Let $Tf(x,t) = e^{2\pi it\phi(D)}f(x)$ be the solution of the general dispersive equation with phase $\phi$ and initial data $f$ in the Sobolev space $H^s$. We prove a weighted $L^2$ estimate for the global maximal operator $T^{**}$ defined by taking the supremum over the time variable $t \in \mathbb{R}$ so that $ \|T^{**}f\|_{L^2(w\,dx)} \le C\|f\|_{H^s}$. The exponent $s$ depends on the phase function $\phi$, whose gradient may vanish or have singularities.

We give sharp endpoint estimates for the decay rates of $L^p$ operator norms of oscillatory integral operators with some real homogeneous polynomial phases.

Let $k \ge 2$ be an integer and let ${\mc A}$ and ${\mc B}$ be two sets of integers. We give upper bounds for the number of perfect $k$-th powers of the form $ab+1$, with $a$ in ${\mc A}$ and $b$ in ${\mc B}$. We further investigate several related questions.

Let $G$ be a locally compact group and $\mu$ a probability measure on $G$. Given a unitary representation $\pi$ of $G$, let $P_\mu$ denote the $\mu$-average $\int_G\pi(g)\,\mu(dg)$. $\mu$ is called neat if for every unitary representation $\pi$ and every $a$ in the support of $\mu$, $\slim_{n\to\infty}\bigl(P_\mu^n -\pi(a)^n E_\mu\bigr) =0$, where $E_\mu$ is a canonically defined orthogonal projection. $G\/$ is called neat if every almost aperiodic probability measure on $G$ is neat. Previously known results show that every almost aperiodic spread out probability measure is neat, in particular, every discrete group is neat; furthermore, identity excluding groups, in particular, compact groups and nilpotent groups, are neat. In this work neatness of solvable Lie groups, connected algebraic groups, Euclidian motion groups, [SIN] groups, and extensions of abelian groups by discrete groups is established. Neatness of ergodic probability measures on any locally compact group is also proven. The key to these results is the result that when $\{X_n\}_{n=1}^\infty$ is the left random walk of law $\mu$ on $G$ and $\pi$ a unitary representation in a separable Hilbert space, then for every $k=0,1,\dots$\,, the sequence $\pi(X_n)^{-1}P_\mu^{n-k}$ converges almost surely in the strong operator topology.

A singular foliation on a complete riemannian manifold is said to be riemannian if every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. In this paper, we study singular riemannian foliations with sections. A section is a totally geodesic complete immersed submanifold that meets each leaf orthogonally and whose dimension is the codimension of the regular leaves.

We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidean space. This result provides local information about the singular foliation and in particular about the singular stratification of the foliation. It allows us to describe the plaques of the foliation as level sets of a transnormal map (a generalization of an isoparametric map). We also prove that the regular leaves of a singular riemannian foliation with sections are locally equifocal. We use this property to define a singular holonomy. Then we establish some results about this singular holonomy and illustrate them with a couple of examples.

A continuous mapping between compact topological groups which is "almost" a homomorphism need not be uniformly close to a homomorphism. The distance of the almost homomorphism from a strict homomorphism can be shown to depend not just on the degree of its nonhomomorphy but also on some "continuity scale."

We prove the Focal Index Lemma and the Rauch--Berger Comparison Theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop space $\Omega (M^n)$ with the $L^2$ (weak) Riemannian structure.

Let $G$ be a locally compact unimodular group and $\mu$ an adapted spread out probability measure on $G$. We relate the rate of decay of the concentration functions associated with $\mu$ to the growth of a certain subgroup $N_{\mu}$ of $G$. In particular, we show that when $\mu$ is strictly aperiodic (i.e., when $N_{\mu}=G$) and $G$ satisfies the growth condition $V_G(m)\geq Cm^D$, then for any compact neighborhood $K\subset G$ we have $\sup_{g\in G}\mu^{*n}(gK) \leq C'n^{-D/2}$. This extends recent results of Retzlaff \cite{Retzlaff} on discrete groups for adapted probability measures.

We show that the complex dilatation of the Douady-Earle extension of a strongly quasisymmetric homeomorphism produces a Carleson measure. As an application, we study the BMO-Teichmüller theory compatible with a Fuchsian group.