It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space to (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type . As a consequence we show that the dyadic integral of a ∞ function is dyadically differentiable and its derivative is f a.e.
In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs defined as inductive limits of increasing sequences of Gelfand pairs . By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element of the set of -biinvariant continuous functions of positive type on .
We construct an example of a cancellative amenable semigroup which is the ascending union of semigroups, none of which are amenable.
The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.
Let G be a locally compact abelian group and M(G) its measure algebra. Two measures μ and λ are said to be equivalent if there exists an invertible measure ϖ such that ϖ*μ = λ. The main result of this note is the following: A measure μ is invertible iff |μ̂| ≥ ε on Ĝ for some ε > 0 and μ is equivalent to a measure λ of the form λ = a + θ, where a ∈ L¹(G) and θ ∈ M(G) is an idempotent measure.
For a locally compact, abelian group , we study the space of functions on belonging locally to the Fourier algebra and with -behavior at infinity. We give an abstract characterization of the family of spaces abelian by its hereditary properties.