A complete characterization of $R$-sets in the theory of differentiation of integrals

Tom 181 / 2007

G. A. Karagulyan Studia Mathematica 181 (2007), 17-32 MSC: Primary 42B25. DOI: 10.4064/sm181-1-2


Let ${\mathcal R}_s$ be the family of open rectangles in the plane $\mathbb{R}^2$ with a side of angle $s$ to the $x$-axis. We say that a set $S$ of directions is an $R$-set if there exists a function $f\in L^1(\mathbb{R}^2)$ such that the basis ${\mathcal R}_s$ differentiates the integral of $f$ if $s\not\in S $, and $ \overline D_sf(x)=\limsup_{\mathop{\rm diam}\nolimits(R)\to 0,\, x\in R\in{\mathcal R}_s} |R|^{-1}\int_R f=\infty $ almost everywhere if $s\in S$. If the condition $\overline D_s f(x)=\infty $ holds on a set of positive measure (instead of a.e.) we say that $S$ is a $WR$-set. It is proved that $S $ is an $R$-set (resp. a $WR$-set) if and only if it is a $G_\delta $ (resp. a $G_{\delta\sigma}$).


  • G. A. KaragulyanDepartment of Computer Science
    Yerevan State University
    Institute of Mathematics
    Armenian National Academy of Sciences
    Marshal Baghramian ave. 24b
    Yerevan, 375019, Armenia

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