Let (ℝ+, Δ(x)dx) be a Jacobi hypergroup with weight function Δ(x) = c(sinh x)2α+1(cosh x)2β+1. As in the Euclidean case, the real Hardy space H1(Δ) for (ℝ+, Δ(x)dx) is defined as the set of all locally integrable functions on ℝ+ whose radial maximal functions belong to L1(Δ). In this paper we give a characterization of H1(Δ) in terms of weighted Triebel–Lizorkin spaces on ℝ via the Abel transform. As an application, we introduce three types of atoms for (ℝ+, Δ), one of them is smooth, and give an atomic decomposition of H1(Δ).

We outline an approach of the Kirillov map for exponential groups based on deformations, and suggest a possible way towards an alternative proof of the Leptin–Ludwig bicontinuity theorem [Unitary representation theory of exponential Lie groups, W.De Gruyter, 1994.] along these lines.

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In this paper we prove the absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual. Next we present some applications of this result.

We prove Beurling's theorem and L p – L q Morgan's theorem for Damek–Ricci spaces. These two theorems exhaust a family of theorems which illustrate a well-known paradigm that a function and its Fourier transform cannot be simultaneously localized.

We define and study a [0, 1] m -valued process depending on three positive real parameters p, q, β that specializes for β = 1, 2 to the eigenvalues process of the real and complex matrix Jacobi processes on the one hand and that has the distribution of the β -Jacobi ensemble as stationary distribution on the other hand. We first prove that this process, called β -Jacobi process, is the unique strong solution of the stochastic differential equation defining it provided that β &gt; 0, p ∧ q &gt; m – 1 + 1/ β . When specialized to β = 1, 2, our results actually improve well known results on eigenvalues of matrix Jacobi processes. While proving the strong uniqueness, the generator of the β -Jacobi process is mapped into the radial part of the Dunkl–Cherednik Laplacian associated with the non reduced root system of type BC . The transformed process is then valued in the principal Weyl alcove and this allows to define the Brownian motion in the Weyl alcove corresponding to all multiplicities equal one. Second, we determine, using stochastic calculus and a comparison theorem, the range of β, p, q for which the m components of the β -Jacobi process first collide, the smallest one reaches 0 and the largest one reaches 1. This is equivalent to the first hitting time of the boundary of the principal Weyl alcove by the transformed process. Finally, we write down its semi group density.

We study non-linear integrable partial differential equations naturally arising as bi-Hamiltonian Euler equations related to the looped cotangent Virasoro algebra. This infinite-dimensional Lie algebra (constructed in [Ovsienko and Roger, Commun. Math. Phys. 273: 357–378, 2007]) is a generalization of the classical Virasoro algebra to the case of two space variables. Two main examples of integrable equations we obtain are quite well known. We show that the relation between these two equations is similar to that between the Korteweg–de Vries and Camassa–Holm equations.

In this paper we derive and study a nonlinear boundary integral equation for calculating the boundary value of the conformal mapping from a starlike simply connected region G onto another starlike simply connected region Ω. The integral equation can be interpreted as a generalization of the classical Theodorsen integral equation. Numerical results are presented to illustrate the performance of the proposed method.