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Transactions of the American Mathematical Society

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The spaces of functions of finite upper $ p$-variation

Author: Robert R. Nelson
Journal: Trans. Amer. Math. Soc. 253 (1979), 171-190
MSC: Primary 46E30; Secondary 28B05
MathSciNet review: 536941
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Abstract: Let y be a Banach space, $ 1\, \leqslant \,p\, < \,\infty $, and $ {U_p}$ be the semi-normed space of Y-valued Bochner measurable functions of a real variable which have finite upper p-variation. Let $ {\tilde U_p}$ be the space of $ {U_p}$-equivalence classes. An averaging operator is defined with the aid of the theory of helixes in Banach spaces, which enables us to show that the spaces $ {\tilde U_p}$ are Banach spaces, to characterize their members, and to show that they are isometrically isomorphic to Banach spaces of Y-valued measures with bounded p-variation.

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Keywords: Upper p-variation, average vector, helixes in Banach spaces, translation operators, Lebesgue-Bochner integral, chordal length function, absolutely continuous function, bounded variation
Article copyright: © Copyright 1979 American Mathematical Society