We study boundedness and compactness properties of the Hardy integral operator Tf(x)=∫xAf from a weighted Banach function space X(v) into L∞ and BMO. We give a new simple characterization of compactness of T from X(v) into BMO. We construct examples of spaces X(v) such that T(X(v)) is (a) bounded in L∞ but not compact in BMO; (b) compact in BMO but not bounded in L∞; (c) bounded in BMO but neither bounded in L∞ nor compact in BMO; (d) bounded in L∞, compact in BMO and yet not compact in L∞. In order to obtain the last of the counterexamples we construct a new weighted Banach function space.

We investigate classes of uniform pseudodifferential operators on Riemannian manifolds with bounded geometry. We prove that operators belonging to the classes are bounded in function spaces of Hardy–Sobolev–Besov type defined on the manifold. The classes contain the powers of the Laplace–Beltrami operators so we get the fractional lift property for the function spaces.

A simple geometrical argument shows that every pair of projections on a finite-dimensional complex vector space has a common invariant subspace of dimension 1 or 2. The idea extends to certain pairs of projections on an infinite-dimensional Hilbert space H. In particular every projection on H has a reducing subspace, although a finite-dimensional one need not exist. In a final section, the results are extended to the existence of hyperinvariant subspaces for pairs of projections.

Given two self-adjoint Hilbert space operators A, B, and a continuous function f, we prove several inequalities of the form

formula here

involving the Lp-norm of the derivative f′ and the Besov Br∞,1-norm of f.

It is well known that there are bounded domains Ω⊂ℝn whose boundaries ∂Ω are not smooth enough for there to exist a bounded linear extension for the Sobolev space W1p(Ω) into W1p(ℝn), but the embedding W1p(Ω)⊂ Lp(Ω) is nevertheless compact. For the Lipγ boundaries (0<γ<1) studied in [3, 4], there does not exist in general an extension operator of W1p(Ω) into W1p(ℝn) but there is a bounded linear extension of W1p(Ω) into Wγp(ℝn) and the smoothness retained by this extension is enough to ensure that the embedding W1p(Ω)⊂ Lp(Ω) is compact. It is natural to ask if this is typical for bounded domains which are such that W1p(Ω)⊂ Lp(Ω) is compact, that is, that there exists a bounded extension into a space of functions in ℝn which enjoy adequate smoothness. This is the question which originally motivated this paper. Specifically we study the ‘extension by zero’ operator on a space of functions with given ‘generalized’ smoothness defined on a domain with an irregular boundary, and determine the target space with respect to which it is bounded.

The following result is well known and easy to prove (see [14, Theorem 2.2.6]).

Theorem 0. If A is a primitive associative Banach algebra, then there exists a Banach space X such that A can be seen as a subalgebra of the Banach algebra BL(X) of all bounded linear operators on X in such a way that A acts irreducibly on X and the inclusion A[rarrhk ]BL(X) is continuous.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A[rarrhk ]BL(X) is contractive.

Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0.

We investigate the closability of those derivations D defined on a (non necessarily closed) subalgebra B of a complex Banach algebra A for which the conditions

BAB⊂B and dim[Bk∩Rad(A)]<∞

hold for some k∈ℕ, where Rad(A) stands for the Jacobson radical of A. In this situation we show that the separating subspace [Sscr ](D) for D satisfies the property

B[B∩[Sscr ](D)]B⊂Rad(A).

Furthermore, we demonstrate several specially relevant situations in which we get a ‘closability property’ which is more precise than the former one.

We establish a Weitzenböck formula for harmonic morphisms between Riemannian manifolds and show that under suitable curvature conditions, such a map is totally geodesic. As an application of the Weitzenböck formula we obtain some non-existence results of a global nature for harmonic morphisms and totally geodesic horizontally conformal maps between compact Riemannian manifolds. In particular, it is shown that the only harmonic morphisms from a Riemannian symmetric space of compact type to a compact Riemann surface of genus at least 1 are the constant maps.

This paper is mainly devoted to the study of the index of a map at a zero, and the index of a polynomial map over ℝn. For semi-quasi-homogeneous maps we prove that the index at a zero coincides with the index at this zero of its quasi-homogeneous part. For a class of polynomial maps with finite zero set we provide a method which makes easier the computation of its index over ℝn. Finally we relate the index and the multiplicity.

For a wide class of infinitely divisible laws μ numbers R(μ) are evaluated satisfying the property: if R>R(μ) then μ is uniquely determined by its values on the half-line (−∞, R); if R<R(μ) then μ is not determined by its values on (−∞, R). Necessary and sufficient conditions are given for ‘half-line’ convergence to μ on (−∞, R) to be equivalent to weak convergence to μ.

Let Y be a topological space; and suppose that the circle group, S1, which will be denoted by G throughout this paper, is acting on Y. Then the equivariant cohomology H*G(Y) of Y is defined to be H*(YG), where YG=(EG×Y)/G, the Borel construction. Because of the bundle map YG→BG, the cohomology H*G(Y) is an H*(BG)-module. With rational coefficients H*(BG; ℚ)=ℚ[u], where deg(u)=2. And, when Y is a reasonably nice finite-dimensional space, for example, a finite-dimensional G-CW-complex, then the localized module H*G(Y; ℚ)[u−1] is isomorphic to H*G (YG; ℚ)[u−1] via the restriction homomorphism, where YG⊆Y is the fixed point set. (See, for example, [2].) That this very useful Localization Theorem does not generalize to infinite dimensional spaces can be seen by taking Y=EG, for example.

A noninjective bounded-to-one factor map from an irreducible shift of finite type onto a sofic system can be factored as a composition of other such maps in only finitely many ways (up to isomorphism). This generalizes to factor maps from systems with canonical coordinates to finitely presented dynamical systems.

Let Xn for n[ges ]1 be independent random variables with EXn=0 and EX2n=1. Set Sk,n= [sum ]i1<…<ik[les ]nXi1…Xik. Define Tk,c,m= inf{n[ges ]m[ratio ][mid ]k!Sk,n[mid ] >cnk/2}. We study critical values ck,p for k[ges ]2 and p>0, such that ETpk,c,m<∞ for c<ck,p and all m, and ETpk,c,m=∞ for c>ck,p and all sufficiently large m. In particular, c1,1=c2,1=1, c3,1=2 and c4,1=3 under certain moment conditions on X1, when Xn are identically distributed. We also investigate perturbed stopping rules of the form Th,m=inf{n[ges ]m[ratio ] h(S1,n/n1/2) <ζn or >ζn} for continuous functions h and random variables ζn∼a and ζn∼b with a<b. Related stopping rules of the Wiener process are also considered via the Uhlenbeck process.