Integrals and Banach spaces for finite order distributions

Abstract

summary:Let Bc\mathcal B_c denote the real-valued functions continuous on the extended real line and vanishing at -\infty . Let Br\mathcal B_r denote the functions that are left continuous, have a right limit at each point and vanish at -\infty . Define Acn\mathcal A^n_c to be the space of tempered distributions that are the nnth distributional derivative of a unique function in Bc\mathcal B_c. Similarly with Arn\mathcal A^n_r from Br\mathcal B_r. A type of integral is defined on distributions in Acn\mathcal A^n_c and Arn\mathcal A^n_r. The multipliers are iterated integrals of functions of bounded variation. For each nNn\in \mathbb N, the spaces Acn\mathcal A^n_c and Arn\mathcal A^n_r are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to Bc\mathcal B_c and Br\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 Ac1\mathcal A_c^1 is the completion of the L1L^1 functions in the Alexiewicz norm. The space Ar1\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

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