Giorgi Oniani

On the Strong Differentiation of Multiple Integrals along Different Frames

Abstract:
Let $E$ be a set consisting of rectangular frames in $\mathbb{R}^{n}$. The following question connected with one problem of A. Zygmund is studied in the paper: Does there exists a function the integral of which is: 1) non-differentiable almost everywhere in a strong sense along every frame from $E$, and 2) strongly differentiable along every frame not belonging to $E$? In particular, the question is solved on the existence of a non-empty set $E$ different from the set of all rectangular frames for which there is a function with the properties 1) and 2).

Keywords:
Strong differentiation, Lebesgue multiple integral, differentiation basis, frame.

MSC 2000: 28A15