Comparison of the P-integral with Burkill's integrals and some applications to trigonometric series

Musial, Paul; Скворцов, В. А.; Tulone, Francesco

doi:10.1016/j.jmaa.2023.127019

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R -Differentiability Of Acg R -Functions1

Introduction1

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2025

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Cited by 2 publications

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“…In the same way using the definition of the upper-left L r -derivate, we obtain a contradiction to the assumption that (17) D − r F (x) > F ′ ap (x) ≥ 0 holds on a set of positive measure. In this case, instead of equality (11), the equality (18)…”

Section: R -Differentiability Of Acg R -Functionsmentioning

confidence: 99%

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On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

2022

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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 L r , 1 ≤ r < ∞. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKrintegral, which serves to recover functions from their L rderivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that a continuous ACGfunction can fail to be L r -differentiable almost everywhere.

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Section: R -Differentiability Of Acg R -Functionsmentioning

confidence: 99%

“…It was also shown in [6] that the indefinite HK r -integral is L r -differentiable almost everywhere and belongs to a Lusin-type class of ACG r -functions. Some other properties of these integrals are investigated in [9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%