Given weight functions $\theta$, $w$, $\rho$ and $v$, the weighted modular inequality $$ Q^{-1}\bigg(\int_{0}^{\infty} Q(\theta(x)Tf(x)) w(x)\, dx\bigg) \leq P^{-1}\bigg(\int_{0}^{\infty} P(C \rho(x) f(x)) v(x)\, dx\bigg)$$ is characterized. Here $Q$ is a strictly increasing function with $Q(0) = 0$, $Q(\infty) = \infty$ and $2Q(x) \leq Q(C x)$, $P$ is a Young's function, and $T$ is the Hardy operator or a Hardy type operator. In particular, a characterizing condition for the Hardy type operator to map $L^{p}(w)$ to $L^{q}(v)$ when $0 < q < 1 \leq p < \infty$ is deduced. In addition, a new proof for the Maz'ja-Sinnamon theorem is given, and weighted Lorentz norm inequalities for Hardy type operators are established.

1991 Mathematics Subject Classification: primary 26D15, 42B25; secondary 26A33, 46E30.

We study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets, typical examples being the Meixner polynomials and the Charlier polynomials. Following ideas of Rakhmanov, Dragnev and Saff, weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials, which can be solved explicitly. We give formulas for the contracted zero distributions, the $n$th root asymptotics and the asymptotics of the largest zeros.

Let $f$ be an orientation-preserving homeomorphism of a compact orientable manifold. Sufficient conditions are given for the persistence of a collection of periodic points under isotopy of $f$ relative to a compact invariant set $A$. Two main applications are described. In the first,~$A$ is the closure of a single discrete orbit of~$f$, and~$f$ has a Smale horseshoe, all of whose periodic orbits persist; in the second,~$A$ is a minimal invariant Cantor set obtained as the limit of a sequence of nested periodic orbits, all of which are shown to persist under isotopy relative to~$A$.

1991 Mathematics Subject Classification: 58F20, 58F15.

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centreline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions.

The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the $m$-fold covered circle to the $n$-fold covered circle for any relatively prime $m$ and $n$. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centrelines.

1991 Mathematics Subject Classification: primary 53A04, 73C02; secondary 57M25.

Denote by $\Bbb A$ the class of all absolutely continuous contractions whose associated Sz. Nagy-Foias functional calculus is isometric. Starting from the fact that if $u$ is a non-constant inner function and if $T\in {\Bbb A}$, then so does $u(T)$, we study how inner functions operate on the classes ${\Bbb A}_{n,m}$, subclasses of the class ${\Bbb A}$. For this purpose, we use standard dual algebra techniques and a decomposition of the algebra $H^\infty$ into a weak*-topological direct sum of copies of itself. We also discuss mapping theorems for the support of the spectral measures associated with the unitary parts of the minimal isometric extension and the minimal co-isometric extension of an absolutely continuous contraction $T$.

We prove global existence and uniqueness of solutions to the Rarita-Schwinger evolution equations compatible with the constraints. We use a gauge fixing for the Rarita-Schwinger equations for helicity 3/2 fields in curved space that leads to a straightforward Hilbert space framework for their study. We explain how these results might be applied to the global analysis of the full Einstein vacuum equations and provide a complete analysis as a basis for such applications. These and a programme for developing a scattering/inverse scattering transform for the full Einstein equations are discussed.

1991 Mathematics Subject Classification: 83C60, 35Q75, 83C05, 35L45.

Let $\cal V$ be a complete connected hyperbolic 3-manifold of finite volume, with Liouville measure $m$, geodesic flow $\Gamma_t$ and Brownian motion $Z_t$. Let $\omega$ be a smooth 1-form, closed in the cusps of $\cal V$. Then the limit laws as $t \to \infty$ of $(t\log t)^{-1/2}\int_0^t\omega(\Gamma_s)$ under $m$ and of $(t\log t)^{-1/2}\int_0^t\omega(Z_s)$ are calculated, and seen to be Gaussian, and equal. The geodesic flow case is studied via the Brownian case.

1991 Mathematics Subject Classification: 60J65, 58F17, 51M10.