In this paper we study the notion of joint functional calculus associated with a couple of resolvent commuting sectorial operators on a Banach space $X$. We present some positive results when $X$ is, for example, a Banach lattice or a quotient of subspaces of a $B$-convex Banach lattice. Furthermore, we develop a notion of a generalized $H^\infty$-functional calculus associated with the extension to $\Lambda(H)$ of a sectorial operator on a $B$-convex Banach lattice $\Lambda$, where $H$ is a Hilbert space. We apply our results to a new construction of operators with a bounded $H^\infty$-functional calculus and to the maximal regularity problem.

1991 Mathematics Subject Classification: 47A60, 47D06, 46C15.

Let $1 \le q \le p <\infty$ and let $\mathcal{C}$ be the class of all bounded convex domains $\Omega$ in $\mathbb{R}^n$. Following the approach in `An optimal Poincaré inequality in $L^1$ for convex domains', by G. Acosta and R. G. Durán (Proc. Amer. Math. Soc. 132 (2003) 195–202), we show that the best constant $C$ in the weighted Poincaré inequality

$$ \| f - f_{av} \|_{L^q_w (\Omega)} \le C w(\Omega)^{\frac{1}{q} - \frac{1}{p}} \mbox{diam}(\Omega) \| \nabla f \|_{L^p_w(\Omega)} $$

for all $\Omega \in \mathcal{C}$, all Lipschitz continuous functions $f$ on $\Omega$, and all weights $w$ which are any positive power of a non-negative concave function on $\Omega$ is the same as the best constant for the corresponding one-dimensional situation, where $\mathcal{C}$ reduces to the class of bounded intervals. Using facts from `Sharp conditions for weighted 1-dimensional Poincaré inequalities', by S.-K. Chua and R. L. Wheeden (Indiana Math. J. 49 (2000) 143–175), we estimate the best constant. In the case $q = 1$ and $1 <\infty$, our estimate is between the best constant and twice the best constant. Furthermore, when $p = q = 1$ or $p = q = 2$, the estimate is sharp. Finally, in the case where the domains in $\mathbb{R}^n$ are further restricted to be parallelepipeds, we obtain a slightly different form of Poincaré's inequality which is better adapted to directional derivatives and the sidelengths of the parallelepipeds. We also show that this estimate is sharp for a fixed rectangle.

The author investigates the solubility in rationals of equations of the form $$a_0X_0^4 + a_1X_1^4 + a_2X_2^4 + a_3X_3^4 = 0 $$ where $a_0a_1a_2a_3$ is a square, building on the ideas which Colliot-Thène, Skorobogatov and he have developed; see {\em Invent. Math.} 134 (1998) 579--650. He obtains sufficient conditions for solubility, which appear to be related to the absence of a Brauer--Manin obstruction. This represents the first large family of K3 surfaces which almost satisfy the Hasse principle, in the sense that the auxiliary condition which ensures that local solubility everywhere implies global solubility is nearly always satisfied. 1991 Mathematics Subject Classification: 10B10.

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree $d$ in $n$ variables over the field of $q$ elements, one desires an algorithm whose running time is a polynomial function of $d^n \log(q)$. (Here we assume $d \geq 2$, for otherwise the problem is easy.) The case $n = 1$ is related to univariate polynomial factorisation and is comparatively straightforward. When $n = 2$ one is counting points on curves, and the method of Schoof and Pila yields a complexity of $\log(q)^{C_d}$, where the function $C_d$ depends exponentially on $d$. For arbitrary $n$, the theorem of the author and Wan gives a complexity which is a polynomial function of $(pd^n \log(q))^n$, where $p$ is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of $pd^n\log(q)$. In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form $Z^p - Z = f$, where the polynomial $f$ has a diagonal leading form. The method utilises a relative $p$-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let $f$ be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime $p$, outputs the number of solutions to the congruence equation $f = 0 \bmod{p}$, and runs in ${\mathcal O}(p^{2 + \varepsilon})$ bit operations, for any $\varepsilon > 0$. This improves upon the elementary estimate of ${\mathcal O}(p^{n-1 + \varepsilon})$ bit operations, where $n$ is the number of variables, which can be achieved using Berlekamp's root counting algorithm.

Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number $\epsilon$, one has $E(N) \ll N^{13/30 + \epsilon}$, thereby exploiting effectively the 'excess' fourth square of a prime so as to improve the recent bound $E(N) \ll N^{13/15 + \epsilon}$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.

2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m > 2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m > 2$, on three real parameters along with an arbitrary Kähler–Einstein metric $h$ in complex dimension $m - 1$. We provide an explicit description of all these local-isometry types, for any given $h$. This result is derived from a more general local classification theorem for metrics admitting functions that we call special Kähler–Ricci potentials.

Suppose that $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential annulus, and $M(r_2)$ contains an essential torus. It is known that $\Delta = \Delta (r_1, r_2) \leq 5.$ We will show that if $\Delta = 5$ then $M$ is the Whitehead sister link exterior, and if $\Delta = 4$ then $M$ is the exterior of either the Whitehead link or the 2-bridge link associated to the rational number $\frac{3}{10}$. There are infinitely many examples with $\Delta = 3$.

The center symmetry set (CSS) of a smooth hypersurface $S$ in an affine space $\mathbf{R}^n$ is the envelope of lines joining pairs of points where $S$ has parallel tangent hyperplanes. The idea stems from a definition of Janeczko, in an alternative version due to Giblin and Holtom. For $n = 2$ the envelope is always real, while for $n \ge 3$ the existence of a real envelope depends on the geometry of the hypersurface. In this paper we make a local study of the CSS, some results applying to $n \le 5$ and others to the cases $n = 2,3$. The method is to construct a generating function whose bifurcation set contains the CSS and possibly some other redundant components. Focal sets of smooth hypersurfaces are a special case of the construction, but the CSS is an affine and not a euclidean invariant. Besides the familiar local forms of focal sets there are other local forms corresponding to boundary singularities, and yet others which do not appear to have arisen elsewhere in a geometrical context. There are connections with Finsler geometry. This paper concentrates on the theory and the proof of the local normal forms for the CSS.

In this paper we characterize those Banach spaces $E$ that admit a locally uniformly rotund renorming by means of a linear topological condition that is a particular case of the notion, introduced by J. E. Jayne, I. Namioka and C. A. Rogers, called countable cover by sets of small local diameter. This allows us to develop a Decomposition Method and a Transfer Technique to obtain locally uniformly rotund renormings.

1991 Mathematics Subject Classification: 46B20.

The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are ‘amenable at the identity’, and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable groups.

The isomorphism types of finite Lie primitive subgroups of the complex Lie groups $E_{6} ( \C )$ and $F_4 ( \C )$ are determined. Here, we call a finite subgroup of a complex Lie group $G$ {\sl Lie primitive\/} if it is not contained in a proper closed subgroup of $G$ of positive dimension. Induction can be used to investigate subgroups which are not Lie primitive. Some additional information is provided, such as the characters of these finite subgroups on some small-dimensional modules for the Lie groups.

In studying these groups, we mainly use two rational linear representations of the universal covering group $\widetilde E$ of $E_6(\C)$, namely a 27-dimensional module (there are two inequivalent ones), denoted by $\K$, and the adjoint module. In particular, we make heavy use of the characters of $\widetilde E$ on these modules. The group $F_4(\C)$ occurs in $\widetilde E$ as the stabilizer subgroup of a vector in $\K$. The finite simple groups of which a perfect central extension occurs in $F_4(\C)$ or $E_6(\C)$ are:

via $G_2$: $Alt_5$, $Alt_6$, $L(2,7)$, $L(2,8)$, $L(2,13)$, $U(3,3)$,

via $F_4$: $Alt_7$, $Alt_8$, $Alt_9$, $L(2,17)$, $L(2,25)$, $L(2,27)$, $L(3,3)$, ${}^3D_4(2)$, $U(4,2)$, $O(7,2)$, $O^{+}(8,2)$,

via $E_6$: $Alt_{10}$, $Alt_{11}$, $L ( 2 , 11)$, $L(2,19)$, $L(3,4)$, $U(4,3)$, ${}^2F_4(2)'$, $M_{11}$, $J_2$.

This list has been found using the classification of the finite simple groups. On the basis of this list, the finite Lie primitive subgroups are found to be either the normalizers of one of these subgroups or of one of the two elementary abelian $3$-groups found by Alekseevskii.

1991 Mathematics Subject Classification: 20K47, 20G40, 17B45, 20C10.

In this paper we define and study Hopf C$^*$-algebras. Roughly speaking, a Hopf C$^*$-algebra is a C$^*$-algebra $A$ with a comultiplication $\phi : A \rightarrow M(A \otimes A)$ such that the maps $a \otimes b \mapsto \phi(a)(1 \otimes b)$ and $a \otimes b \mapsto (a \otimes 1)\phi(b)$ have their range in $A \otimes A$ and are injective after being extended to a larger natural domain, the Haagerup tensor product $A \otimes_h A$. In a purely algebraic setting, these conditions on $\phi$ are closely related to the existence of a counit and antipode. In this topological context, things turn out to be much more subtle, but nevertheless one can show the existence of a suitable counit and antipode under these conditions.

The basic example is the C$^*$-algebra $C_0(G)$ of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication. But also the reduced group C$^*$-algebra $C^*_r(G)$ of a locally compact group with the well-known comultiplication falls in this category. In fact all locally compact quantum groups in the sense of Kustermans and the first author (such as the compact and discrete ones) as well as most of the known examples are included.

This theory differs from other similar approaches in that there is no Haar measure assumed.

2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant $K > 0$ such that if $\mathcal{M}$ is a semi-finite von Neumann algebra and $( \mathcal{M}_n )^{ \infty }_{n = 1}$ is an increasing filtration of von Neumann subalgebras of $\mathcal{M}$ then for any given martingale $x = ( x_n )^{\infty}_{n = 1}$ that is bounded in $L^2 ( \mathcal{M} ) \cap L^1 ( \mathcal{M} )$, adapted to $( \mathcal{M}_n )^{\infty}_{n = 1}$, there exist two martingale difference sequences, $a = ( a_n )_{n = 1}^\infty$ and $b = ( b_n )_{n = 1}^\infty$, with $dx_n = a_n + b_n$ for every $n \geq 1$,

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{2} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{2} \leq 2 \| x \|_2,

and

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{1, \infty} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{1, \infty} \leq K \| x \|_1.

As an application, we obtain the optimal orders of growth for the constants involved in the Pisier–Xu non-commutative analogue of the classical Burkholder–Gundy inequalities.

We study densities $\rho$ on the unit ball in euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure associated with $\rho$. For these densities a geometric theory can be developed which captures many features of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring-Hayman theorem, the radial limit theorem and find analogues of compression and expansion phenomena on the boundary.

1991 Mathematics Subject Classification: 30C65.

A famous theorem of Donaldson describes a correspondence between $\mbox{SU}(2)$ monopoles over three-dimensional Euclidean space and maps from $\Bbb{CP}^1$ to itself. This paper generalises this to monopoles with arbitrary gauge group $G,$ and new maps from $\Bbb{CP}^1$ to flag manifolds $G/H$ are produced. This is in line with a conjecture of Atiyah and Murray, following a similar result of Atiyah's on hyperbolic monopoles.

Donaldson's approach, also followed by Hurtubise and Murray in previous proofs of many important cases of the current result, depends on a description of monopoles in terms of a system of ordinary differential equations known as Nahm's equations. In contrast, our approach is more direct, and the bulk of the paper is concerned with describing the rational map associated to a particular framed monopole, via solutions to a scattering equation (first introduced by Hitchin) along parallel lines.

A subsidiary section of the paper analyses rational maps into flag manifolds, constructing canonical lifts into larger flag manifolds, and into the (complexified) Lie group. The main result is dependent upon additional work in a companion paper, ‘Construction of Euclidean monopoles’, to appear in the same journal, where a procedure is described for recovering a monopole from its rational map.

1991 Mathematics Subject Classification: 53C80, 58D27, 58E15, 58G11.

We study Rouquier blocks of symmetric groups and Schur algebras in detail, and obtain explicit descriptions for the radical layers of the principal indecomposable, Weyl, Young and Specht modules of these blocks. At the same time, the Jantzen filtrations of the Weyl modules are shown to coincide with their radical filtrations. We also address the conjectures of Martin, Lascoux–Leclerc–Thibon–Rouquier and James for these blocks.

We employ techniques from the theory of disintegration of measures to study the Lifting Problem for commuting $n$-tuples of subnormal weighted shifts. We obtain a new necessary condition for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift. For subnormal $2$-variable weighted shifts, we then find the precise relation between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights.

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.

Let $N$ be a natural number and $A \subseteq [1, \dots, N]^2$ be a set of cardinality at least $N^2 / (\log \log N)^c$, where $c > 0$ is an absolute constant. We prove that $A$ contains a triple $\{(k, m), (k+d, m), (k, m+d) \}$, where $d > 0$. This theorem is a two-dimensional generalization of Szemerédi's theorem on arithmetic progressions.