A remark on the spaces $V^{p}_{\Lambda ,\alpha }$

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.