It is proved that any function of a Lusin-type class, the class of ACGr-functions, is differentiable almost everywhere in the sense of a derivative defined in the space Lr, 1 ≤ r < ∞. This leads to a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKr-integral, which serves to recover functions from their Lr-derivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that continuous ACG-functions can fail to be Lr-differentiable almost everywhere.
Musial, P., Skvortsov, V.A., Sworowski, P., Tulone, F. (2025). On the Lr-differentiability of two Lusin classes and a full descriptive characterization of the HKr-integral. SBORNIK MATHEMATICS, 216(6), 46-58 [10.4213/sm10129].
On the Lr-differentiability of two Lusin classes and a full descriptive characterization of the HKr-integral
Tulone, Francesco
2025-01-01
Abstract
It is proved that any function of a Lusin-type class, the class of ACGr-functions, is differentiable almost everywhere in the sense of a derivative defined in the space Lr, 1 ≤ r < ∞. This leads to a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKr-integral, which serves to recover functions from their Lr-derivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that continuous ACG-functions can fail to be Lr-differentiable almost everywhere.
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