**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