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In this paper we study harmonic analysis operators in Dunkl settings associated with finite reflection groups on Euclidean spaces. We consider maximal operators, Littlewood-Paley functions, σ -variation, and oscillation operators involving time derivatives of the heat semigroup generated by Dunkl operators. We establish the boundedness properties of these operators in L p (R d , ω K) , 1 ≤ p < ∞ , Hardy spaces, BMO, and BLO-type spaces in the Dunkl setting. The study of harmonic analysis operators associated to reflection groups need different strategies from the ones used in the Euclidean case since the integral kernels of the operators admit estimations involving two different metrics, namely, the Euclidean and the orbit metrics. [ABSTRACT FROM AUTHOR]

The orthonormal Strichartz estimates for the Schrödinger equation associated to the Dunkl Laplacian and the Dunkl-Hermite operator are derived in Senapati et al. (J Geom Anal 34:74, 2024) and Mondal and Song (Israel J Math, 2023). In this article we construct a set of coherent states in the Dunkl setting and apply semi-classical analysis to derive a necessary condition on the Schatten exponent for the aforementioned orthonormal Strichartz estimates, which turns out to be optimal for the Schrödinger equations associated with Laplacian and Hermite operator as a particular case. [ABSTRACT FROM AUTHOR]

In this paper, we construct the general Hamiltonian for position-dependent mass systems within the Dunkl formalism. The associated Schrödinger equation is shown to admit closed-form solutions of bound state type for two particular Dunkl–Liénard systems, given by the Dunkl–Higgs oscillator, and a spiked mass Dunkl oscillator. Differences to the conventional scenario are discussed. [ABSTRACT FROM AUTHOR]

In this paper, utilizing the Dunkl transform and a generalized translation operator, we derive an analogue of the Riesz-Feller fractional derivative for a class of functions satisfying Lipschitz conditions. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform. [ABSTRACT FROM AUTHOR]

In this paper, we consider two specific quantum systems governed by the one-dimensional Klein–Gordon equation within the Dunkl formalism. Our systems are characterized by a symmetrized Coulomb potential and an inverse quadratic interaction, respectively. For both systems, we construct closed-form solutions of bound state type. [ABSTRACT FROM AUTHOR]

We generate solvable cases of the two angular equations resulting from variable separation in the three-dimensional Dunkl-Schrödinger equation expressed in spherical coordinates. It is shown that the Dunkl formalism interrelates these angular equations with trigonometric Pöschl-Teller systems. Based on this interrelation, we use point transformations and Darboux-Crum transformations to construct new solvable cases of the angular equations. Instead of the stationary energy, we use the constants due to the separation of variables as transformation parameters for our Darboux-Crum transformations.

The polynomial null solutions of the k-hyperbolic Dirac operator are investigated by the osp (1 | 2) approach. These solutions are then utilized to construct the (fractional) Fourier transform associated to the k-hyperbolic Dirac operator. The resulting integral kernels are found to be a specific kind of Dunkl kernels. Additionally, we give tight uncertainty inequalities for three distinct fractional Fourier transforms that we have defined. These inequalities are new even for the ordinary fractional Hankel and Weinstein transforms. [ABSTRACT FROM AUTHOR]

In this paper, we study non–local in time evolution type equations generated by the Dunkl operator. Direct and inverse problems are investigated with the Caputo time-fractional heat equation with the parameter 0 < γ ≤ 1 . In particular, well-posedness properties are established for the forward problem. To adopt techniques of the harmonic analysis, we solve the problems in the Sobolev type spaces associated with the Dunkl operator. Our special interest is an inverse source problem for the Caputo–Dunkl heat equation. As additional data, the final time measurement is taken. Since our inverse source problem is ill-posed, we also show the stability result. Moreover, as an advantage of our calculus used here, we derive explicit formulas for the solutions of the direct and inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness for Cauchy problem for multi-term time-fractional heat equation associated with Dunkl operator. The equation under consideration includes a linear combination of Caputo derivatives in time with decreasing orders in (0, 1) and positive constant coefficients and one-dimensional Dunkl operator. To show solvability of this problem we use several important properties of multinomial Mittag-Leffler functions and Dunkl transforms, since various estimates follow from the explicit solutions in form of these special functions and transforms. Then we prove the uniqueness and existence results. To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, hypoelliptic operators theory and functional analysis. In particular, we use the direct and inverse Dunkl transform to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The generalized solutions of this problem are studied.

The main purpose of this article is to establish the Lp{L}^{p} Hardy’s identities and inequalities for Dunkl operator on any finite balls and the entire space RN{{\mathbb{R}}}^{N}. We also prove Hardy’s identities and inequalities on certain domains with distance function to the boundary ∂Ω\partial \Omega . In particular, we use the notion of Bessel pairs introduced in Ghoussoub and Moradifam to extend Hardy’s identities for the classical gradients obtained by Lam et al., Duy et al., Flynn et al. to Dunkl gradients introduced by Dunkl. Our Hardy’s identities with explicit Bessel pairs significantly improve many existing Hardy’s inequalities for Dunkl operators.

We apply generalized Darboux transformations to Schrödinger equations at fixed energy that contain Dunkl derivatives, and are equipped with a position-dependent mass. Our Darboux transformations, originating from coupled Korteweg–De Vries systems, consist of two algorithms that generate new solutions, potentials and position-dependent mass functions for the Dunkl–Schrödinger system. [ABSTRACT FROM AUTHOR]

Dunkl operators may be regarded as differential‐difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in Rd$\mathbb {R}^d$ is introduced by employing the full "gradient" induced by the corresponding carré du champ operator and then the Lp$L^p$ boundedness is studied for all p∈(1,∞)$p\in (1,\infty)$. For p∈(1,2]$p\in (1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the Lp$L^p$ boundedness, while for p∈[2,∞)$p\in [2,\infty)$, we restrict to a particular case when the corresponding Weyl group is isomorphic to Z2d$\mathbb {Z}_2^d$ and apply a probabilistic method to prove the Lp$L^p$ boundedness. In the latter case, the curvature‐dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension‐free. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the Dunkl—Hilbert transform Hk, with k ≥ 0, induced by the Dunkl differential operator and associated with the reflection group ℤ2. For this end, we establish that the Dunkl—Poisson kernel and the conjugate Dunkl—Poisson kernel satisfy the Cauchy—Riemann equations in the Dunkl context. We prove the continuity of Hk on Lp(wk) for 1 < p < ∞, where wk(x) = ∣x∣2k. Finally, we introduce the maximal Hilbert operator H*k and establish an analogue of Cotlar's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce matrix operator algebra involving a universal curvature constant and using the Dunkl derivative. Consequently, the Dirac equation can be written without spin connections. Iterating the Dirac equation gives the Klein–Gordon equation in its canonical form without first-order Dunkl derivatives. This leads to a new form for the static metric based on the Dunkl operator in curved space. [ABSTRACT FROM AUTHOR]

In this paper, we give a new characterization for the Dunkl-classical orthogonal polynomials. The previous characterization has been illustrated by some examples.

The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

We use a symmetrized Hamiltonian containing Dunkl operators to generate a Dunkl–Schrödinger equation with position-dependent mass. Our equation is shown to admit arbitrary-order Darboux transformations if the mass either has odd parity or is an inverse-square function. We construct the Darboux transformations and apply them to a Dunkl–Schrödinger system featuring a rational double well potential. [ABSTRACT FROM AUTHOR]

We construct a quantum Hamiltonian for the isotonic Mathews–Lakshmanan oscillator system within the Dunkl formalism. It is shown that the Dunkl Hamiltonian does not admit bound states if the isotonic term is present. If the isotonic term is not present, then the Dunkl Hamiltonian admits bound states of even parity only. These results are in contrast to the findings for the conventional Mathews–Lakshmanan oscillator system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the raising operator Rξ = ξTμ +xI, ξ ̸= 0, where Tμ and I are the Dunkl operator and the identity operator respectively. Our purpose is to determine all monic orthogonal polynomials sequences {Pn(x)}n≥0 such that the sequence of polynomials {(RξPn)(x)}n≥0 is also orthogonal. We prove that the only sequence of polynomials satisfying this condition is, up to a dilation, the generalized Hermite polynomial sequence. Then, we explore our result to deduce a Rodrigues formula for the generalized Hermite polynomials sequence. [ABSTRACT FROM AUTHOR]

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In this paper, we consider two fundamental problems in the framework of the Dunkl derivative: the Bosonic oscillator model of spin 0 under magnetic field: Landau levels and the Duffin–Kemmer–Petiau equation with the Coulomb potential. In both cases, we obtain the exact analytical solutions for the bound states in the general case for the different eigenvalues of the reflection operator. [ABSTRACT FROM AUTHOR]

The Dunkl–Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl–Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl–Dirac equation are given as a realisation of one family of such irreducible unitary representations. [ABSTRACT FROM AUTHOR]

Let N be symmetric group of degree N. In the representation of N corresponding to trapeziform partition (precisely τ = (m 2 k , a , b) , 0 ≤ b ≤ a < m), we propose nonsymmetric Jack polynomials in 2 m k + a + b variables which are singular at 1 m + 2 . Moreover those nonsymmetric Jack polynomials span a module of N which is isomorphic to the representation of type σ = (m k + a , m k + b). [ABSTRACT FROM AUTHOR]

We introduce in this paper the concept of Dunkl-coherent pair of forms (linear functionals) in the symmetric case. We prove that if { u , v } is a Dunkl-symmetrically coherent pair of symmetric forms, then at least one of them has to be a symmetric Dunkl-classical form, i.e. Generalized Hermite or Generalized Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

In the present work, we construct a Dunkl generalization of the modified Szász-Mirakjan operators of integral form defined by Pǎltanea [1]. We study the approximation properties of these operators including weighted Korovkin theorem, the rate of convergence in terms of the modulus of continuity, second order modulus of continuity via Steklov-mean, the degree of approximation for Lipschitz class of functions and the weighted space. Furthermore, we obtain the rate of convergence of the considered operators with the aid of the unified Ditzian-Totik modulus of smoothness and for functions having derivatives of bounded variation.

In this paper, we are interested in estimates of the Dunkl kernels on some special sets. We improving results of M.F.E. de Jeu and M. Rösler [6]. [ABSTRACT FROM AUTHOR]

On the R d the Dunkl operators { D k , j } j = 1 d are the differential-difference operators associated with the reflection group Z 2 d on R d . In this paper, in the setting R d we find necessary and sufficient conditions for the boundedness of the fractional maximal operator M α , k on Orlicz spaces L Φ , k (R d) . As an application of this result we show that b ∈ BMO k (R d) if and only if the maximal commutator M b , k is bounded on Orlicz spaces L Φ , k (R d) . [ABSTRACT FROM AUTHOR]

In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation.

In this paper we study certain relations between the Dunkl–Sobolev classes of fractional order Lα,kp(IR) and the generalized Dunkl–Lipschitz classes ∧α,p,qk(IR). Next, we show a Dunkl–Hardy–Littlewood theorem for k‐temperatures and its dual. Also, we give some theorems on Dunkl convolutions of Dunkl–Lipschitz functions. Finally, we establish some results concerning the Dunkl transforms of functions and distributions belonging to generalized Dunkl–Lipschitz spaces. [ABSTRACT FROM AUTHOR]

Using a generalized spherical mean operator, we obtain a generalization of two theorems 84 and 85 of Titchmarsh for the Dunkl transform for functions satisfying the Dunkl—Lipschitz condition in the space Lp(ℝd, wk(x)dx), where 1 < p ≤ 2. [ABSTRACT FROM AUTHOR]

In the paper we study the Schrödinger oscillatory integrals Tλ,atf(x) (λ≥0, a>1) associated with the one-dimensional Dunkl transform Fλ. If a=2, the function u(x,t):=Tλ,2tf(x) solves the free Schrödinger equation associated to the Dunkl operator, with f as the initial data. It is proved that, if f is in the Sobolev spaces Hλs(R) associated with the Dunkl transform, with the exponents s not less than 1 / 4, then Tλ,atf converges almost everywhere to f as t→0. A counterexample is constructed to show that 1 / 4 can not be improved for a=2, and when 1/4≤s≤1/2, the Hausdorff dimension of the divergence set of Tλ,atf for f∈Hλs(R) is proved to be 1-2s at most. [ABSTRACT FROM AUTHOR]

We consider the Dunkl differential-difference operator associated with a reflection group W and a nonnegative parameter function k. We exhibit and study a new family of sharp integral inequalities related to the Dunkl heat semigroup generated by the Dunkl-Laplacian . These estimations are a natural extension and refinement of the Poincaré inequality established in [1] and used to derive other applications. [ABSTRACT FROM AUTHOR]

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner-Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials. [ABSTRACT FROM AUTHOR]

We consider a generalization of the classical Laplace operator, which includes the Laplace-Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai-Ito algebra. [ABSTRACT FROM AUTHOR]

We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S 3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A 2 , with corresponding Weyl group S 3 , the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra s o ( 3 ) , incorporating elements of S 3 . This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials. [ABSTRACT FROM AUTHOR]

Using a generalized Dunkl translation, we obtain an analog of Theorem 5.2 in Younis' paper [2] for the Dunkl transform for functions satisfying the $(\delta, \gamma)$-Dunkl Lipschitz condition in the space $\mathrm{L}^{2}(\mathbb{R}, |x|^{2\alpha+1}dx)$.}