Search

This paper deals with the Schrödinger equation $i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where $ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where ${q}_{s} $ is the heat kernel associated to $ \mathcal{L} . $ If in addition $\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some ${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that $u(\mathbf{z} , t; s)= 0$ for all $s\in \mathbb{R} $ whenever $\alpha \beta \lt { 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} . $ [ABSTRACT FROM AUTHOR]

Abstract: We construct a two-parameter family of actions of the Lie algebra by differential-difference operators on . Here, k is a multiplicity-function for the Dunkl operators, and arises from the interpolation of the Weil representation and the minimal unitary representation of the conformal group. The action lifts to a unitary representation of the universal covering of , and can even be extended to a holomorphic semigroup . Our semigroup generalizes the Hermite semigroup studied by R. Howe (, ) and the Laguerre semigroup by T. Kobayashi and G. Mano (, ). The boundary value of our semigroup provides us with -generalized Fourier transforms , which includes the Dunkl transform () and a new unitary operator () as a Dunkl-type generalization of the classical Hankel transform. To cite this article: S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

By taking an appropriate limit, we obtain the Bessel functions related to root systems as limits of Heckman-Opdam hypergeometric functions. A more general class of Bessel functions is also investigated, which we will call the Θ-Bessel functions. Explicit formulas for the Θ-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of noncompact causal symmetric spaces, where an integral representation for these special functions is shown. [ABSTRACT FROM AUTHOR]

We prove a Fatou-type theorem on a Riemannian symmetric space of the noncompact type and characterize the image of the Poisson transform of Lp-functions on the maximal boundary as a Hardy-type space. [ABSTRACT FROM AUTHOR]

We consider a deformed wave equation where the Laplacian operator has been replaced by a differential-difference operator. We prove that this equation does not satisfy Huygens' principle. Our approach is based on the representation theory of the Lie algebra s l (2 , R) . [ABSTRACT FROM AUTHOR]