To the problem of a strong differentiability of integrals along different directions |
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Authors: | G Lepsveridze |
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Institution: | (1) Faculty of Mechanics and Mathematics, I. Javakhishvili Tbilisi State University, 2, University St., 380043 Tbilisi, Georgia |
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Abstract: | It is proved that for any given sequence (σ
n
,n ∈ ℕ)=Γ0 ⊂ Γ, where Γ is the set of all directions in ℝ2 (i.e., pairs of orthogonal straight lines) there exists a locally integrable functionf on ℝ2 such that: (1) for almost all directionsσ ∈ Γ\Γ0 the integral ∫f is differentiable with respect to the familyB
2σ
of open rectangles with sides parallel to the straight lines fromσ: (2) for every directionσ
n
∈ Γ0 the upper derivative of ∫f with respect toB
2σ
n
equals +∞; (3) for every directionσ ∈ Γ the upper derivative of ∫ |f| with respect toB
2σ
equals +∞. |
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Keywords: | Strong derivative of an integral field of directions negative result in the theory of differentiation of integrals |
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