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

Studia Mathematica (2007)

• Volume: 181, Issue: 1, page 17-32
• ISSN: 0039-3223

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## Abstract

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Let ${}_{s}$ be the family of open rectangles in the plane ℝ² 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 ∈ L¹(ℝ²) such that the basis ${}_{s}$ differentiates the integral of f if s ∉ S, and $D{̅}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in }_{s}}{|R|}^{-1}{\int }_{R}f=\infty$ almost everywhere if s ∈ S. If the condition $D{̅}_{s}f\left(x\right)=\infty$ 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 ${G}_{\delta \sigma }$).

## How to cite

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G. A. Karagulyan. "A complete characterization of R-sets in the theory of differentiation of integrals." Studia Mathematica 181.1 (2007): 17-32. <http://eudml.org/doc/284928>.

@article{G2007,
abstract = {Let $_\{s\}$ be the family of open rectangles in the plane ℝ² 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 ∈ L¹(ℝ²) such that the basis $_\{s\}$ differentiates the integral of f if s ∉ S, and $D̅_\{s\}f(x) = lim sup_\{diam(R)→0, x∈R∈_\{s\}\} |R|^\{-1\} ∫_\{R\} f = ∞$ almost everywhere if s ∈ S. If the condition $D̅_\{s\}f(x) = ∞$ 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 $G_\{δ\}$ (resp. a $G_\{δσ\}$).},
author = {G. A. Karagulyan},
journal = {Studia Mathematica},
keywords = {differentiation of integrals; maximal functions; Zygmund's problem},
language = {eng},
number = {1},
pages = {17-32},
title = {A complete characterization of R-sets in the theory of differentiation of integrals},
url = {http://eudml.org/doc/284928},
volume = {181},
year = {2007},
}

TY - JOUR
AU - G. A. Karagulyan
TI - A complete characterization of R-sets in the theory of differentiation of integrals
JO - Studia Mathematica
PY - 2007
VL - 181
IS - 1
SP - 17
EP - 32
AB - Let $_{s}$ be the family of open rectangles in the plane ℝ² 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 ∈ L¹(ℝ²) such that the basis $_{s}$ differentiates the integral of f if s ∉ S, and $D̅_{s}f(x) = lim sup_{diam(R)→0, x∈R∈_{s}} |R|^{-1} ∫_{R} f = ∞$ almost everywhere if s ∈ S. If the condition $D̅_{s}f(x) = ∞$ 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 $G_{δ}$ (resp. a $G_{δσ}$).
LA - eng
KW - differentiation of integrals; maximal functions; Zygmund's problem
UR - http://eudml.org/doc/284928
ER -

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