A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric spaces Eigenspace representations Solutions to exercises Bibliography Symbols frequently used Index.

Geometric Analysis on Symmetric Spaces

I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II. The boundary at infinity.- 1. Closure of X via Busemann functions.- 2. Closure of X via rays.- 3. Classification of isometries.- 4. The cone at infinity and the Tits metric.- III. Weak hyperbolicity.- 1. The duality condition.- 2. Geodesic flows on Hadamard spaces.- 3. The flat half plane condition.- 4. Harmonic functions and random walks on ?.- IV. Rank rigidity.- 1. Preliminaries on geodesic flows.- 2. Jacobi fields and curvature.- 3. Busemann functions and horospheres.- 4. Rank, regular vectors and flats.- 5. An invariant set at infinity.- 6. Proof of the rank rigidity.- Appendix. Ergodicity of geodesic flows.- 1. Introductory remarks.- Measure and ergodic theory preliminaries.- Absolutely continuous foliations.- Anosov flows and the Ho continuity of invariant distributions.- Proof of absolute continuity and ergodicity.

Lectures on Spaces of Nonpositive Curvature

Entropies et rigidités des espaces localement symétriques de courbure strictement négative

We prove the l 2 Decoupling Conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which implies the full range of expected L p Strichartz estimates for both the rational and (up to N " losses) the irrational torus. Another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere. Various applications to Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.

http://annals.math.princeton.edu/wp-content/uploads/Bourgain_Demeter.pdf

The proof of the l 2 Decoupling Conjecture

We consider the multidimensional aggregation equation ut − ∇ (u∇K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717–35), whereas for C2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K(x) = |x| and some open problems.

/pdf/blow-up-in-multidimensional-aggregation-equations-with-369qta94xp.pdf

Blow-up in multidimensional aggregation equations with mildly singular interaction kernels !

H-type groups and Iwasawa decompositions☆

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra This makesS an example of a harmonic manifold, not necessarily symmetric In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context We finally determine the spherical functions for the groupsS and their Martin boundary

Harmonic analysis on solvable extensions of H-type groups

Certain solvable extensions of $H$-type groups provide noncompact counterexamples to the so-called Lichnerowicz conjecture, which asserted that ``harmonic'' Riemannian spaces must be rank 1 symmetric spaces.

/pdf/a-class-of-nonsymmetric-harmonic-riemannian-spaces-18nr2ca6vx.pdf

Fulvio Ricci

Papers

H-type groups and Iwasawa decompositions☆

Harmonic analysis on solvable extensions of H-type groups

A class of nonsymmetric harmonic Riemannian spaces

Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, I

Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds