Differentiation of Integrals

TL;DR

This paper identifies specific subspaces of functions in L^s that have differentiable integrals over measurable sets in R^n, answering a question by Stein, and provides a continuous nowhere example.

Contribution

It introduces new subspaces of L^s functions with differentiable integrals over measurable sets, addressing an open problem posed by Stein.

Findings

Established subspaces of L^s with differentiable integrals

Provided an example of a continuous nowhere function in these classes

Extended understanding of differentiation of integrals in harmonic analysis

Abstract

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.