Integrals and Banach spaces for finite order distributions

Abstract

Let B c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A n c to be the space of tempered distributions that are the nth distributional derivative of a unique function in B c . Similarly with A n r from B r . A type of integral is defined on distributions in A n c and A n r . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A n c and A n r are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B c and B r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A 1 c is the completion of the L 1 functions in the Alexiewicz norm. The space A 1 r contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.