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A remark on the spaces $ V\sp{p}\sb{\Lambda ,\alpha }$

Authors: Casper Goffman, Fon Che Liu and Daniel Waterman
Journal: Proc. Amer. Math. Soc. 82 (1981), 366-368
MSC: Primary 26A45; Secondary 28A20, 42B05
MathSciNet review: 612720
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Abstract: A function $ f \in {L^p}$, $ p \geqslant 1$, over an interval in $ {R^n}$, is in $ V_{\Lambda ,\alpha }^p$ if, corresponding to the $ i$th coordinate direction, $ i = 1, \ldots ,n$, there is an equivalent function which is of $ \Lambda $-bounded variation on a.e. line $ {l_i}$ in that direction and whose $ \Lambda $-variation on those lines is in $ {L^\alpha }$, $ \alpha \geqslant 1$, as a function of the other $ (n - 1)$ variables. For each $ i$, another equivalent function may be chosen so that on a.e. $ {l_i}$ it has an internal saltus at each point. It is shown that for this function, the $ \Lambda $-variation on the lines $ {l_i}$ is a measurable function of the other variables. This was known for $ n = 2$; for $ n > 2$, the measurability was previously assumed as an additional hypothesis. The classes $ V_{\Lambda ,\alpha }^p$ are Banach spaces and have been shown to be of interest in the study of localization of multiple Fourier series.

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Keywords: Measurable function, $ \Lambda $-bounded variation, approximate lower semicontinuity
Article copyright: © Copyright 1981 American Mathematical Society

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