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To the problem of a strong differentiability of integrals along different directions. (English) Zbl 0895.28003

Summary: It is proved that for any given sequence \((\sigma_n, n\in \mathbb{N})= \Gamma_0\subset\Gamma\), where \(\Gamma\) is the set of all directions in \(\mathbb{R}^2\) (i.e., pairs of orthogonal straight lines) there exists a locally integrable function \(f\) on \(\mathbb{R}^2\) such that: (1) for almost all directions \(\sigma\in\Gamma\backslash\Gamma_0\) the integral \(\int f\) is differentiable with respect to the family \(B_{2\sigma}\) of open rectangles with sides parallel to the straight lines from \(\sigma\); (2) for every direction \(\sigma_n\in\Gamma_0\) the upper derivative of \(\int f\) with respect to \(B_{2\sigma_n}\) equals \(+\infty\); (3) for every direction \(\sigma\in\Gamma\) the upper derivative of \(\int| f|\) with respect to \(B_{2\sigma}\) equals \(+\infty\).

MSC:

28A15 Abstract differentiation theory, differentiation of set functions
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References:

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