This paper is devoted to the study on the $L^p$-mapping properties for a class of multilinear oscillatory singular integrals with polynomial phase and rough kernel. By means of the method of block decomposition for the kernel function, the authors show that for any non-trivial polynomial phase, the $L^p(\rz)$ boundedness of the multilinear oscillatory singular integral operators and that of the corresponding local multilinear singular integral operators are equivalent; and for any real-valued polynomial phase, the $L^p(\rz)$ boundedness of the multilinear oscillatory integral operators can be deduced from that of the corresponding multilinear singular integral operators.

We study smooth foliations of compact manifolds where the set of noncompact leaves has zero Lebesgue measure (and is nonempty). We review a simple example due to Reeb and provide two new examples that show more complicated behavior that may occur.

In the context of a strongly local Dirichlet space we show that if a function mapping the real line to itself (and fixing the origin) operates by composition on the left to map the Dirichlet space into itself, then the function is necessarily locally Lipschitz continuous. If, in addition, the Dirichlet space contains unbounded elements, then the function must be globally Lipschitz continuous. The proofs rely on a co-area formula for condenser potentials.

It is proved that, for a complex minimal smooth projective surface $S$ of general type, any automorphism group of $S$, inducing trivial actions on the second rational cohomology of $S$, is isomorphic to a cyclic group of order less than five or the product of two groups of order two, provided that the Euler characteristic of the structure sheaf of $S$ is larger than $188$.

We prove that the unit tangent sphere bundle of a Riemannian manifold is semi-symmetric if and only if it is locally symmetric, i.e., the base manifold is either flat or it is two-dimensional with constant sectional curvature 1.

In the previous note we calculated the rank of the group of relative units for a Galois extension of number fields. In this note the calculation is concluded.

Two-weight $L^p$ norm inequalities, uniform with respect to the order of the involved Bessel function, are proved for the Bochner-Riesz means of the first order for the Hankel transform. Both sufficient and necessary conditions for parameters used in the two weights are determined. The proof relies on uniform pointwise asymptotic estimates for the Bessel functions that were shown by Barceló and Córdoba.

We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, then strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.

The continuous wavelet transform is extended to certain distributions and continuity results are obtained. Boundedness results in a generalized Sobolev space, Besov space and Lizorkin-Triebel space are given.

We construct codimension-one, Lorentzian geodesible foliations of closed three-manifolds having Heegaard splittings of genus one. We prove that all the inner leaves of a Reeb component of a codimension-one, totally geodesic foliation of a Lorentzian three-manifold are spacelike, and the boundary leaf of a Reeb component is lightlike.

Given a Jordan domain $\Omega$ in the extended complex plane $\overline{\kern-1.5pt\Bbb C}$, denote by $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ the boundary quasiextremal distance constant, quasiextremal distance constant and quasiconformal reflection constant of $\Omega$, respectively. It is known that $M_b(\Omega)\le M(\Omega)\le R(\Omega)+1$. In this paper, we will give some further relations among $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ by introducing and studying some other closely related constants. Particularly, we will give a necessary and sufficient condition for $M_b(\Omega)=R(\Omega)+1$ and show that $M(\Omega)<R(\Omega)+1$ for all asymptotically conformal extension domains other than disks. This gives an affirmative answer to a question asked by Yang, showing that the conjecture $M(\Omega)=R(\Omega)+1$ by Garnett and Yang is not true for all asymptotically conformal extension domains other than disks. Our discussion relies heavily on the theory of extremal quasiconformal mappings, which in turn gives some interesting results in the extremal quasiconformal mapping theory as well.