We introduce coarse cohomology for families of metric spaces and develop the properties of this cohomology theory. Foliations provide the primary examples of such families and for these there are de~Rham, Čech and Alexander-Spanier versions of this theory, which are all isomorphic. We compute the coarse cohomology for a number of important examples.

We derive weighted smoothing inequalities for solutions of the free Schrödinger equation. As an application, we give a new proof of the endpoint Strichartz estimates in the radial case. We also consider general dispersive equations and obtain similar estimates in this case.

We give a geometric classification of regular ends with constant mean curvature $1$ and finite total curvature, embedded in hyperbolic space. We prove that each such end is either asymptotic to a catenoid cousin or asymptotic to a horosphere. We also study symmetry properties of constant mean curvature $1$ surfaces in hyperbolic space associated to minimal surfaces in Euclidean space. We describe the constant mean curvature $1$ surfaces in $\hi3$ associated to the family of surfaces in $\m3$ that is isometric to the helicoid.

Consider monotone functions $u\colon \mathbb{B}^n\to \mathbb{R}$ in the weighted Sobolev space $W^{1,p}(\mathbb{B}^n;w)$, where $n-1<p\leq n$ and $w$ is a weight in the class $A_q$ for some $1\leq q<{{p}/({n-1})}$ which has a certain symmetry property with respect to $\partial \mathbb{B}^n$. We prove that $u$ has nontangential limits at all points of $\partial \mathbb{B}^n$ except possibly those on a set $E$ of weighted $(p,w)$-capacity zero. The proof is based on a new weighted oscillation estimate (Theorem 1) that may be of independent interest. In the special case $w(x)=|1-|x||^\alpha$, the weighted $(p,w)$-capacity of a ball can be easily estimated to conclude that the Hausdorff dimension of the set $E$ is smaller than or equal to $\alpha+n-p$, where $0\leq \alpha<{(p-(n-1))}/{(n-1)}$.

We investigate relationships between classes of harmonic functions corresponding to the meromorphic $Q_{p}^{\#}$ classes. We give many analogs to the situations in the corresponding analytic and meromorphic classes, and we give some examples in which the behavior is different in the harmonic classes.

We determine when the Hopf vector fields on orientable real hypersurfaces $(M,g)$ in complex space forms are minimal or harmonic. Furthermore, we determine when these vector fields give rise to harmonic maps from $(M,g)$ to the unit tangent sphere bundle $(T_1M,g_S)$. In particular, we consider the special case of Hopf hypersurfaces and of ruled hypersurfaces. The Hopf vector fields on Hopf hypersurfaces with constant principal curvatures provide examples. The minimal ruled real hypersurfaces form another class of particular examples.

We compare Green's function $g$ on an infinite volume, hyperbolic Riemann surface $X$ with an analogous discrete function $g_{\disc}$ on a graphical caricature $\Gamma$ of $X$. The main result, modulo technical hypotheses, is that $g$ and $g_{\disc}$ differ by at most an additive constant $C$ which depends only on the Euler characteristic of $X$. In particular, the estimate of $g$ by $g_{\disc}$ remains uniform as the geometry (i.e., the conformal structure) of $X$ varies.

In this paper we obtain some uniqueness theorems for $p$-adic holomorphic curves intersecting hyperplanes.

In this paper we define the loose block independence property for positive entropy $\mathbb Z^d$ actions and extend some of the classical results to higher dimensions. In particular, we prove that two loose block independent actions are even Kakutani equivalent if and only if they have the same entropy. We also prove that for $d > 1$ the ergodic, isometric extensions of the positive entropy loose block independent $\mathbb Z^d$ actions are also loose block independent.

Given a compact Riemannian manifold on which a compact Lie group acts by isometries, it is shown that there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension of the orbit may be different from the dimension of the leaf closure. Conversely, given a Riemannian foliation, there is a metric on the basic manifold (an $O(q)$-manifold associated to the foliation) such that the leaf closure space is isometric to the $O(q)$-orbit space of the basic manifold via an isometry that preserves the volume of the leaf closures of maximal dimension. Thus, the orbit space of any Riemannian G-manifold is isometric to the orbit space of a Riemannian $O(q)$-manifold via an isometry that preserves the volumes of orbits of maximal dimension. Consequently, the spectrum of the Laplacian restricted to invariant functions on any $G$-manifold may be identified with the spectrum of the Laplacian restricted to invariant functions on a Riemannian $O(q)$-manifold. Other similar results concerning the spectrum of differential operators on sections of vector bundles over Riemannian foliations and $G$-manifolds are discussed.

If $\ohm$ is a ball in $\Real ^n$ $(n\geq 2)$, then the boundary integral operator of the double layer potential for the Laplacian is self-adjoint on $L^2({\partial}{\ohm})$. In this paper we prove that the ball is the only bounded Lipschitz domain on which the integral operator is self-adjoint.

In this note the authors prove the $L^p(\mathbb{R}^n \times \mathbb{R}^m)$-boundedness for a class of maximal singular integral operators with rough kernel on product spaces. This extends a result obtained by Chen and Wang in 1992.

We prove a Glivenko-Cantelli type strong law of large numbers for the pair correlation of independent random variables. Except for a few powers of logarithms the results obtained are sharp. Similar estimates hold for the pair correlation of lacunary sequences $\{n_k\omega\}\mod 1$.

The purpose of this paper is to introduce and study the notion of a vector-valued $\pi\text{-invariant}$ mean associated to a unitary representation $\pi$ of a locally compact group $G$ on $\mathcal{S},$ a self-adjoint linear subspace containing $I$ of $\mathcal{B}(H_\pi).$ We obtain, among other results, an extension theorem for $\pi\text{-invariant}$ completely positive maps and $\pi\text{-invariant}$ means which characterizes amenability of $G.$ We also study vector-valued means on $\mathcal{S}$ of $\pi\text{-(weakly)}$ almost periodic operators on $H_\pi.$

Let $F:M\to N$ be a harmonic map between complete Riemannian manifolds. Assume that $N$ is simply connected with sectional curvature bounded between two negative constants. If $F$ is a quasiconformal harmonic diffeomorphism, then $M$ supports an infinite dimensional space of bounded harmonic functions. On the other hand, if $M$ supports no non-constant bounded harmonic functions, then any harmonic map of bounded dilation is constant.

We prove that the manifold $M^n$ of minimal radial curvature $K^{\min}_o\geq 1$ is homeomorphic to the sphere $S^n$ if its radius or volume is larger than half the radius or volume of the round sphere of constant curvature $1$. These results are optimal and give a complete generalization of the corresponding results for manifolds of sectional curvature bounded from below.

We present an elementary proof of two a priori estimates for Schrödinger type multipliers on the circle. The first is an $L^4 - L^2$ inequality of Bourgain, while the second is a new $L^6 - L^{3/2}$ inequality. Estimates of this type are useful for the study of the Cauchy problem for Schr\"odinger type equations. The proofs are based on a counting argument and standard real and harmonic analysis techniques.

We study the relationship between the Assouad dimension and quasisymmetric mappings, showing that spaces of dimension strictly less than one can be quasisymmetrically deformed onto spaces of arbitrarily small dimension. We conjecture that this fact holds also for the Hausdorff dimension, and our results yield several corollaries which provide partial support for this conjecture. The proofs make use of connections between Assouad dimension, porosity, and ultrametrics.

In this paper we study the unstability and nonexistence of $F$-harmonic maps. We introduce the notion of $F$-strongly unstable and $F$-unstable manifolds and discuss properties of such manifolds. We classify all compact irreducible $F$-unstable symmetric spaces.

We examine conditions under which a point in the stable set of a hyperbolic invariant set for a $C^1$ surface diffeomorphism is accessible via a path from the complement of the stable set. Let $M$ be a surface, and let $\Lambda$ be a compact saturated hyperbolic locally stably closed invariant set possessing a local product structure. Denote the stable set of $\Lambda$ by $W^s(\Lambda)$. Our main result states that $z \in W^s(\Lambda)$ is accessible from $M \setminus W^s(\Lambda)$ if and only if $z$ lies on the stable manifold of a periodic point $p$, and there is a branch of a local unstable manifold of $p$ disjoint from $W^s(\Lambda)$.