A Clifford algebra approach to real simple Lie algebras,II: The algebras with root system BCr

Received: 14 September 2001 / in final form: 28 April 2002 // Published online: 20 March 2003     

On the norms of quaternionic harmonic projection operators

As a consequence of integral bounds for three classes of quaternionic spherical harmonics, we prove some bounds from below for the $(L^p,L^2)$ norm of quaternionic harmonic projectors, for $p\in [1,2]$.     

A new proof of the -condition for real rank one simple Lie algebras and their classification

In this paper a new purely algebraic proof of the -condition for the nilpotent Iwasawa algebras in real rank one simple Lie algebras is presented, yielding the classification of real rank one simple Lie algebras.

     

$$L^p$$ Joint Eigenfunction Bounds on Quaternionic Spheres

We prove some sharp \(L^p-L^2\) estimates for joint spectral projections \(\pi _{\ell \ell '}\), with \(\ell ,\ell '\in {\mathbb {N}}\), \(\ell \ge \ell '\ge 0\), \(1\le p\le 2\), associated to the Laplace–Beltrami operator and to a suitably defined subLaplacian on the unit quaternionic sphere.     

Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G  . In particular, we show that the operators _Tα:f?|⋅|^−αL^−α/2f$T_{\alpha }:f?{|\cdot |}^{-\alpha }{L}^{-\alpha /2}f$, where |⋅|$|\cdot |$ is a homogeneous norm, 0<α<Q/p$0<\alpha <Q/p$, and L   is the sub-Laplacian, are bounded on the Lebesgue space L^p(G)$L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the L^p$L^p$ norm of a function f   to the L^q$L^q$ norm of |⋅|^βf${|\cdot |}^{\beta }f$ and the L^r$L^r$ norm of L^δ/2f$L^{\delta /2}f$.     

Heisenberg-Pauli-Weyl uncertainty inequalities and polynomial volume growth

In its simpler form, the Heisenberg-Pauli-Weyl inequality says that

     

A restriction theorem for Métivier groups

In the spirit of an earlier result of D. Müller on the Heisenberg group we prove a restriction theorem on a certain class of two step nilpotent Lie groups. Our result extends that of Müller also in the framework of the Heisenberg group.     

Scalar products on clifford modules and pseudo-H-type lie algebras

In 1980 A. Kaplan introduced the so called generalised Heisenberg algebras, which are two step nilpotent algebras endowed with an inner product satisfying a compatibility condition with the Lie algebra structure. In this paper we generalize the definition of A. Kaplan to the case of a nonpositive definite scalar product. In the non-positive definite case the proof of the existence and the classification raise entirely new problems. The natural setting to solve them is that of the theory of Clifford modules. Entrata in Redazione il 27 ottobre 1995.