We study the Riesz means for the eigenfunction expansions of a class of hypoelliptic differential operators on the Heisenberg group. The operators we consider are homogeneous with respect to dilations and invariant under the action of the unitary group. We obtain convergence results in $L^p$ norm, at Lebesgue points and almost everywhere. We also prove localization results.

We show that the $L^p$ boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho$ behaves like a polynomial in $\rho$, and consider the mapping properties between amalgams of kernel operators where the kernel $kerK(x,y)$ behaves like $d(x,y{)}^{-a}$ when $d(x,y)\le 1$ and like $d(x,y{)}^{-b}$ when $d(x,y)\ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

We study the validity of the $L^p$ inequality for the Riesz transform when $p\>2$ and of its reverse inequality when $1\<p\<2$ on complete riemannian manifolds under the doubling property and some Poincaré inequalities.

We prove $L^p$-bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for $p$ in a certain interval depending on the Lipschitz character of the domain.

Assume that $M_0$ is a complete Riemannian manifold with Ricci curvature bounded from below and that $M_0$ satisfies a Sobolev inequality of dimension $\nu \>3$. Let $M$ be a complete Riemannian manifold isometric at infinity to $M_0$ and let $p\in (\nu /(\nu -1),\nu )$. The boundedness of the Riesz transform of $L^p\left(M_0\right)$ then implies the boundedness of the Riesz transform of $L^p\left(M\right)$

A G-shift of finite type (G-SFT) is a shift of finite type which commutes with the continuous action of a finite group G. For irreducible G-SFTs we classify right closing almost conjugacy, answering a question of Bill Parry.

We study rigid paths of generic 2-distributions with degenerate points on 3-manifolds. A complete description of such paths is obtained. For the proof, we construct separating surfaces of paths admissible for distributions.

Using the description of non solvable dynamics by Nakai, we give in this paper a new proof of the rigidity properties of some sub-groups of Diff(C, O). The Cinfinity case is also considered here.

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M,\omega )$ with $c_1{{|}_{{\pi }_2\left(M\right)}=\left[\omega \right]|}_{{\pi }_2\left(M\right)}=0$. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^0$ close generic function/hamiltonian....

When $X=\Gamma \setminus {ℍ}^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.

Any Kähler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.