EXPLICIT FORMULAS AND MEROMORPHIC EXTENSION OF BESSEL FUNCTIONS ON TANGENT SPACES TO NONCOMPACTLY CAUSAL SYMMETRIC SPACES

Assume that $G/H$ is a noncompactly causal symmetric space with restricted root system of the non-exceptional type and the multiplicity of the short roots is even. Using shift operators we obtain explicit formulas for the Bessel function on the tangent space to $G/H$ at the origin. This enable us to investigate the nature and order of the singularities of the Bessel function, and to formulate a conjecture on this matter

The source operator method: an overview

This is an overview on the {source operator method} which leads to the construction of symmetry breaking differential operators (SBDO) in the context of tensor product of two principals series representations for the conformal group of a simple real Jordan algebra. This method can be applied to other geometric contexts: in the construction of SBDO for differential forms and for spinors, and also for the construction of Juhl's operators corresponding to the restriction from the sphere $S^n$ to $S^{n-1}$

Laguerre semigroup and Dunkl operators

We construct a two-parameter family of actions \omega_{k,a} of the Lie algebra sl(2,R) by differential-difference operators on R^N \setminus {0}. Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from the interpolation of the Weil representation of Mp(N,R) and the minimal unitary representation of O(N+1,2) keeping smaller symmetries. We prove that this action \omega_{k,a} lifts to a unitary representation of the universal covering of SL(2,R), and can even be extended to a holomorphic semigroup \Omega_{k,a}. In the k\equiv 0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup by the second author with G. Mano (a=1). One boundary value of our semigroup \Omega_{k,a} provides us with (k,a)-generalized Fourier transforms F_{k,a}, which includes the Dunkl transform D_k (a=2) and a new unitary operator H_k (a=1), namely a Dunkl-Hankel transform. We establish the inversion formula, and a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty inequality for F_{k,a}. We also find kernel functions for \Omega_{k,a} and F_{k,a} for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.Comment: final version (some few typos, updated references

A Paley-Wiener theorem about the spectral parameter, and a support theorem for general types of Dunkl spherical means

For $s\in \mathbb R,$ denote by $\cal P_k^s$ the ``projections" of a function $f$ in $\mathcal D(\mathbb R^d)$ into the eigenspaces of the Dunkl Laplacian $\Delta_k$ corresponding to the eigenvalue $-s^2.$ The parameter $k$ comes from Dunkl's theory of differential-difference operators. Wa shall characterize the range of $\cal P_k^s$ on the space of functions $f\in \mathcal D(\mathbb R^d)$ supported inside the closed ball $\overline{B(O,R)}.$ As a first application of this Paley-Wiener type theorem, we provide a spectral version of de Jeu's Paley-Wiener theorem for the Dunkl transform. The second application concerns a support theorem for general types of Dunkl spherical means

Strichartz estimates for Schrödinger-Laguerre operators

International audienceIn $\mathbb R^n\times \mathbb R$ we consider the Schr\"odinger equation \begin{equation}\tag{$\mathcal E_1$}i\partial_t u (x,t)+{\rm H}_{k,\varsigma} u(x,t)=F (x,t)\end{equation} with given boundary values on $\mathbb R^n.$ Here ${\rm H}_{k,\varsigma}=(\Vert x\Vert^{2-\varsigma}\Delta_k-\Vert x\Vert^\varsigma)/\varsigma$ is a differential-difference operator on $\mathbb R^n,$ where $k$ is a multiplicity function for the Dunkl Laplacian $\Delta_k,$ and $\varsigma$ is either $1$ or $2.$ In the $k\equiv 0$ case, ${\rm H}_{0,1}$ is the Laguerre operator and ${\rm H}_{0,2}$ is the Hermite operator. In this paper we obtain Strichartz estimates for the Schr\"odinger equation $(\mathcal E_1)$. We then prove that Strichartz estimates for the Schr\"odinger equation \begin{equation}\tag{$\mathcal E_2$} i \partial_t \psi(x,t)+(1/\varsigma) \Vert x\Vert^{2-\varsigma} \Delta_k \psi(x,t)=F(x,t) \end{equation} can be obtained from those for $(\mathcal E_1).$ With the specializations $k\equiv 0$ and $\varsigma=2$, the equation $(\mathcal E_2)$ reduces to the Euclidean Schr\"odinger equation, while the case where $k\equiv 0$ and $\varsigma=1$ is already new

Uniqueness of solutions to Schrödinger equations on H-type groups

International audienceThis paper deals with the Schrodinger equation $i\partial_s u({\bf z},t;s)-\cal L u({\bf z}, t;s)=0,$ where $\cal L$ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $| f({\bf z},t)| \lesssim q_\alpha({\bf z},t),$ where $q_s$ is the heat kernel associated to $\cal L.$ If in addition $|u({\bf z},t;s_0)|\lesssim q_\beta({\bf z},t),$ for some $s_0\in \mathbb R\setminus\{0\},$ then we prove that $u({\bf z},t;s)=0$ for all $s\in \mathbb R$ whenever $\alpha\beta<s_0^2.$ This result holds true in the more general context of $H$-type groups. We also prove an analogous result for the Grushin operator on $ \mathbb R^{n+1}.

Lp harmonic analysis for differential-reflection operators

International audienceWe introduce and study differential-reflection operators$\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Sturm-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). As, by construction, the operators $\Lambda_{A, \varepsilon}$ are mixture of ${\rm d}/{\rm d}x$and reflection operators, we prove the existence of an operator $V_{A,\varepsilon}$ so that $\Lambda_{A, \varepsilon}\circ V_{A,\varepsilon}=V_{A,\varepsilon}\circ {\rm d}/{\rm d}x.$ The positivity of the intertwining operator $V_{A,\varepsilon}$ is also established. Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce a generalized Fourier transform $\mathcal F_{A,\varepsilon}.$ For $-1\leq \varepsilon\leq 1$ and $0 < p \leq \frac{2}{1+\sqrt{1-\varepsilon^2}},$ we develop an $L^p$-Fourier analysis for $\mathcal F_{A,\varepsilon},$ and then we prove an $L^p$-Schwartz space isomorphism theorem for $\mathcal F_{A,\varepsilon}$.Details of this paper will be given in another article