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arXiv · 2310.18520

On the $L^r$-differentiability of Two Lusin Classes and a Full Descriptive Characterization of the $HK_r$-integral

Paul Musial·Valentin A. Skvortsov·Piotr Sworowski·Francesco Tulone
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Abstract

It is proved that any function of a Lusin-type class, the class of $ACG_r$-functions, is differentiable almost everywhere in the sense of a derivative defined in the space~$L^r$, $1\le r<\infty$. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the $HK_r$-integral, which serves to recover functions from their $L^r$-derivatives. The class $ACG_r$ is compared with the classical Lusin class $ACG$ and it is shown that a continuous $ACG$-function can fail to be $L^r$-differentiable almost everywhere.

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Paul Musial, Valentin A. Skvortsov, Piotr Sworowski, Francesco Tulone. 2023-10-27. On the $L^r$-differentiability of Two Lusin Classes and a Full Descriptive Characterization of the $HK_r$-integral. https://arxiv.org/abs/2310.18520

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