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

G. A. Karagulyan¹

¹ Department of Computer Science Yerevan State University and Institute of Mathematics Armenian National Academy of Sciences Marshal Baghramian ave. 24b Yerevan, 375019, Armenia

Studia Mathematica, Tome 181 (2007) no. 1, pp. 17-32

Résumé

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 $G_{\delta\sigma}$).

Détail

DOI : 10.4064/sm181-1-2

Keywords: mathcal family rectangles plane mathbb side angle x axis say set directions r set there exists function mathbb basis mathcal differentiates integral overline limsup mathop diam nolimits mathcal int infty almost everywhere condition overline x infty holds set positive measure instead say wr set proved r set resp wr set only delta resp delta sigma