On the complete characterization of differentiation sets of integrals

Details On the complete characterization of differentiation sets of integrals On the complete characterization of differentiation sets of integrals

2005-11-19

arxiv.org/abs/math/0511479v3

Mathematics

Classical Analysis and ODEs

Published in: Studia Mathematica, 2007, vol. 181, No 1, pp. 17-32

Scientific paper

Let $B_\theta $ be the family of rectangles in the plane $R^2$, having slope $\theta $ with the abscissa. We say a set of slopes $\Theta $ is $D$-set if there exists a function $f\in L(R^2)$, such that the basis $B_\theta $ differentiates integral of $f$ if $\theta\not\in\Theta $ and $\bar D_\theta f(x)=\infty $ almost everywhere if $\theta\in\Theta $. If the condition $\bar D_\theta f(x)=\infty $ holds on a set of positive measure (instead of a.e.) we shall say it is $WD$-set. It is proved, that $\Theta $ is $D$-set($WD$-set) if and only if it is $G_\delta $($G_{\delta\sigma}$).

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