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

• Autorzy: G. A. Karagulyan
• Języki publikacji: EN

Abstrakty

EN

Let $𝓡_{s}$ 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 $𝓡_{s}$ differentiates the integral of f if s ∉ S, and $D̅_{s}f(x) = lim sup_{diam(R)→0, x∈R∈𝓡_{s}} |R|^{-1} ∫_{R} f = ∞$ 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 $G_{δσ}$).

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 181
• Numer: 1
• Strony: 17-32

Daty

wydano

2007

Twórcy

autor

G. A. Karagulyan

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

Identyfikatory

DOI

10.4064/sm181-1-2