# 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

## Access Full Article

top## Abstract

top## How to cite

topG. 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.