Fourier and Poisson transformation associated to a semisimple Symmetrie space.
Frame characterizations of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications.
Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program
For the scalar holomorphic discrete series representations of and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside . We construct a Cayley transform between the Ol’shanskiĭ semigroup having as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for . This allows calculating the composition series in terms of harmonic analysis on . In particular we show that the Ol’shanskiĭ Hardy space for is different...
Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups
Généralisation d'un théorème de Haagerup
Let G be a group of automorphisms of a tree X (with set of vertices S) and H a kernel on S × S invariant under the action of G. We want to give an estimate of the -operator norm (1 ≤ r ≤ 2) of the operator associated to H in terms of a norm for H. This was obtained by U. Haagerup when G is the free group acting simply transitively on a homogeneous tree. Our result is valid when X is a locally finite tree and one of the orbits of G is the set of vertices at even distance from a given vertex; a technical...
Generalisations of some properties of invariant polynomials with matrix arguments
Some extensions of the properties of invariant polynomials proved by Davis (1980), Chikuse (1980), Chikuse and Davis (1986) and Ratnarajah et al. (2005) are given for symmetric and Hermitian matrices.
Generalized coefficients of spherical principal series for split symmetric spaces. (Coefficients généralisés de séries principales sphériques relatifs aux espaces symétriques déployés.)
Geometric realization of discrete series for semisimple symmetric spaces.
Geometry of Carnot-Carathéodory spaces, quasiconformal analysis, and geometric measure theory.
estimates for weakly strongly singular integral operators on spaces of homogeneous type
Hardy spaces on affine symmetric spaces.
Hardy spaces on two-sheeted covering semigroups.
Hardy-type spaces of harmonic functions on symmetric spaces on noncompact type.
Harmonic analysis for spinors on real hyperbolic spaces
We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...
Harmonic analysis of spherical functions on
Denote by the algebra of spherical integrable functions on , with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in . In particular, we are interested in conditions on an ideal that ensure that it is all of , or that it is . Spherical functions on are naturally represented as radial functions on the unit disk in the complex plane. Using this representation, these results are applied to characterize harmonic...
Harmonic analysis on .
Harmonic functions on homogeneous spaces of commutator induced extensions of compact groups
Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces
We study harmonic functions for the Laplace-eltrami operator on the real hyperbolic space . We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball . We then study the Hardy spaces , 0
Henkin measures, Riesz products and singular sets
The mutual singularity problem for measures with restrictions on the spectrum is studied. The -pluriharmonic Riesz product construction on the complex sphere is introduced. Singular pluriharmonic measures supported by sets of maximal Hausdorff dimension are obtained.
High order representation formulas and embedding theorems on stratified groups and generalizations
We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable to Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and to Poincaré...