We formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

Let X be a geometrically irreducible smooth projective curve defined over ℝ, of genus at least 2, that admits a nontrivial automorphism, σ. Assume that X does not have any real points. Let τ be the antiholomorphic involution of the complexification xℂ of X. We show that if the action of σ on the set 𝒮(X) of all real theta characteristics of X is trivial, then the order of σ is even, say 2k, and the automorphism of Xℂ has a fixed point, where is the automorphism of X×ℝℂ defined by σ. We then show that there exists X with a real point and admitting a nontrivial automorphism σ, such that the action of σ on 𝒮(X) is trivial, while X/〈σ〉≠ℙ1ℝ. We also give an example of X with no real points and admitting a nontrivial automorphism σ, such that the automorphism has a fixed point, the action of σ on 𝒮(X) is trivial, and X/〈σ〉≠ℙ1ℝ.

Finite groups with abelian Sylow p-subgroups for certain primes p are characterized in terms of arithmetical properties of commutators.

Given a metrizable locally convex-solid Riesz space of measurable functions we provide a procedure to construct a minimal Fréchet (function) lattice containing it, called its Fatou completion. As an application, we obtain that the Fatou completion of the space L1(ν) of integrable functions with respect to a Fréchet-space-valued measure ν is the space L1w(ν) of scalarly ν-integrable functions. Further consequences are also given.

We study some classes of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small in 1984. The first section reviews the basic concepts and describes the shear construction. The main body of the paper deals with the geometry of the classes constructed.

We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

Let X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous functions.

Following R. A. Rankin’s method, D. Zagier computed the nth Rankin–Cohen bracket of a modular form g of weight k1 with the Eisenstein series of weight k2, computed the inner product of this Rankin–Cohen bracket with a cusp form f of weight k=k1+k2+2n and showed that this inner product gives, up to a constant, the special value of the Rankin–Selberg convolution of f and g. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).