The L r -Variational Integral

. We deﬁne the L r -variational integral and we prove that it is equivalent to the HK r -integral deﬁned in 2004 by P. Musial and Y. Sagher in the Studia Mathematica paper The L r -Henstock–Kurzweil integral . We prove also the continuity of L r -variation function. 26A39, 28C99.


History and Aim
At the beginning of the 1900s, Denjoy and Perron developed descriptive processes for recovering a function from its derivative that solved known problems of classical Riemann and Lebesgue integrals. Many years later, an equivalent constructive Riemann-type integral process was developed by Henstock and Kurzweil. Both integration processes were generalized quite recently for many different spaces (see [1,11] and [12]) solving the problem of recovering Fourier coefficients in Haar, Walsh and Vilenkin systems (see [9,10,14,15] and [16]). Many properties of these non-absolute integrals were investigated, for example, the Hake property was studied with an abstract differential basis in a topological spaces, in terms of variational measure and in Riesz spaces (see [13,17] and [2]).
To establish pointwise estimates for solutions of elliptic partial differential equations, in 1961 Calderon and Zygmund introduced the L r -derivative (see [3]) and in 1968 L. Gordon described a Perron-type integral, the P rintegral, that recovers a function from its L r -derivative (see [4]). In 2004, Musial and Sagher extended the P r -integral to the L r -Henstock-Kurzweil integral, the HK r -integral, that recovers also a function from its L r -derivative (see [6]). Quite recently the integration by parts formula for the HK r -integral was investigated by Musial and Tulone (see [7]) and the same authors described a norm on the space of HK r -integrable functions and studied the dual and completion of this space (see [8]).

Introduction
We will assume that r ≥ 1 and we will consider the case of the closed interval [a, b].
if there exists a function F ∈ L r [a, b] with the following property: for each ε > 0 there exist a non-decreasing function φ defined on [a, b] and a gauge δ, i.e., a positive function, defined on [a, b] We will use the following definition given in [6] so that for any ε > 0 there exists a gauge δ so that for any finite collection of nonoverlapping δ-fine tagged intervals By Theorem 5 in [6], the function F in the Definition 2.2 is unique up to an additive constant, so we can state that for each x ∈ (a, b] We need the following definition in a later theorem.
If F is L r -continuous for all x ∈ E, we say that F is L r -continuous on E.
The Henstock-Kurzweil integral primitive is continuous in the usual sense. In [6] is proved an equivalent result for L r -Henstock-Kurzweil indefinite integral.
Remark 2.6. Throughout this paper, if an interval function is said to be continuous, it is to be considered continuous in the sense of Definition 2.5.

Definition 2.7. Let δ be a gauge and let
The main tool we need to get the L r -variational integral is the following definition of L r -variation function.
where the supremum is taken over all δ-fine partitions P of [c, d] .
Now we can prove the following theorem that extends Theorem 11.9 in [5] 96 Page 4 of 10 F. Tulone and P. Musial MJOM

Main Results
Proof. Suppose there exists a function F with the property stated in the theorem. Let ε > 0 and choose Φ and δ according to the hypotheses. If for each x ∈ (a, b]. Let ε > 0. By hypothesis, there exists a gauge δ on [a, b] such that and let W (P) be defined as in (2.2) and let Φ be defined on the subintervals of [a, b] as in (2.3). By Theorem 2.9, Φ is superadditive. Also, This completes the proof.

MJOM
The L r -Variational Integral Page 5 of 10 96 Proof. Suppose first that f is L r -variational integrable on [a, b] . Let ε > 0 and let F , δ and φ satisfy the conditions in Definition 2.1.
Now suppose that f is L r -Henstock-Kurzweil integrable on [a, b] and that for each x ∈ (a, b], and so φ is non-decreasing. In addition, Hence, the function f is L r -variational integrable on [a, b] . This completes the proof. We now prove the continuity of the interval function Φ. Consequently, for each ξ ∈ [a, c) , there exists P ξ , a δ-fine tagged partition of [ξ, c] such that W (P ξ ) > η.
We now prove that we can make the following three assumptions about P x : 1. P x contains at least two tagged intervals, 2. c is a tag of P x , and 3. the interval containing c is arbitrarily small. Fix x and ε > 0. Choose y ∈ (max (x, c − ε) , c). By Cousin's Lemma there exists Q, a δ-fine tagged partition of [x, y]. Define P x = Q ∪ P y . We then have If c is the tag of its interval, then P x has the desired properties. Now suppose that c is not the tag of its interval. Let s and t be such that (t, [s, c]) is the tagged interval which contains c. It is possible that s = t but we assume that t < c.
It suffices to show that The L r -Variational Integral Page 7 of 10 96 Using Minkowski's inequality, we have By Theorem 2.4 the function F is L r -continuous at each point of [a, b], and so we have that We also have that . We now prove the proposition. Set x 1 = a and write . . .
By the result proved above, we may assume that for each k, c−x k < 1/k and, therefore, that x k → c.
This shows that the series We then have for each k, By (3.1), the term on the right goes to zero; therefore, the entire right side of the equality goes to zero. This contradiction completes the proof.
Funding Open access funding provided by Università degli Studi di Palermo within the CRUI-CARE Agreement.
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