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.