Integrals and Banach spaces for finite order distributions

TL;DR

This paper introduces new Banach spaces of finite order distributions with integral and algebraic structures, generalizing classical integrals and including measures, with many properties analogous to standard integrals.

Contribution

It defines and analyzes Banach spaces of finite order distributions, establishing their structure, duality, and integral properties, extending classical integration theory.

Findings

Spaces are Banach lattices and algebras isometrically isomorphic to function spaces.

Integrability implies absolute integrability of the absolute value.

Includes classical results like Hölder inequality and convergence theorems.

Abstract

Let $\Bc$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty$. Let $\Br$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty$. Define $\acn$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\Bc$. Similarly with $\arn$ from $\Br$. A type of integral is defined on distributions in $\acn$ and $\arn$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in\N$, the spaces $\acn$ and $\arn$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\Bc$ and $\Br$, 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 $\ac^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\ar^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: H\"older inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.