Differentiation of Integrals

Shunchao Long

Published 2014-07-08Version 1

No functions class for general measurable sets classes are known whose functions have the property of differentiability of integrals associated to such sets classes. In this paper,we give some subspaces of $L^s$ with $1<s<\infty$, whose functions are proven to have the differentiability of integrals associated to measurable sets classes in ${\bf R}^n $, this gives an answer to a question stated by Stein in his book Harmonic Analysis. We give also a example of some functions in these classes on ${\bf R}^2 $, which is continuous nowhere.