1. Definition of the A-polynomial

The A-polynomial was introduced in [3] (see also [5]), and we present an alternative definition here. Let M be a compact 3-manifold with boundary a torus T. Pick a basis λ,μ of π1T, which we shall refer to as the longitude and meridian. Consider the subset RU of the affine algebraic variety R = Hom (π1M, SL2[Copf ]) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic subset of R, since one just adds equations stating that the bottom-left entries in certain matrices are zero. There is a well-defined eigenvalue map

formula here

given by taking the top-left entries of ρ(λ) and ρ(μ).

1. Introduction

In 1978, Atiyah, Hitchin and Singer [1] introduced the twistor space as a ℙ1-fibration over a half-conformally flat 4-manifold, and thereby established a beautiful correspondence between Yang–Mills fields on 4-manifolds and holomorphic vector bundles on complex 3-folds. Hitchin gave the Penrose correspondence between solutions of anti-self-dual zero rest-mass field equations on a half-conformally flat manifold M and the sheaf cohomology groups H1(Z, F(k)), k[les ]−2, on its twistor space Z [4]. Here the holomorphic bundle F is the pull-back bundle of an anti-self-dual bundle over M, and we denote by [Oscr ](−1) the tautological line bundle associated to the half-spin bundle. Hitchin further showed how H1(Z, F(−1)) corresponds to solutions of the self-dual Dirac equations, and interpreted H1(Z, F(k)), k[ges ]0, in terms of the cohomology of an elliptic complex on M.

Klass has established decoupling inequalities for discrete time processes with independent increments. We extend his result to continuous time processes with independent increments. This extension enables us to obtain BDG inequalities for exponential functions instead of moderate functions.

The first part of the proof of Lemma 3 given in the author's paper [1] is obviously incomplete. Namely, [Kfr ]v(ζ) in the fifth line from the bottom of page 245 of [1] is not necessarily a cyclic extension over [Kfr ]v−1(ζ) in the next line. To complete the proof, we shall prove an assertion after simple preliminaries, as follows. Let p be an odd prime. For any CM-field k of finite or infinite degree, let k+ denote the maximal real subfield of k, A(k) the p-class group of k (that is, the p-primary component of the ideal class group of k), A(k+) the p-class group of k+, and A(k)− the kernel of the norm map A(k)→A(k+). Since p>2, A(k)−={a∈A(k)[mid ] aj=a−1} where j denotes the complex conjugation of k. For any extension F′/F of CM-fields, let ϕF′/F denote the homomorphism A(F)−→ A(F′)− induced by the natural map A(F)→A(F′). Now, the following completes the proof in question.

The author thanks Martin Wursthorn, who pointed out that the supposedly distinct 2-groups H(m, 1) and H(m, 3), defined on page 588 of [1], are in fact isomorphic to each other. This is an error (rather than a misprint), but may be corrected by changing the presentations of H(m, q) and H˜ as follows. Replace the relation B4=1 in the presentation for H(m, q) by the relation B8=1, and replace the relation Y4=1 in the presentation for H˜ by the relation Y8=1. With these modifications, the paragraph concerning the groups H(m, q) becomes valid.