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...

We prove that transplantations for Jacobi polynomials can be derived from representation of a special integral operator as fractional Weyl’s integral. Furthermore, we show that, in a sense, Jacobi transplantation can be reduced to transplantations for ultraspherical polynomials. As an application of these results, we obtain transplantation theorems for Jacobi polynomials in ReH1 and BMO. The pa...

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...

In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld [BD] have recently given a notion of “chiral algebra” in terms of D-modules on algebraic curves. This definition consists of a “skew-symmetry” relation and a “Jacobi identity” relation in a categorical setting, and it leads to the operator product expansion...

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As a special case we obtain a generalization of Kenser’s criterion for Sturm-Liouville operators to Jacobi operators.

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...

We determine the matrix of the balanced metric of the Siegel–Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace–Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel–Jacobi ball.

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...