It is shown that the Schatten–von Neumann p-norm of a Volterra integral operator of the form

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is sandwiched, when 2 [les ] p < ∞, by an integral expression similar to the well-known Hilbert–Schmidt norm.

Let α > 0. The operator of the form

formula here

is considered, where the real weight function v(x) is locally integrable on R+ := (0, ∞). In case v(x) = 1 the operator coincides with the Riemann–Liouville fractional integral, Lp → Lq estimates of which with power weights are well known. This work gives Lp → Lq boundedness and compactness criteria for the operator Tα in the case 0 < p, q < ∞, p > max(1/α, 1).

My paper [2], which appeared in the special issue dedicated to the late Professor Brian Hartley, contained an error in Lemma 2.2 as a result of overlooking the case that the exponents ki on p. 359 may be divisible by arbitrary powers of p, which was pointed out by Professor H. Smith. In order to overcome this situation all of Section 2, except Lemma 2.1, has been rewritten. However this does not cause any change in the rest of the paper. In particular the original assertions remain correct.

It is shown, by a simple and direct proof, that if a bounded valuation on a monotone convergence space is the supremum of a directed family of simple valuations, then it has a unique extension to a Borel measure. In particular, this holds for any directed complete partial order with the Scott topology. It follows that every bounded and continuous valuation on a continuous directed complete partial order can be extended uniquely to a Borel measure. The last result also holds for σ-finite valuations, but fails for directed complete partial orders in general.

The autonomous composition operator is the nonlinear map which takes a pair of functions into its composite function. The composition operator often appears in problems of nonlinear analysis and to analyse such problems it is often important to know whether the composition operator is continuous or differentiable. A fairly large number of papers in the literature have been devoted to the study of composition operators. For fullscale references, we refer the reader to the extensive monographs of Appell and Zabrejko [1] and Runst and Sickel [8]. To exemplify a typical situation, we consider the semilinear Dirichlet boundary value problem

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where Ω denotes a sufficiently regular bounded open subset of ℝN, and h0 a map of ℝ to ℝ, and where u is the unknown of the problem. We assume that we know that a certain function u0 belonging to a certain function space [Xscr ] solves (1.1). Then if we wish to know whether by perturbing h0 in a certain function space, say [Yscr ], the solutions u depend on h continuously, with differentiability, with analyticity or bifurcate, we could set G[h, u] ≡ Δu+h ∘ u, recast problem (1.1) into the abstract form

formula here

and study the solution set of equation (1.2) around the pair (h0, u0) by means of the implicit function theorem or by local bifurcation theorems in a Banach space setting.

All references unless otherwise stated are to [1]. The lemma in Subsection 2.3 of [1] is incorrect as stated. The statements and proofs of the lemmas in Subsections 3.2 and 3.4 are in need of revision. These revisions do not affect any other results in [1].

The authors thank H.-J. Schneider for pointing out the errors.

Let T be the compact real torus, and TC its complexification. Fix an integral weight α, and consider the α-weighted T-action on TC. If ω is a T-invariant Kähler form on TC, it corresponds to a pre-quantum line bundle L over TC. Let Hω be the square-integrable holomorphic sections of L. The weighted T-action lifts to a unitary T-representation on the Hilbert space Hω, and the multiplicity of its irreducible sub-representations is considered. It is shown that this is controlled by the image of the moment map, as well as the principle that ‘quantization commutes with reduction’.

A full characterization is given of those compact Lie groups G with the property that every G-map X→X on a finite-dimensional G-complex X of finite orbit type, XG = Ø, is (non-equivariantly) essential. For arbitrary G, conditions are given on the G-space X which guarantee this property. Finally, conditions are given for the non-existence of a G-map X→Y inducing a homotopy equivalence XG≃YG on the fixed point sets. These results have applications to critical point theory of almost G-invariant functionals.