Search

In this work we extend the theory of the classical Hardy space $$H^1$$ to the rational Dunkl setting. Specifically, let $$\Delta $$ be the Dunkl Laplacian on a Euclidean space $$\mathbb {R}^N$$ . On the half-space $$\mathbb {R}_+\times \mathbb {R}^N$$ , we consider systems of conjugate $$(\partial _t^2+\Delta _{\mathbf {x}})$$ -harmonic functions satisfying an appropriate uniform $$L^1$$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $$H^1_{\Delta }$$ , can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood–Paley square functions.

V. Ignatowski (1910) showed that assumptions about light are not necessary to obtain Lorentzian kinematics as one of only few possibilities. We give a much simplified proof of his result as formulated by V. Gorini (1971) for $n$+1-dimensional space-time., 9 pages. Accepted version, to appear in Journal of Geometry and Physics

International audience; Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was closely connected with certain classes of special functions in one variable. During the eighties, several attempts were made, mainly by the Dutch school, to extend these results in higher rank (i.e. in several variables), until the discovery of Dunkl operators in the rational case and Cherednik operators in the trigonometric case. Together with q-special functions introduced by Macdonald, this has led to a beautiful theory, developed by several authors, which encompasses in a unified way harmonic analysis on all Riemannian symmetric spaces and spherical functions thereon.In this series of lectures, delivered at the Summer School AAGADE 2015 (Analytic, Algebraic and Geometric Aspects of Differential Equations, Mathematical Research and Conference Center, Bedlewo, Poland, September 2015), we aimed at giving an updated overview of Dunkl theory, with an emphasis on its analytic aspects.

This paper is perhaps the first attempt at a study of the Hardy space $$H^1$$ in the rational Dunkl setting. Following Uchiyama’s approach, we characterize $$H^1$$ atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for $$H^1$$ . These results are proved here in the one-dimensional case and in the product case.

This note is devoted to the intertwining operator in the one--dimensional trigonometric Dunkl setting. We obtain a simple integral expression of this operator and deduce its positivity., A para{\^i}tre dans Proc. Amer. Math. Soc

In this paper we consider the Laplace-Beltrami operator \Delta on Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(\tau \Delta), when \tau \in C* with Re(\tau) \geq 0. When \tau \in iR*, we obtain in particular pointwise estimates of the Schr\"odinger kernel associated with \Delta. We then prove Strichartz estimates for the Schr\"odinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schr\"odinger equation associated with a distinguished Laplacian on Damek-Ricci spaces, showing that in this case the standard dispersive estimate fails while suitable weighted Strichartz estimates hold.

In this article, we establish first a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the $L^p\to L^p$ norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

International audience; We solve explicitly the shifted wave equation on Damek--Ricci spaces, using Asgeirsson's theorem and the inverse dual Abel transform. As an application, we investigate Huygens' principle. A similar analysis is carried out in the discrete setting of homogeneous trees.

Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. R\"osler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda}^{(\alpha,\beta)}$., Comment: Adv. Pure Appl. Math. (2011) 27 pp

International audience; In this paper, we study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on real hyperbolic spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain local well-posedness results for the nonlinear wave equation.

Using pointwise upper bounds recently obtained by the first author, we first show that the heat kernel ht(x, y) on a Riemannian symmetric space of the noncompact type GK is asymptotically concentrated in an annulus centered at y and moving to infinity with finite speed 2 ¦ϱ¦, ϱ being as usual the half sum of all positive roots of GK. In the higher rank case we prove moreover that heat not only concentrates in an annulus but also along the (K-orbit) of the ϱ-axis. By applying wave equation techniques developed by M. E. Taylor, we partially extend the above results to Riemannian manifolds with exponential volume growth and L2 spectrum of the Laplacian bounded away from 0. Some extensions to the vector bundle case are also considered.

International audience; In this paper, we show the inclusion and the density of the Schwartz space in Besov-Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov-Dunkl space.

We consider the Schr\"odinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we get strong well-posedness results for NLS. Specifically, for small intial data, we prove $L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (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 the 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 & Staffilani, for small radial $L^2$ data and for large radial $H^1$ data. The second important 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, in the Euclidean case, this is only possible for the critical power $\gamma=1+\frac4n$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1, Comment: Version 1 : 18 January 2008. Version 2 : 29 February 2008

The infinite Brownian loop $\{B_t^0,t\ge 0\}$ on a Riemannian manifold $\mathbb M$ is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin $0$, when $T\to+\infty$. It has no spectral gap. When $\mathbb M$ has nonnegative Ricci curvature, $B^0$ is the Brownian motion itself. When $\mathbb M=G/K$ is a noncompact symmetric space, $B^0$ is the relativized $\Phi_0$--process of the Brownian motion, where $\Phi_0$ denotes the basic spherical function of Harish--Chandra, i.e. the $K$--invariant ground state of the Laplacian. In this case, we consider the polar decomposition $B_t^0=(K_t,X_t)$, where $K_t\in K/M$ and $X_t\in\overline{\mathfrak a_+}$, the positive Weyl chamber. Then, as $t\to+\infty$, $K_t$ converges and $d(0,X_t)/t\to 0$ almost surely. Moreover the processes $\{X_{tT}/\sqrt{T},t\ge 0\}$ converge in distribution, as $T\to+\infty$, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that $d(0,X_{tT})/\sqrt{T}$ converges to a Bessel process of dimension $D=\operatorname{rank}\mathbb M+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on $\Phi_0$.

In a recent joint work with the same title, we have obtained optimal upper and lower bounds for the heat kernel $h_t(x,y)$ on a noncompact symmetric space $G/K$, under the assumption that $d(x,y)=O(1+t)$. In this article, we reprove our main result in a simpler way, using Harnack's inequality and avoiding this way hard analysis along faces.