In this note we consider a stability property of p-harmonic functions in general metric measure spaces. Despite the fact that there may not be a corresponding differential equation, it is shown here that a family of uniformly convergent p-harmonic functions converges to a p-harmonic function; that is, the collection of p-harmonic functions forms a closed subspace in a certain Ho¨lder class. As a consequence, it is shown that a family of p-harmonic functions bounded in a Sobolev-type space (called the Newtonian space) has a sequence that converges locally uniformly in the domain of harmonicity to a p-harmonic function. This result is used to construct p-harmonic functions on unbounded domains. We also use this convergence result to prove a characterization of a parabolicity property of metric measure spaces. This characterization has been given for Riemannian manifolds by Holopainen, and the result here is a generalization of Holopainen's result.

Let $G$ be a locally compact group, and let $L^1 (G)$ be the Banach algebra which is the group algebra of $G$. We consider a variety of Banach left $L^1 (G)$ -modules over $L^1 (G)$ , and seek to determine conditions on $G$ that determine when these modules are either projective or injective or flat in the category. The answers typically involve $G$ being compact or discrete or amenable. For example, in the case where $G$ is discrete and $1 < p < \infty$, we find that the module $\ell^p (G)$ is injective whenever $G$ is amenable, and that, if it is amenable, then $G$ is ‘pseudo-amenable’, a property very close to that of amenability.

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.

The relation between $\mathbb{Q}$-curves and certainabelian varieties of $\operatorname{GL}_2$-type was established by Ribet (`Abelian varieties over $\mathbb{Q}$ and modular forms', {\em Proceedings ofthe KAIST Mathematics Workshop} (1992) 53--79)and generalized to building blocks, the higher-dimensional analogues of $\mathbb{Q}$-curves,by Pyle in her PhD Thesis (University of California at Berkeley, 1995).In this paper we investigate some aspectsof $\mathbb{Q}$-curves with no complex multiplicationand the corresponding abelian varieties of $\operatorname{GL}_2$-type, for which we mainly use the ideas and techniques introduced by Ribet(op. cit. and `Fields of definition of abelian varieties with real multiplication', {\em Contemp.\ Math.} 174(1994) 107--118).After the Introduction,in Sections 2 and 3 we obtain a characterization of thefields where a $\mathbb{Q}$-curve and all the isogeniesbetween its Galois conjugates can be defined up to isogeny, and we apply it to certain fields of type $(2,\dots,2)$.In Section 4 we determine the endomorphism algebras of all the abelian varieties of $\operatorname{GL}_2$-type having as a quotient a given $\mathbb{Q}$-curve in easily computable terms.Section 5 is devoted to a particular case ofWeil's restriction of scalars functor applied to a $\mathbb{Q}$-curve,in which the resulting abelian variety factors over $\mathbb{Q}$ up to isogeny as a product of abelianvarieties of $\operatorname{GL}_2$-type.Finally, Section 6 contains examples:we parametrize the $\mathbb{Q}$-curves coming fromrational points of the modular curves $X^*(N)$ having genus zero, and we apply the results of Sections 2--5 to some of the curves obtained.We also give results concerning the existence of quadratic $\mathbb{Q}$-curves. 1991 Mathematics Subject Classification:primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.

We give a new non-capacitary characterization of positive Borel measures $\mu$ on ${\Bbb R}^n$ such that the potential space $I_\alpha*L^p$ is imbedded in $L^q(d\mu)$ for $1< q < p<+\infty$, that is, the trace inequality $\|I_\alpha f\|_{L^q(d\mu)} \le C \, \|f\|_{L^p(d x)}$ holds, for Riesz potentials $I_\alpha = (- \Delta)^{-\alpha/2}$. A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere ${\Bbb S}^n$ is studied, as well as the holomorphic case for Hardy--Sobolev spaces $H_\alpha^p$ in the ball. 1991 Mathematics Subject Classification: primary 31C15, 42B20; secondary 32A35.

We prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds $P^m$ in ambient Riemannian manifolds $N^n$ with a pole $p$. The upper bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained via previously established isoperimetric inequalities, which are here extended to hold for this more general setting based on warped product comparison spaces. We also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for Brownian motion at infinity is finite.

The Hardy operator $T_a$ on a tree $\Gamma$ is defined by \[ (T_af)(x):=v(x) \int^x_a f(t)u(t)\,dt \quad \mbox{for } a, x\in \Gamma. \] Properties of $T_a$ as a map from $L^p(\Gamma)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy \begin{equation*} \tag{$*$} \lim_{n\rightarrow \infty} na_n(T_a) = \alpha_p\int_{\Gamma}|uv|\,dt \end{equation*} for a specified constant $\alpha_p$ and $1 p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, $(*)$ was unknown even when $\Gamma$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined.

2000 Mathematical Subject Classification: 47G10, 47B10.

Let $G$ be a geometrically finite Kleinian group with parabolic elements and let $p$ be any parabolic fixed point of $G$. For each positive real $\tau$, let ${\cal W}_{p}(\tau)$ denote the set of limit points of $G$ for which the inequality $$ | x-g(p) | \leq |g^\prime(0)|^{\tau}$$ is satisfied for infinitely many elements $g$ in $G$. This subset of the limit set is precisely the analogue of the set of $\tau$-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the `size' of the set ${\cal W}_{p}(\tau)$ expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for $\tau \geq 1$,$$ \dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$$ where $\mbox{rk}(p)$ denotes the rank of the parabolic fixed point $p$.

1991 Mathematics Subject Classification: 11K55, 11K60, 11F99, 58D20.

We prove that, if $M > 4(1+2\sqrt{3})$ and $\varepsilon > 0$, if $\mathcal{V}$ and $\mathcal{W}$ are complex JBW*-triples (with preduals $\mathcal{V}_{*}$ and $\mathcal{W}_{*}$, respectively), and if $U$ is a separately weak*-continuous bilinear form on $\mathcal{V} \times \mathcal{W}$, then there exist norm-one functionals $\varphi_{1},\varphi_{2}\in \mathcal{V}_{*}$ and $\psi_{1},\psi_{2}\in \mathcal{W}_{*}$ satisfying |U(x,y)| \leq M \,\|U\| ( \|x\|_{\varphi_{2}}^{2} + \varepsilon^{2} \, \|x\|_{\varphi_{1}}^{2} )^{\frac 12} ( \|y\|_{\psi_{2}}^{2} + \varepsilon^{2} \, \|y\|_{\psi_{1}}^{2} )^{\frac 12} for all $(x,y)\in \mathcal{V} \times \mathcal{W}$. Here, for a norm-one functional $\varphi$ on a complex JB*-triple $\mathcal{V}$, $\|\cdot\|_{\varphi}$ stands for the prehilbertian seminorm on $\mathcal{V}$ associated to $\varphi$ given by $\|x\|_{\varphi}^{2} := \varphi \{x,x,z\}$ for all $x\in \mathcal{W}$, where $z\in \mathcal{V}^{**}$ satisfies $\varphi (z) = \|z\|=1$. We arrive at this form of ‘Grothendieck's inequality’ through results of C.-H. Chu, B. Iochum, and G. Loupias, and an amended version of the ‘little Grothendieck's inequality’ for complex JB*-triples due to T. Barton and Y. Friedman. We also obtain extensions of these results to the setting of real JB*-triples. 2000 Mathematical Subject Classification: 17C65, 46K70, 46L05, 46L10, 46L70.

Following the work of Chevalley and Warning, Ax obtained a bound on the $p$-divisibility of exponential sums involving multi-variable polynomials of fixed degree $d$ over a finite field of characteristic $p$. This bound was subsequently improved by Katz. More recently, Moreno and Moreno, and Adolphson and Sperber, derived bounds that in many instances improved upon the Ax–Katz result. Here we derive a tight bound on the $p$-divisibility of the exponential sums. While exact computation of this bound requires the solution of a system of modular equations, approximations are provided which in several classes of examples, improve on the results of Chevalley and Warning, Ax and Katz, Adolphson and Sperber, and Moreno and Moreno. All of the above results readily translate into bounds on the $p$-divisibility of the number of zeros of multi-variable polynomials.

An important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight. In particular, we give classes of examples for which the Moreno–Moreno bound is tight.

It is important to note that we have also found applications of our results to coding theory (computation of the covering radius of certain codes) and to Waring's problem over finite fields. These will be described elsewhere.

The paper extends to complex Hamiltonian systems previous work of the authors on the Sims extension of the Titchmarsh–Weyl theory for Sturm–Liouville equations with complex potentials, and analyses the spectral properties of associated non-self-adjoint operators.

Let $V(\mathbb{R})$ denote the Banach algebra of absolutely continuous functions of bounded total variation on $\mathbb{R}$, and let $\mathcal{B}_p$ be the Banach algebra of bounded linear operators acting on the Lebesgue space $L^p(\mathbb{R})$ for $1 < p < \infty$. We study the Banach algebra $\mathfrak{A}\subset\mathcal{B}_p$ generated by the pseudodifferential operators of zero order with slowly oscillating $V(\mathbb{R})$-valued symbols on $\mathbb{R}$. Boundedness and compactness conditions for pseudodifferential operators with symbols in $L^\infty (\mathbb{R}, V(\mathbb{R}))$ are obtained. A symbol calculus for the non-closed algebra of pseudodifferential operators with slowly oscillating $V(\mathbb{R})$-valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. As a result, the quotient Banach algebra $\mathfrak{A}^\pi = {\mathfrak A} / \mathcal{K}$, where $\mathcal{K}$ is the ideal of compact operators in $\mathcal{B}_p$, is commutative and involutive. An isomorphism between the quotient Banach algebra $\mathfrak{A}^\pi$ of pseudodifferential operators and the Banach algebra $\widehat{\mathfrak{A}}$ of their Fredholm symbols is established. A Fredholm criterion and an index formula for the pseudodifferential operators $A \in \mathfrak{A}$ are obtained in terms of their Fredholm symbols.

Suppose a group $G$ acts on a Gromov-hyperbolic space $X$ properly discontinuously. If the limit set $L(G)$ of the action has at least three points, then the second bounded cohomology group of $G$,$H^2_b(G;{\Bbb R})$ is infinite dimensional. For example, if $M$ is a complete, pinched negatively curved Riemannian manifold with finite volume, then $H_b^2(\pi _1(M); {\Bbb R})$ is infinite dimensional. As an application, we show that if $G$ is a knot group with $G \not\simeq{\Bbb Z}$, then $H^2_b(G;{\Bbb R})$ is infinite dimensional.

1991 Mathematics Subject Classification: primary 20F32; secondary 53C20, 57M25.

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analogous to the dyadic partition $[2^j, 2^{ j + 1})$ of the positive real line, in this paper we develop an adapted Littlewood–Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p, q}_\nu$, where the role of the usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. We show that $B^{p, q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p, q}_\nu (T_\Omega)$, at least in a 'good range' of indices $1 \leq q < q_{\nu, p}$. We obtain the sharp $q_{\nu, p}$ when $p \leq 2$, and conjecture a critical index for $p > 2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu \colon L^{p, q}_\nu \to A^{p, q}_\nu$, for which our result implies a positive answer when $q_{\nu, p}' < q < q_{\nu, p}$. This extends, to general cones, previous work of the authors on the light-cone.

Finally, we focus on light-cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light-cones when $p$ is sufficiently large.

Schwarz's lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances. In this paper, inner functions which decrease hyperbolic distances as much as possible, when one approaches the unit circle, are constructed. Actually, it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function. This is related to a result of L. Carleson on the existence of singular symmetric measures. As a consequence, some results on composition operators are obtained, bringing out the importance of the Bloch spaces in this connection. Another consequence is a uniform way of producing singular measures which are simultaneously symmetric and Kahane.

1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.

The absolutely continuous spectrum of a very general class of differential operators of order $2n$ is determined, for operators whose coefficients satisfy conditions that combine smoothness and decay properties. The main methods are asymptotic integration and the analysis of the associated $M$-matrix. The form of the solutions precludes the absence of a singular continuous spectrum.

A regular map of type $\{m,n\}$ is a 2-cell embedding of a graph in an orientable surface, with the property that for any two directed edges $e$ and $e'$ there exists an orientation-preserving automorphism of the embedding that takes $e$ onto $e'$, and in which the face length and the vertex valence are $m$ and $n$, respectively. Such maps are known to be in a one-to-one correspondence with torsion-free normal subgroups of the triangle groups $T(2,m,n)$. We first show that some of the known existence results about regular maps follow from residual finiteness of triangle groups. With the help of representations of triangle groups in special linear groups over algebraic extensions of ${\mathbb Z}$ we then constructively describe homomorphisms from $T(2,m,n)=\langle y,z|\ y^m=z^n=(yz)^2=1\rangle$ into finite groups of order at most $c^r$ where $c=c(m,n)$, such that no non-identity word of length at most $r$ in $x,y$ is mapped onto the identity. As an application, for any hyperbolic pair $\{m,n\}$ and any $r$ we construct a finite regular map of type $\{m,n\}$ of size at most $C^r$, such that every non-contractible closed curve on the supporting surface of the map intersects the embedded graph in more than $r$ points. We also show that this result is the best possible up to determining $C=C(m,n)$. For $r\ge m$ the graphs of the above regular maps are arc-transitive, of valence $n$, and of girth $m$; moreover, if each prime divisor of $m$ is larger than $2n$ then these graphs are non-Cayley. 2000 Mathematics Subject Classification: 05C10, 05C25, 20F99, 20H25.

In this paper we expand the theory of connected components in non-archimedean discrete dynamical systems. We define two types of components and discuss their uses and applications in the study of dynamics of a rational function $\phi\in K(z)$ defined over a non-archimedean field K. We prove that some fundamental conjectures, including the No Wandering Domains conjecture, are equivalent, regardless of which definition of 'component' is used. We derive several results on the geometry of our components and the existence of periodic points within them. We also give a number of examples of p-adic maps with interesting or pathological dynamics.

2000 Mathematical Subject Classification: primary 37B99; secondary 11S99, 30D05.

Non-singular toric embeddings with dual defect are classified. The associated polytopes, called defect polytopes, are proven to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes. The structure of a defect polytope is described.

Let $F$ be a non-archimedean local field of residual characteristic different from $2$, and let $G$ be a unitary, symplectic or orthogonal group, considered as the fixed point subgroup in $\tilde G={\rm GL}(N,F)$ of an involution $\sigma$. We generalize the notion of a {\it simple character} for $\tilde G$, which was introduced by Bushnell and Kutzko [Annals of Mathematics Studies 129 (Princeton University Press, 1993)], to define semisimple characters. Given a semisimple character $\theta$ for $\tilde G$ fixed by $\sigma$, we transfer it to a character $\theta_{-}$ for $G$ and calculate its intertwining. If the torus associated to $\theta_{-}$ is maximal compact, we obtain supercuspidal representations of $G$, which are new if the torus is split only over a wildly ramified extension. 2000 Mathematics Subject Classification: 22E50.