The Levi Decomposition of a Graded Lie Algebra

We show that a graded Lie algebra admits a Levi decomposition that is compatible with the grading

On derivations of subalgebras of real semisimple Lie algebras

Let g be a real semisimple Lie algebra with Iwasawa decomposition k+a+n. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra n that preserves its restricted root space decomposition is of the form ad( W), where W belongs to m+a

Uniformly bounded representations and completely bounded multipliers of SL(2,R)

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal

Pointwise convergence and semigroups acting on vector-valued functions

A submarkovian C0 semigroup (Tt)t2R+ acting on the scale of complex-valued functions Lp(X;C) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X;E), when E is a Banach space. It is known that, if f 2 Lp(X;C), where 1 < p < 1, then Ttf ! f pointwise almost everywhere. We show that the same holds when f 2 Lp(X;E

Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

A sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation