Hausdorff Operators

In the present paper, we introduce the multidimensional Dunkl-Hausdorff operator ℋκ and we give simple sufficient conditions so that these operators be bounded on the weighted lebesgue spaces Lκpℝn and in the Hardy space Hκ1ℝn associated with the Dunkl operators. We also determine the Dunkl-Hausdorff operator ℋκ∗ that is adjoint to ℋκ.

Sonine Formulas and Intertwining Operators in Dunkl Theory

Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$, we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.

Dunkl jump processes: relaxation and a phase transition

Dunkl processes are multidimensional Markov processes defined through the use of Dunkl operators. Their paths show discontinuities, and so they can be separated into their continuous (radial) part, and their discontinuous (jump) part. While radial Dunkl processes have been studied thoroughly due to their relationship with families of stochastic particle systems such as the Dyson model and Wishart–Laguerre processes, Dunkl jump processes have gone largely unnoticed after the initial work of Gallardo, Yor and Chybiryakov. We study the dynamical properties of the latter processes, and we derive their master equation. By calculating the asymptotic behavior of their total jump rate, we find that the jump processes of types AN−1 and BN undergo a phase transition when the parameter decreases toward one in the bulk scaling limit. In addition, we show that the relaxation behavior of these processes is given by a non-trivial power law, and we derive an asymptotic relation for the relaxation exponent in order to discuss its -dependence.

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

W-exponentials, Schur elements and the support of the spherical representation of the rational Cherednik algebra

Given a complex reflection group W we compute the support of the spherical irreducible module of the rational Cherednik algebra of W in terms of the simultaneous eigenfunction of the Dunkl operators and Schur elements for finite Hecke algebras.

Fractional maximal commutator on Orlicz spaces for the Dunkl operator on the real line

Fractional maximal commutator on Orlicz spaces for the Dunkl operator on the real line