Integrals and Banach spaces for finite order distributions

Talvila, Erik

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 77-104.

Résumé

Let ${\mathcal{B}}_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\mathrm{\infty }$. Let ${\mathcal{B}}_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\mathrm{\infty }$. Define ${\mathcal{A}}_c^n$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in ${\mathcal{B}}_c$. Similarly with ${\mathcal{A}}_r^n$ from ${\mathcal{B}}_r$. A type of integral is defined on distributions in ${\mathcal{A}}_c^n$ and ${\mathcal{A}}_r^n$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb{N}$, the spaces ${\mathcal{A}}_c^n$ and ${\mathcal{A}}_r^n$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to ${\mathcal{B}}_c$ and ${\mathcal{B}}_r$, 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 ${\mathcal{A}}_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space ${\mathcal{A}}_r^1$ 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.

Détail

DOI : 10.1007/s10587-012-0018-5

Classification : 26A39, 46B42, 46E15, 46F10, 46G12, 46J10
Mots-clés : regulated function; regulated primitive integral; Banach space; Banach lattice; Banach algebra; Schwartz distribution; generalized function; distributional Denjoy integral; continuous primitive integral; Henstock-Kurzweil integral; primitive