It is an interesting open question when Dehn surgery on a knot in the 3-sphere S3 can produce a lens space (see [10, 12]). Some studies have been made for special knots; in particular, the question is completely solved for torus knots [21] and satellite knots [3, 29, 31]. It is known that there are many examples of hyperbolic knots which admit Dehn surgeries yielding lens spaces. For example, Fintushel and Stern [8] have shown that 18- and 19-surgeries on the (−2, 3, 7)-pretzel knot give lens spaces L(18, 5) and L(19, 7), respectively. However, there seems to be no essential progress on hyperbolic knots. It might be a reason that some famous classes of hyperbolic knots, such as 2-bridge knots [26], alternating knots [5], admit no surgery yielding lens spaces.

In this paper we focus on the genera of knots to treat the present condition methodically and show that there is a constraint on the order of the fundamental group of the resulting lens space obtained by Dehn surgery on a hyperbolic knot. Also, this new standpoint enables us to present a conjecture concerning such a constraint, which holds for all known examples.

In a recent paper [5], the classical Bernoulli and Euler polynomials were expressed as finite sums involving the Hurwitz zeta function. The object of this sequel is first to give several remarkably shorter proofs of each of these summation formulas. Various generalizations and analogues, which are relevant to the present investigation, are also considered.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings–Jaco theorem which established a similar result for the Poincaré Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

Let S be a smooth hypersurface in ℝn with surface area measure ds and Gaussian curvature κ(s). Define the convolution operator T by

formula here

for suitable functions f on ℝn. We are interested in the Lp − Lq mapping properties of T. Write [Sscr ] for the type set of T, the set

formula here

It is well known (see, e.g. [O1]) that [Sscr ] is contained in the closed triangle [Tscr ] with vertices (0, 0), (1, 1) and (n/(n+1), 1/(n+1)). This paper is concerned with estimates of the form

formula here

The estimate (1) is interesting because it serves as a weak substitute for the L(n+1)/n − Ln+1 boundedness of T. For example, if S is compact and (1) holds, then well-known arguments show that [Sscr ] differs from the full triangle [Tscr ] by at most the point (n/(n + 1), 1/(n + 1)). Our main result is a condition sufficient to imply (1). Its statement requires the following definition.

We prove global and local weighted stability theorems for representations of abelian semigroups on Banach spaces under countable spectral conditions and certain growth assumptions on the weights. In the global case, we use a limit semigroup construction together with an extension theorem for weighted semigroup representations. In the local case, we adapt a definition of Albrecht to introduce a local spectrum of a representation which is no larger than the usual notion of local spectrum for representations of Z+ and R+, and we establish the corresponding local stability theorem. We give an application to the asymptotic theory of functions on abelian semigroups.

0·1. Conical domains

The aim of this paper is to improve the results of my previous work in the area [1]. Let us start by recalling some of the notations and definitions of [1].

Let Ω ⊂ ℝd be some convex open cone with vertex at 0 such that [emptyv ] &≠; Ω &≠; ℝd. Let u ∈ C(Ω) be the reduite, i.e.

formula here

and let α = degΩ[ges ] 1 be the homogeneity degree of u. Let us consider the random walk

formula here

controlled by some compactly supported measure

formula here

that is centred with covarience = I, i.e.

formula here

We then denote:

formula here

We give necessary and sufficient conditions for a conformal foliation locally generated by conformal vector fields to produce harmonic morphisms. Natural constructions of harmonic maps and morphisms are thus obtained. Also we obtain reducibility results for harmonic morphisms induced by (infinitesimal) conformal actions on Einstein manifolds.

Let X be a Banach space. In this paper we show that, assuming that X does not contain a copy of [lscr ]∞, then ba (Σ,X) contains a complemented copy of c0 whenever X contains a copy of c0. Conversely, if ba (Σ,X) contains a complemented copy of c0, then X contains a copy of c0.

We introduce non-homogeneous equilibrium states which include the classical equilibrium states for subshifts of finite type, Riesz products and G-measures. We prove a rather precise estimate for ∥Pnf∥∞ where Pn are the averaging operators. Applications are given to estimate ∥Lngf∥∞ (Lg being a transfer operator on a subshift of finite type) and then to prove a central limit theorem for f(Tnx) (T being the shift), to study the almost everywhere convergence of some lacunary series with respect the equilibrium state and then to compute the Hausdorff dimension of the equilibrium state etc.

Let h be a harmonic function on ℝn of suitably restricted growth. It is known that if h vanishes, or is bounded, on the lattice ℤn−1 × {0}, then the same is true on ℝn−1 × {0}. This paper presents sharp results which show that, if n [ges ] 3, then the same conclusions can be drawn even if information about h is missing on a substantial proportion of the lattice points. As corollaries we obtain uniqueness and Liouville-type theorems for harmonic, and also polyharmonic, functions which improve results by several authors.

(Volume 129 (2000), 263–275)

It is with very great regret that we record the death of one of the authors, Frank Peterson, shortly before the appearance of the paper. The Authors and Editor wish to express their great personal and professional respect for him.

A function f, analytic in the unit disc Δ, is said to be a Bloch function if supz∈Δ(1 − [mid ]z[mid ]2)[mid ]f′(z)[mid ] < ∞. In this paper we study the zero sequences of non-trivial Bloch functions. Among other results we prove that if f is a Bloch function with f(0) ≠ 0 and {zk} is the sequence of ordered zeros of f. then

formula here

and

formula here

We will also prove that (ii) is best possible even for the little Bloch space [Bscr ]0. To this end we construct a function f ∈ [Bscr ]0 whose zero sequence {zk} satisfies

formula here

We also consider analogous problems for some other related spaces of analytic functions.

The purpose of this short note is to reformulate theorem 9·3 in [5] which is not correct as stated. We note that all other results in [5] are independent of that statement.

The notation is the same as [5] with the sole exception that C0(S) will always denote the space of all real-valued continuous functions on the locally compact space S vanishing at infinity. As usual, αS stands for the one-point compactification of S. Recall that a norm ‖ · ‖ on a Banach space X is said to be maximal if there is no equivalent norm on X whose isometry group contains properly that of ‖ · ‖.

A necessary condition for the existence and uniqueness of mild solutions of first- and second-order differential equations on the real line is given. It is known that the first-order problem is related to the operator equation AX − XD = −δ0. Here, it is shown how the second-order problem on the real line is connected to the operator equation AX − XD2 = −δ0 and also to some other operator equations.

A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of Δj+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary.

The purpose of this paper to characterize the existence of nontrivial solutions of nonlinear Volterra integral equations and to prove the uniqueness of this kind of solutions. To this end, different properties of the solutions are presented, in particular some referred to their attracting character.

Based on the characterization of periodic eigenvalues using rotation numbers, we analyse the second and the third periodic eigenvalues of one-dimensional Schrödinger operators with certain step potentials. This gives counter-examples to the Alikakos–Fusco conjecture on the second periodic eigenvalues. Using this simple model, we can also construct infinitely many resonance pockets, which are much like calabashes emanating from a cane, of one-parameter Hill's equations.

We prove sharp L2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first space derivatives, using only the Hardy inequality. These imply L2 boundary decay properties of the heat kernel and spectral density. We deduce bounds on the rate of convergence of the eigenvalues when the region is slightly reduced in size. It is remarkable that several of the bounds do not involve the space dimension.

A theorem is established for generating by differentiation and integration the Airy stress function due to the presence of any arbitrary two-dimensional singularity near a thick layer separating two other dissimilar semi-infinite solids directly from the corresponding Airy stress function for the unbounded homogeneous solid. A systematic application of the theorem yields the dominant distant effect associated with any arbitrary influencing line singularity in a well-structured physically interpretable form. As an illustration, the problem of a concentrated line force of arbitrary orientation near the interface layer is examined from this standpoint. It is found that a force aligned normal to the interface layer produces a far-field effect of an edge dislocation with Burgers vector parallel to the interface layer, together with a semi-infinite plane of centres of dilatation, whereas a force aligned parallel to the interface layer produces a far-field effect of an edge dislocation with Burgers vector perpendicular to the interface layer together with a semi-infinite plane of concentrated couples.