Jean-Philippe Anker

Nonlinear Schrödinger equation on real hyperbolic spaces

Abstract We consider the Schrodinger equation with no radial assumption on real hyperbolic spaces H n . We obtain in all dimensions n ⩾ 2 sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong well-posedness results for NLS. Specifically, for small initial data, we prove L 2 and H 1 global well-posedness for any subcritical power (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity F. On the other hand, if F is gauge invariant, L 2 charge is conserved and hence, as in the Euclidean case, it is possible to extend local L 2 solutions to global ones. The corresponding argument in H 1 requires conservation of energy, which holds under the stronger condition that F is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles and Staffilani, for small radial L 2 data or for large radial H 1 data. The second application of our global Strichartz estimates is scattering for NLS both in L 2 and in H 1 , with no radial or gauge invariance assumption. Notice that, on Euclidean spaces R n , this is only possible for the critical power γ = 1 + 4 n and can be false for subcritical powers while, on hyperbolic spaces H n , global existence and scattering of small L 2 solutions hold for all powers 1 γ ⩽ 1 + 4 n . If we restrict to defocusing nonlinearities F, we can extend the H 1 scattering results of Banica, Carles and Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearities: the geometry of hyperbolic spaces makes every power-like nonlinearity short range.

Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

Abstract. Let G (,)

Wave and Klein–Gordon equations on hyperbolic spaces

\operatorname{G}_\lambda ^{(\alpha ,\beta )}

Schrödinger Equations on Damek–Ricci Spaces

be the eigenfunctions of the Dunkl–Cherednik operator T (,)

Besov-Type Spaces on R d and Integrability for the Dunkl Transform ?

\operatorname{T}^{(\alpha ,\beta )}

The Shifted Wave Equation on Damek–Ricci Spaces and on Homogeneous Trees

on

An Introduction to Dunkl Theory and Its Analytic Aspects

\mathbb {R}

The wave equation on hyperbolic spaces

. In this paper we express the product G (,) (x)G (,) (y)

Three results in Dunkl analysis

\operatorname{G}_\lambda ^{(\alpha ,\beta )}(x)\,\operatorname{G}_\lambda ^{(\alpha ,\beta )}(y)

The Hardy Space H^1 in the Rational Dunkl Setting

as an integral in terms of G (,) (z)