The cohomology of [Mscr ](n, d), the moduli space of stable holomorphic bundles of coprime rank n and degree d and fixed determinant, over a Riemann surface Σ of genus g [ges ] 2, has been widely studied from a wide range of approaches. Narasimhan and Seshadri [17] originally showed that the topology of [Mscr ](n, d) depends only on the genus g rather than the complex structure of Σ. An inductive method to determine the Betti numbers of [Mscr ](n, d) was first given by Harder and Narasimhan [7] and subsequently by Atiyah and Bott [1]. The integral cohomology of [Mscr ](n, d) is known to have no torsion [1] and a set of generators was found by Newstead [19] for n = 2, and by Atiyah and Bott [1] for arbitrary n. Much progress has been made recently in determining the relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of [Mscr ](2, 1) is now known to be complete [14]; recently several authors have found a minimal complete set of relations for the ‘invariant’ subring of the rational cohomology of [Mscr ](2, 1) [2, 13, 20, 25].

Unless otherwise stated all cohomology in this paper will have rational coefficients.

Inequalities involving classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals of convolutive type have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [4, 20, 24]. Generalizations of these results to Zygmund spaces are presented in [4]. An exhaustive treatment of the problem of boundedness of such operators in Lorentz and Lorentz–Zygmund spaces is given in [3]. See also [8, 9] for further extensions in the framework of generalized Lorentz–Zygmund spaces.

As far as Orlicz spaces are concerned, a characterization of Young functions A having the property that the Hardy–Littlewood maximal operator or the Hilbert and Riesz transforms are of weak or strong type from the Orlicz space LA into itself is known (see for example [13]). In [17, 23] conditions on Young functions A and B are given for the fractional integral operator to be bounded from LA into LB under some restrictions involving the growths and certain monotonicity properties of A and B.

The main purpose of this paper is to find necessary and sufficient conditions on general Young functions A and B ensuring that the above-mentioned operators are of weak or strong type from LA into LB. Our results for (fractional) maximal operators are presented in Section 2, while Section 3 deals with fractional and singular integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood maximal operator which has recently been proved in [2, 11, 12]. Finally, in Section 4, the resolvent operator of some differential problems is taken into account and a priori bounds for Orlicz norms of solutions to elliptic boundary value problems in terms of Orlicz norms of the data are established. Let us mention that part of the results of the present paper were announced in [6].

It is proved that, for a Banach space X, the following properties are equivalent: (a) (X, weak) is fragmentable by a metric d(., .) that majorizes the norm topology (that is, the topology generated by d contains the norm topology), (b) (X, weak) is fragmentable by a metric d(., .) that majorizes the weak topology, and (c) (X, weak) is sigma-fragmentable by the norm. The paper gives a game characterization of these equivalent properties and uses it to show that a large class of Banach spaces (including the spaces that are Čech-analytic in their weak topology) enjoy the properties (a)–(c). In particular, if the unit ball B of X is a Borel subset of the second dual ball (B**, weak*) then X possesses the properties (a)–(c). This provides an alternative approach to the proofs of some of the results of J. E. Jayne, I. Namioka and C. A. Rogers on sigma-fragmentability of Banach spaces. It is also proved that C(T), with the topology p of pointwise convergence, is sigma-fragmentable by the norm whenever T=∪i[ges ]1Ti and (C(Ti), p), i[ges ]1, is sigma-fragmentable by the norm. This answers a question of R. Haydon. Finally, a simple proof is given that l∞ does not have any of the properties (a)–(c) and that (l∞(Γ), weak) is not sigma-fragmentable by any metric, provided that the cardinality of Γ is bigger than the continuum.

A family of transcendental meromorphic functions, fp(z), p ∈ N is considered. It is shown that, if p [ges ] 6, then the Hausdorff dimension of the Julia set of λfp satisfies dim J(λfp) [les ] 1/p, for 0 < λ < 1/6p, and dim J(λfp) [ges ] 1−(30 ln ln p/ln p), for p4p−1/105 ln p < λ < p4p−1/104 ln p. These results are used elsewhere to show that, for each d ∈ (0, 1), there exists a transcendental meromorphic function for which dim J(f) = d.

The subject is the 2-point boundary value problem for a second order ordinary differential equation

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the simplest and oldest of all boundary value problems, which arose in Euler's work on calculus of variations in the eighteenth century. Accordingly, there is a vast literature, mostly in the context of applied mathematics, where one frequently uses the methods of functional analysis.

Here we adopt a different point of view: we look for the number of solutions of problem (1), if any, and how this number varies with the endpoints (t1, x1) and (t2, x2). The concept of focal decomposition, introduced in [15] and developed by Peixoto and Thom in [19], expresses precisely this point of view (see §2). It was further developed by Kupka and Peixoto in [10] in the context of geodesics. From there, one is led naturally to relationships with the arithmetic of positive definite quadratic forms, a line that is considered in [16]. In both [10] and [16] attention is drawn to the close formal relationship between focal decomposition and the Brillouin zones of solid state physics. In [17] this line is pursued further and it is pointed out that the focal decomposition associated to (1) appears naturally as a prerequisite for the semiclassical quantization of this equation via the Feynman path integral method.

The main goal of the present paper is to prove Theorem 2, stated at the end of §4. There we give a functional ∫t2t1L (t, x, x˙) dt and consider its Euler equation. To this second order differential equation we associate the corresponding focal decomposition. Theorem 2 gives a criterion on the Lagrangian L from which we get a rough geometric description of this focal decomposition. It turns out to be somewhat similar to the one associated to the pendulum equation x¨+sin x = 0, worked out by Peixoto and Thom in [19, pp. 631, 197]. In the case L = (x˙2/2)−V(x) this type of focal decomposition is generic in some topological sense.

The paper establishes estimates for ideal measures of operators interpolated by the real method in terms of the measures of their restrictions to a sequence of spaces modelled on the intersection. It also shows that the estimates are optimal.

Let Γ be an anisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalues of the fractal differential operator

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acting in the classical Sobolev space W˚12(Ω) where Ω is a bounded C∞ domain in the plane ℝ2 with Γ⊂Ω. Here −Δ is the Dirichlet Laplacian in Ω and trΓ is closely related to the trace operator trΓ.

A new proof is given of a theorem of Menšov which states that a measurable function can be changed on a set of arbitrarily small measure to obtain a function with uniformly convergent Fourier series. Other results of Menšov on the representation of functions by trigonometric series are obtained by using a related result.

The paper presents diverse methods for estimating the covering number of a precompact subset of a Banach space when the entropy of the set of its extremal points is already known. In the case of a Hilbert space, the Gelfand diameters of the subset are also estimated.

Let 2 [les ] m [les ] n. The paper gives necessary and sufficient conditions on the parameters s1, s2, …, sm, p1, p2, …, pm such that the Jacobian determinant extends to a bounded operator from [Hscr ]s1p1 × [Hscr ]s2p2 × … × [Hscr ]smpm into [Sscr ]′. Here all spaces are defined on ℝn or on domains Ω⊂ℝn. In almost all cases the regularity of the Jacobian determinant is calculated exactly.

The restrictions Bspq(Ω) and Fspq(Ω) of the Besov and Triebel–Lizorkin spaces of tempered distributions Bspq(ℝn) and Fspq(ℝn) to Lipschitz domains Ω⊂ℝn are studied. For general values of parameters (s∈ℝ, p>0, q>0) a ‘universal’ linear bounded extension operator from Bspq(Ω) and Fspq(Ω) into the corresponding spaces on ℝn is constructed. The construction is based on a new variant of the Calderón reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of Bspq(Ω) and Fspq(Ω) in terms of their values in Ω are also obtained.

Some constrained open mapping theorems are obtained via Ekeland's variational principle. The constraint need only be a closed subset when the mapping is assumed to be Lipschitz, or a closed convex cone when the mapping is assumed to be closed. Generalizations of some previous results of Welsh and others are obtained. Apart from the presence of a constraint and a different method, the differentiability assumptions made are weaker. As applications, two results on the constrained controllability of nonlinear systems are given.

The Volterra convolution operator [Vscr ]f(x) = ∫x0ϕ (x−y)f(y)dy, where ϕ(·) is a non-negative non-decreasing integrable kernel on [0, 1], is considered. Under certain conditions on the kernel ϕ, the maximal Banach function space on [0, 1] on which the Volterra operator is a continuous linear operator with values in a given rearrangement invariant function space on [0, 1] is identified in terms of interpolation spaces. The compactness of the operator on this space is studied.

Let T = {T(t)}t[ges ]0 be a C0-semigroup on a Banach space X. The following results are proved.

(i) If X is separable, there exist separable Hilbert spaces X0 and X1, continuous dense embeddings j0[ratio ]X0 → X and j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that j0 ∘ T0(t) = T(t) ∘ j0 and T1(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0.

(ii) If T is [odot ]-reflexive, there exist reflexive Banach spaces X0 and X1 , continuous dense embeddings j[ratio ]D(A2) → X0, j0[ratio ]X0 → X, j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that T0(t) ∘ j = j ∘ T(t), j0 ∘ T0(t) = T(t) ∘ j0 and T(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0, and such that σ(A0) = σ(A) = σ(A1), where Ak is the generator of Tk, k = 0, [emptyv ], 1.

In the years 1979–80 Sacks and Uhlenbeck [19] and Schoen and Yau [21] proved independently the following beautiful theorem.

The structure of a collineation group G preserving a unital in a projective plane of order m2 with m ≡ 1(4) is investigated and several results are obtained about G, when G is a 2-group. These results are then used to investigate the structure of G/O(G) in the general case and, in particular, that of F*(G/O(G)).

A diffeomorphism f of the disk has a periodic orbit [Oscr ], and the paper considers the minimal topological entropy l(f, [Oscr ]) in the isotopy class of f with respect to this orbit. The structure of the periodic orbits with l(f, [Oscr ]) = 0 has been studied lately and is now well understood. For l(f, [Oscr ]) non-zero, the paper obtains a lower bound of l(f, [Oscr ]) that depends only on the period of [Oscr ] and which improves previously known estimates. This result is a consequence of a stronger bound in the case where the isotopy class is pseudo-Anosov. It is obtained by studying the induced dynamics on a graph.

Suppose [Cscr ] is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in [Cscr ], and the Picard group Pic([Cscr ]) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with [Cscr ].

There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category [Sscr ]. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere Sn [8, 19]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE. The best-known case is E=K(n), the nth Morava K-theory, considered in [8].

In this paper we study the case E=E(n), where E(n) is the Johnson–Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by [Lscr ]. Our main theorem is the following result.

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.

For bounded Hilbert space operators X, An and Bn, n = 1, 2, …,

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for all 1 [les ] p < ∞ and

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These inequalities involve some estimates for the norm of elementary operators with the range contained in the Schatten p-ideals.