Conical distributions for rank one symmetric spaces

Patterson–Sullivan distributions for rank one symmetric spaces of the noncompact type

There is a remarkable relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold: It was observed in [1] that for compact hyperbolic surfaces XΓ = Γ\H Wigner distributions R S∗XΓ a dWirj = 〈Op(a)φirj , φirj 〉L2(XΓ) and Patterson–Sullivan distributions PSirj are asymptotically equivalent as rj → ∞. We generalize the defi...

Cohomogeneity One Actions on Noncompact Symmetric Spaces of Rank One

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CHn, n ≥ 3. For the quaternionic hyperbolic spaces HHn, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificat...

Singular Integrals on Symmetric Spaces of Real Rank One

In this paper we prove a new variant of the Herz majorizing principle for operators defined by K-bi-invariant kernels with certain large-scale cancellation properties. As an application, we prove L p-boundedness of operators defined by Fourier multipliers which satisfy singular differential inequalities of the Hörmander-Michlin type. We also find sharp bounds on the L p-norm of large imaginary ...

Ford Fundamental Domains in Symmetric Spaces of Rank One

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and domains are essentially subsumed by our work.

Curvature Characterization and Classification of Rank-one Symmetric Spaces