Rotation of coordinate system and differentiation of integrals with respect to translation-invariant bases

We study the dependence of differential properties of the indefinite integral on the rotation of the coordinate system (that is, on the transformation (of the variables) that is a rotation about the origin); in particular, we study Zygmund's problem concerning the possibility of correcting an arbitrary integrable function using a rotation to achieve the differentiability of its integral for general differential bases, and also the problem of invariance for the differentiability property of the integral with respect to rotations. The results obtained in the paper imply negative solutions of the above questions for bases of rather general form.
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