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

100%

Karagulyan G.

Studia Mathematica

2007

tom 181

nr 1

17-32

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 $G_{δσ}$).

Divergence of general operators on sets of measure zero

100%

Karagulyan G.

nr 1

113-119

We consider sequences of linear operators Uₙ with a localization property. It is proved that for any set E of measure zero there exists a set G for which $Uₙ𝕀_{G}(x)$ diverges at each point x ∈ E. This result is a generalization of analogous theorems known for the Fourier sum operators with respect to different orthogonal systems.