View source for Strong differentiation of an indefinite integral

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Finding the [[Strong derivative|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Jessen,   J. Marcinkiewicz,   A. Zygmund,   "Note on the differentiability of multiple integrals"  ''Fund. Math.'' , '''25'''  (1935)  pp. 217–234</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,   "On the strong derivatives of functions of intervals"  ''Fund. Math.'' , '''25'''  (1935)  pp. 235–252</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,   "On the differentiability of multiple integrals"  ''Fund. Math.'' , '''23'''  (1934)  pp. 143–149</TD></TR></table>