Search results for " Absolute continuity."
showing 3 items of 3 documents
THE HKr-INTEGRAL IS NOT CONTAINED IN THE Pr-INTEGRAL
2022
We compare a Perron-type integral with a Henstock-Kurzweiltype integral, both having been introduced to recover functions from their generalized derivatives defined in the metric Lr. We give an example of an HKr-integrable function which is not Pr-integrable, thereby showing that the first integral is strictly wider than the second one.
Regular subclasses in the Sobolev space W_{loc}^{1,n}
2009
Westudy some slight modifications of the class AC^n introduced in [D. Bongiorno, Absolutely continuous functions in Rn, J. Math. Anal. and Appl. 303 (2005) 119 134]. In particular we prove that the classes introduced in [C. Di Bari, C. Vetro, A remark on absolutely continuous functions in R^n, Rend. Circ. Matem. Palermo 55 (2006) 296 304] are independent by and contain properly the class AC^n.
On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative
2022
The notion of Lr-variational measure generated by a function F ∈ Lr[a, b] is introduced and, in terms of absolute continuity of this measure, a descriptive characterization of the HKr -integral recovering a function from its Lr-derivative is given. It is shown that the class of functions generating absolutely continuous Lr-variational measure coincides with the class of ACGr -functions which was introduced earlier, and that both classes coincide with the class of the indefinite HKr-integrals under the assumption of Lr-differentiability almost everywhere of the functions consisting these classes