In this paper, we show the inclusion and the density of the Schwartz space in Besov–Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov–Dunkl space.

In this article, we establish first a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the L p → L norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

Abstract We investigate some spectral properties of a second order differential-difference operator $$J_{\alpha ,\beta }$$ J α , β on $$L^2((-\pi ,\pi ),d\mu _{\alpha , \beta })$$ L <mm...

The $(k,a)$-generalised Fourier transform is the unitary operator defined using $a$-deformed Dunkl harmonic oscillator.The main aim of this paper to prove $L^p$-$L^q$ boundedness $(k, a)$-generalised multipliers. To show we first establish Paley inequality and Hausdorff-Young-Paley for transform. We also demonstrate applications obtained results study well-posedness nonlinear partial differenti...

A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various ne...

We continue a program generalizing classical results from the analysis on symmetric cones to Dunkl setting for root systems of type A A ...

Let G be a noncompact semisimple Lie group with finite centre. $$X=G/K$$ the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections to Laplace-Beltrami operator are given by $$P_{\lambda }f =f*\Phi _{\lambda }$$ , where $$\Phi elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for }f$$ . Moreove...

Abstract Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors this manuscript presented most general forms that depends on three Wigner parameters have a better tuning. In manuscript, we employ latter generalized in relativistic equation examine two dimensional harmonic and anharmonic oscillators solutions. We obtain Nikiforo...

We define the Littlewood-Paley decomposition associated with the Dunkl operators; from this decomposition we give the characterization of the Sobolev, Hölder and Lebesgue spaces associated with the Dunkl operators. We construct the paraproduct operators associated with the Dunkl operators similar to those defined by J.M. Bony in [1]. Using the LittlewoodPaley decomposition we establish the Sobo...