We find a Jacobi identity for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. We prove that intertwining operators for a suitable vertex operator algebra satisfy t...

Homogeneous Banach spaces determined by the Jacobi translation operator are introduced and studied. Based on this translation operator a Jacobi differential operator is analyzed. Approximation procedures in the homogeneous Banach spaces are presented.

In 1961, Bargmann introduced the classical Fock space $$\mathscr {F}(\mathbb {C}^{d})$$ and in 1984, Cholewinsky generalized {F}_{\alpha ,e}(\mathbb . These two spaces are aim of many works, have applications mathematics, physics, quantum mechanics. this work, we introduce study }(\mathbb associated to Dunkl operators $$T_{\alpha _{j}}$$ with $$\alpha _{j}>-1/2$$ for all $$j=1,\ldots ,d$$ This ...

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

In this paper we prove inversion formulas for the Dunkl intertwining operator Vk and for its dual tVk and we deduce the expression of the representing distributions of the inverse operators V −1 k and V −1 k , and we give some applications.

Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. We study when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of definition. In particular, we examine the Jacobi operator, the higher order Jacobi operator, the Szabo operator, and the skew-symmetric curvature operator.

Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lip...

Let M n be a Riemannian manifold and R its curvature tensor. For a point p ∈ M n and a unit vector X ∈ TpM n , the Jacobi operator is defined by RX = R(X, ·)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that...

We obtain a finite-sum representation for the general solution of the equation ∆ (p(n− 1)∆u(n − 1)) + q(n)u(n) = λr(n)u(n) in terms of a nonvanishing solution corresponding to some fixed value of λ = λ0. Applications of this representation to some results on the boundedness of solutions are given as well as illustrating examples.