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An application of shift operators to ordered symmetric spaces

Nils Byrial Andersen, Jérémie M. Unterberger (2002)

Annales de l’institut Fourier

We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ℳ m , n of Cayley type, where m ∈ ℕ denotes the multiplicity of the short roots and n ∈ ℕ the rank of the symmetric space. For m even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ℳ m , n by a reduction to the rank 1 case. Finally we generalize our notions and results to B C n type root systems.

Analysis of two step nilsequences

Bernard Host, Bryna Kra (2008)

Annales de l’institut Fourier

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg’s proof of Szemerédi’s Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for two step nilsequences and give a classification scheme for them.

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

M. Brundin (2007)

Czechoslovak Mathematical Journal

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L p and weak L p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L Φ having the property L ∞ ⊂ L Φ ⊂ L p , 1 ≤ p < ∞ . The second contains spaces L Φ that...

Besov spaces on symmetric manifolds—the atomic decomposition

Leszek Skrzypczak (1997)

Studia Mathematica

We give the atomic decomposition of the inhomogeneous Besov spaces defined on symmetric Riemannian spaces of noncompact type. As an application we get a theorem of Bernstein type for the Helgason-Fourier transform.

BGG sequences on spheres

Petr Somberg (2000)

Commentationes Mathematicae Universitatis Carolinae

BGG sequences on flat homogeneous spaces are analyzed from the point of view of decomposition of appropriate representation spaces on irreducible parts with respect to a maximal compact subgroup, the so called K -types. In particular, the kernels and images of all standard invariant differential operators (including the higher spin analogs of the basic twistor operator), i.e. operators appearing in BGG sequences, are described.

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