Strong differentiation of an indefinite integral

Finding the strong derivative of an indefinite integral

$$F(I)=\int\limits_If(x)dx$$

of a real-valued function $f$ that is summable in an open subset $G$ of $n$-dimensional Euclidean space, considered as a function of the interval $I\subset G$. If

$$|f|(\ln(1+|f|))^{n-1}$$

is summable on $G$ (in particular, if $f\in L_p(G)$, $p>1$), then the integral $F$ of $f$ is strongly differentiable almost-everywhere on $G$. For any $\phi(u)$, $u\geq0$, that is positive, non-decreasing and such that

$$\phi(u)=o(u\ln^{n-1}u)$$

as $u\to\infty$, there is a summable function $f\geq0$ on $G$ such that $\phi\circ f$ is also summable and such that the ratio $F(I)/|I|$ is unbounded at each $x\in G$, as $I$ tends to $x$, that is, $F$ cannot be strongly differentiated.

References

[1]	B. Jessen, J. Marcinkiewicz, A. Zygmund, "Note on the differentiability of multiple integrals" Fund. Math. , 25 (1935) pp. 217–234
[2]	S. Saks, "On the strong derivatives of functions of intervals" Fund. Math. , 25 (1935) pp. 235–252
[3]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[4]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)

References

[a1]	A. Zygmund, "On the differentiability of multiple integrals" Fund. Math. , 23 (1934) pp. 143–149