On integral bounded variation

In this paper we will investigate the concept of the q-integral p-variation introduced in 1970’s by Terehin. This kind of integral variation has been mainly used, until now, to describe the regularity of functions in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} in terms of the Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q$$\end{document}-norm (for q>p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>p$$\end{document}). In the present paper we show that the class of functions of bounded q-integral p-variation appears a very nice tool to work with integral operators, including these corresponding to Riemann–Liouville fractional integration. We also give the acting conditions for autonomous superposition operators and prove some existence theorems for Hammerstein integral equation in this space.


Introduction
The concept of a variation of a function dates back to the last decades of the 19th century and to the works of Jordan. Later the concept was generalized, in different directions, by Wiener ( p-variation), Young (φ-variation), F. Riesz (Riesz variation) and Waterman ( -variation), to name a few (for an overview see e.g. [1]). All of the above-mentioned concepts of variation are defined pointwise and may not be directly applied in L q -spaces. Still, because of the role that L q -spaces play in mathematics and their numerous applications, the question how to measure the variation of a function, so it is not sensitive to a change of values in a zero-measure set, appears to be of vital importance.
B Jacek Gulgowski dzak@mat.ug.edu.pl 1 Faculty of Mathematics Physics and Informatics, Institute of Mathematics, University of Gdańsk, 80-308 Gdańsk, Poland Several interesting attempts have been made. In [19] the Authors suggest the so-called variation in the mean that might be applied to L 1 -functions over a bounded interval. This type of a variation is constant within the equivalence classes belonging to L 1 but, as it was shown in [22], it actually represents the space of functions of bounded Jordan variation. Quite recently the concept of lower -variation was introduced in [8] generalizing the Waterman -variation to functions integrable in the Lebesgue sense. In the mentioned paper the Authors investigate the properties of the space of functions of lower -variation and its elements. One of the important observations is that these spaces are actually subsets of the space L ∞ of essentially bounded functions.
In this paper we are going to deal with the idea of the integral variation that was introduced a long time before the concepts mentioned above-i.e. with the q-integral p-variation introduced by Terehin in 1972 (see [24]). This idea is different from these described above mainly because it does not imply that the functions having bounded q-integral p-variation have to be essentially bounded (cf. Example 3). In a series of papers (see [25][26][27]) Terehin investigated the properties of functions of bounded q-integral p-variation, including the multidimensional case. Such concept of the variation turned out to be useful in the discussion about the smoothness of functions in L p spaces or in Sobolev spaces, described in terms of their L q -properties (for q > p).
At the beginning of 1980's Borucka-Cieślewicz introduced a more general concept of the q-integral M-variation (see [2,3]). She generalized Terehin's concept towards the variation in the sense of Young. In these two papers this new concept was studied and a few applications were suggested (including some estimates for integral operators with the Dirichlet kernel).
We should also note that quite recently another application of the q-integral p-variation has been suggested. As one of the latest, we should mention here the papers by Donchyk concerning the problems of impulse control for the string equation-solved both in the classical space of functions of bounded p-variation and q-integral p-variation (see [12] and [13]).
Quite recently the paper [4] by Brudnyi appeared, where Author investigated the properties of functions of bounded q-integral p-variation (and its generalizations) from the point of view of the approximation theory. Author proves there that functions of bounded q-integral p-variation, for certain values of p and q, may be approximated by piecewise polynomial functions in L q norm, with a given rate.
In the present paper we are going to investigate properties of spaces of functions of bounded q-integral p-variation and of linear integral operators defined in L q -spaces with values being functions of bounded integral variation. We will also study the autonomous superposition operators defined in the space of functions of bounded q-integral 1-variation which will lead us to the existence theorems for certain nonlinear Hammerstein equations in such spaces. The interesting observation will follow, in the last section of the paper, that maps corresponding to the fractional Riemann-Liouville integration may be considered (under certain assumptions) as acting into the space of functions of bounded q-integral 1-variation.

Preliminaries
Notation. By N we denote the set of all positive integers. Moreover, by I we will denote the unit interval [0, 1].
As usual, for q ∈ [1, +∞), by L q (J ) we will denote the Banach space of all the equivalence classes of real-valued functions defined on a bounded interval J ⊆ R which are Lebesgue integrable with qth power, endowed with the norm x L q := J |x(t)| q dt 1/q . Further on, we will write just L q instead of L q (I ). From now on we will always assume that q ∈ (1, +∞) is a conjugate of q ∈ (1, +∞), that is 1/q + 1/q = 1. In the space of bounded functions B(I ) and its subspaces (like the space of continuous functions C(I )) we consider the norm x ∞ = sup t∈I |x(t)| for x ∈ B(I ).
In the next subsection we collect basic definitions and facts which will be needed in the sequel.

Basic definitions and properties
Let us first remind the classical definition of the L q modulus of continuity (see e.g. [28]).
is called the L q -modulus of continuity of the function x on the interval [a, b].
Remark 1 A similar definition can be given in the L ∞ -case (see [24]), but we will not refer to this in the present paper.
The following example illustrates the above notion.
Example 1 Let x : [a, b] → R be a step function given by where u, w are distinct real numbers and c ∈ (a, b).
We are going to show that Let us consider three cases.
We may arrive to similar conclusions when c − a > b − c. Thus Let us now define the integral variation of a measurable function.
where the supremum is taken over all finite partitions a = t 0 < t 1 < · · · < t N = b of the interval [a, b], is called the q-integral p-variation of the function x. If ivar q p (x; a, b) < +∞, then we say that x is a function of bounded q-integral p-variation. The set of all such functions is denoted by I BV The concept of the q-integral p-variation was introduced by Terehin (see [24]). Following Terehin let us note that the case q = +∞ recreates the well-known Wiener p-variation. Now let us pass on to the concept of the q-integral M-variation introduced by Borucka-Cieślewicz (see [2,3]). For a Lebesgue measurable function x : [0, 1] → R we can define where M : [0, +∞) → R is a continuous, non-decreasing function with M(0) = 0 and M(u) > 0 for u > 0, and the supremum is taken over all finite partitions In the following proposition we have collected a few known facts concerning the q-integral p-variation.

Proposition 1 The following properties hold:
(i) (cf. [24, p. 278 Actually the statement (iii) above shows that the Sobolev space W 1, p (I ) ⊂ I BV q p for any q ∈ [1, +∞). However, the results presented in [24] lead to much more sophisticated conclusions: in [24,Theorem 2] the Author gives the precise description of W 1, p (I ) in terms of the integral variation.

Basic examples
Now we are going to present a few examples that seem to be important to understand the nature of the space I BV q p .
Example 2 Let x ∈ L q (a, b) be the step function given by (1). Then x ∈ I BV q p (x; a, b) and To prove this let us take any partition a = t So we may focus on the situation when t j−1 < c < t j for some j = 1, . . . , N . Then Thus, ivar q p (x; a, b) ≤ |u −w| min(c −a, b −c) 1/q . Moreover, as we can see, the inequality turns into equality when we take the partition a = t 0 < t 1 = b.
Another class of functions of bounded q-integral p-variation is indicated by the following Then Moreover, let us note that this inequality may be strict. Indeed, assume that We may also prove the reversed inequality-adding to it one more term.
Of course, when c = t j for some j = 1, . . . , N − 1, then we can see that Let us now assume that and one of its terms . Now we will estimate the quantity for different values of h. We will use the following classical inequality which holds for an arbitrary finite collection of non-negative real numbers a 1 , . . . , a m (see [15]*Inequality 19). Let us consider four cases.
Let us also observe that the length of the interval in the middle term integral is Let us observe that the length of the interval in the first integral is Here the length of the interval in the integral We should also note that whatever h where the last supremum should be seen as the maximum value of the integral of |x| q over the subintervals of [t j−1 , t j ] of length β − α. On the other hand, we may estimate This ends the proof.
Using Proposition 3 we can prove the following Proposition 4 Let 1 ≤ p ≤ q and let x ∈ L q (a, b) be a periodic step function with the basic period T = |b − a|/M, where M ∈ N, and such that where u, w are distinct real numbers. Then Proof First, we will slightly modify the conclusion of Proposition 3. In the case of periodic step functions we may estimate the integral Since x is periodic, it is easy to see that ivar q p (x; a, a + T ) = ivar q p (x; a + kT, a + (k + 1)T ) for k = 0, . . . , M − 1. This lets us write: Thus, by induction, we get On the other hand, Proposition 5 Let 1 ≤ p ≤ q and let x ∈ L q . Moreover, assume that there exists such a non-decreasing function ϕ : where t 0 := 0 and t n : (6) is obvious. So we may assume that +∞ n=1 ivar q p (x; t n−1 , t n ) p < +∞. For any fixed n ≥ 2 by Proposition 3 we can see that Now, we are going to prove that not only the sequence ivar q p (x; 0, t n ) n∈N is uniformly bounded, but also the integral variation ivar q p (x; 0, 1) in the entire interval [0, 1] is bounded. To prove this let us take any partition 0 = s 0 < s 1 < · · · < s N −1 < s N = 1 of the interval I and take n ∈ N big enough that t n ≥ s N −1 . Now we can see that

Moreover
, This shows the inequality (6) and ends the proof. Now, we are going to show that essentially unbounded functions may also belong to I BV q p .
where β ∈ (0, 1/q) and q > 1. Now we are going to show that x ∈ I BV q 1 . First, let us observe that for any (a, b) ⊂ (0, 1] and α > 1 we have This directly leads to We also know that since x| [ 1 2 ,1] is a Lipschitz function, by Proposition 2, the value ivar q 1 (x; 1 2 , 1) is well-defined (and finite). Let us also observe that This proves that for any n ≥ 2 we have ivar q 1 (x; Let us now take any partition 0 = t 0 < t 1 < · · · < t N −1 < t N = 1 and such n ∈ N that 1 2 n < t 1 . Then, similarly as in the proof of Proposition 5, are uniformly bounded and completes the proof.
We have just shown that essentially unbounded functions may be of bounded q-integral 1variation. The next example shows that bounded functions may be of unbounded q-integral 1-variation. The presented example is similar to the one given in [2, Theorem 3], butcontrary to that one-we will consider a bounded function.

Example 4
We are going to show that there exists a bounded function x : Let us also assume that x(1) = 1. In order to show that ivar q 1 (x; 0, 1) = +∞, let us observe that ivar Thus, which proves our claim. for any p < q and n ≥ 2. The right-hand sum is unbounded, so this shows that there exist L ∞ -functions that do not belong to any I BV q p for p < q. Now we are going to ask whether the embedding I BV q 1 ⊂ L q is completely continuous. The answer is negative, which is shown in the next example.

Remark 3 If we replace the series +∞
Example 5 Let the sequence (x n ) n∈N of functions belonging to L q be given by This sequence is uniformly bounded in I BV q 1 since ivar(x n ; 0, 1) = n 1/q · 1 n 1/q = 1 and x n L q = 1, but is not relatively compact in L q .
By the well-known Kolmogorov-Riesz compactness condition in L q (see e.g. [14]) we know that for a relatively compact set (x n ) n∈N and for any ε > 0, there exists δ > 0 that for any |h| ≤ δ and n ∈ N. As we can see Hence, for ε = 1 there is no δ > 0 for which the condition (7) holds uniformly for the entire sequence (x n ) n∈N .

Integral operators
In this section we are going to consider linear integral operators acting between the spaces L q and I BV q 1 . From now on let us assume that p = 1 and q ∈ (1, +∞). Let us take a function k : I × I → R and let us consider the integral operator Such operators, considered on spaces of functions of bounded variation in the sense of Jordan and Wiener, have been extensively studied in [10,11]. In particular in [11] there was presented the characterization of kernels k for which the corresponding map K : BV (I ) → BV (I ) is bounded. Quite recently similar necessary and sufficient conditions for the continuity of the integral operator have been also specified for BV spaces (see [9]). We are going to present now two sets of sufficient conditions for the continuity of the integral operator K : L q → I BV q 1 . Each of them is a consequence of one of the classical inequalities: the first one corresponds the generalized Minkowski inequality, while the second one corresponds to the Hölder inequality. This proves that the map K is well-defined and that K x L q ≤ m 0 L q x L q .
Let us take any partition 0 = t 0 < t 1 < · · · < t N = 1 of the interval I and consider Now, let us focus on one of the terms in the previous sum and apply the generalized Minkowski inequality once more: This allows us to estimate Now, since we cannot be sure that the function is measurable, let us fix positive ε > 0 and choose numbers h i ∈ (0, t i − t i−1 ) is such a way that Now we may continue our estimation (9) Since the inequality above holds true for any ε > 0, we infer that which, in turn, shows that ivar q 1 (K x; 0, 1) ≤ m L q x L q and ends the proof.
Theorem 2 Let 1 < q < +∞ and 1/q + 1/q = 1. Assume that the function k : I × I → R satisfies the following conditions: (B1) k is a Lebesgue measurable function on I × I ; (B2) the function s → k(t, s) belongs to L q for a.e. t ∈ I ; where the supremum is taken over the set of all partitions I of the interval I . Then the linear mapping K , induced by k, acts from L q to I BV q 1 and is continuous.
Proof First, we should observe that due to Hölder's inequality and (B2) for each x ∈ L q and a.e. t ∈ I the integral 1 0 k(t, s)x(s)ds exists and is finite. Hence the map K is well-defined. Moreover, Let us note that where the supremum is taken over the family of all finite partitions 0 = t 0 < · · · < t N = 1 of the interval I . By the Hölder inequality, for a.e. t ∈ I , we have Let us now estimate the sum This actually shows that and proves the continuity of K .

Remark 4
The concept of the variation of a function may be extended to maps with values in any Banach space (see Definition 7.3.4 in [16]). In particular, the value K given in (B4) may be interpreted as the strong q−integral 1−variation of the L q -valued map κ : [0, 1] → L q given by κ(t) = k(t, ·) ∈ L q , since Then we have , for a certain constant C > 0-taking the supremum over h ∈ (0, t i − t i−1 ), similarly to what we did in Proposition 2. That is why Hence the condition (10) implies (B4).

Superposition operator
In this section we are going to answer the question concerning the acting conditions for the autonomous superposition operator defined on the space of functions of bounded q-integral 1-variation, that is we are going to find conditions which should be imposed on a function f : R → R so that its superposition with a function x ∈ I BV q 1 remains in this space. The problem of acting conditions and related problems have been studied extensively for different concepts of variation (see the classical papers [17,20], more recent results [6,[9][10][11]21] and also [1] for an overview). Therefore, it seems to be important to answer this question also in this case.
The first observation in this section is a pretty obvious one.
Proof The proof is the consequence of the following observation: x; a, b).
We are going to show that x ∈ I BV q 1 and f • x / ∈ I BV q 1 . Please note that Therefore, by Proposition 5 we see that x ∈ I BV q 1 . On the other hand, By Lemma 1 we may assume that f is bounded on [−M, M], which implies that |b n −a n | → 0 as n → +∞. Let us observe that we may select a sequence of intervals [a n , b n ] in such a way that |b n − a n | ≤ 1 and | f (b n ) − f (a n )| ≥ n|a n − b n |. Now, let us observe that for each n ∈ N there exists a positive integer M n ≥ 1 such that 1 (M n + 1) 1−1/q < |b n − a n | ≤ Hence we have , Clearly x ∈ L q . Moreover by Proposition 4, we have Therefore, x ∈ I BV q 1 .
On the other hand, the function f • x is also a step function but with f (b n ) and f (a n ) replacing b n and a n respectively. Hence, .

Lemma 3 Let f : R → R be a Borel function which does not satisfy the (global) Lipschitz condition. Then there exists x
Proof Let us take two sequences (a n ) n∈N , (b n ) n∈N such that | f (b n ) − f (a n )| > n|b n − a n |.
For simplicity assume that |b n | = max{|a n |, |b n |} for n ∈ N. By the previous Lemma 2 we may assume that the sequence (b n ) n∈N is unbounded. We may also assume that the sequence |b n | is non-decreasing and |b n | ≥ 1. We can also easily observe that the difference |b n − a n | may be assumed to be arbitrarily small. Indeed, if for some ε 0 > 0 we have for a positive N ∈ N such that |b − a|/N < ε 0 . This clearly implies that f is a Lipschitz function, which is not possible. Therefore, we may assume that |b n − a n | ≤ 1 for n ∈ N. Let us observe that the series ∞ n=1 1 |b n | q 1 n 2q is convergent and denote h n = 1 We can see that not only ∞ n=1 h n is convergent but ∞ n=1 h 1/q n is convergent, too. Let us define the sequence (t n ) n∈N ⊂ (0, 1) as t n = n i=1 h i for n ∈ N and t 0 = 0.
Now, as in the proof of the Lemma 2, we divide each interval , i ∈ N, j = 1, 2, . . . , M i , be given by , We can see that By Proposition 4 we can see that Let us proceed to the estimation of ivar q 1 (x; 0, 1). First, let us observe that for any δ ∈ (0, t n ) and s ∈ [0, t n − δ] s+δ s |x(t)| q dt 1/q ≤ |b n |δ 1/q . We cannot apply Proposition 5 directly, but we may slightly change its proof to show that x ∈ I BV q 1 . Indeed, for any n ≥ 2, we have Therefore, if 0 = σ 0 < σ 1 < · · · < σ N −1 < σ N = 1 is an arbitrary partition of the interval I and n ≥ 2 is such that t n ≥ σ N −1 , we see that This proves that ivar q 1 (x; 0, 1) < +∞. Now, we have to check that ivar( f • x; 0, 1) = +∞. This estimation is much easier as The main result of this section is the following theorem concerning the acting conditions. It is an immediate consequence of Proposition 6 and Lemma 3, so its proof will be omitted.

Existence theorems
In this section we will be interested in the problem of the existence of solutions to the following nonlinear Hammerstein integral equation belonging to the space I BV q 1 where λ ∈ R.
In what follows we will need the following lemma. Proof First, let us note that in view of Proposition 6 the superposition operator F is welldefined, that is, F(x) ∈ I BV q 1 ⊆ L q for each x ∈ I BV q 1 . Moreover, for x, y ∈ I BV q 1 we have which completes the proof.
Theorem 4 Let us admit the following assumptions: Proof The problem (11) corresponds to the operator equation where F : I BV q 1 → L q is an autonomous superposition operator associated to the function f , while K : L q → I BV q 1 is a continuous linear map induced by the kernel k (see Theorem 1 and Theorem 2). By Lemma 4 the superposition operator F : I BV q 1 → L q is a Lipschitz map. This shows that the superposition λK • F is a Lipschitz map with a constant |λ| K · L. Hence, for λ small enough, it is a contraction, proving the existence and uniqueness of the solution to the problem (11).

Fractional Riemann-Liouville operators
In Sect. 3 we have discussed linear integral operators between spaces L q and I BV q 1 and some sufficient conditions for the continuity of these operators have been given. Moreover, in Sect. 5, we have shown some existence theorems for nonlinear Hammerstein equations in I BV q 1 spaces. In this section we are going to show that the class of continuous operators acting from L q to I BV q 1 , considered in Sect. 3, contains Riemann-Liouville fractional-order integrals. Hence, the results from the previous sections may be applied to some problems in which fractional derivatives are involved. To be more precise, let us now fix α ∈ (0, 1) and let us consider the linear map where denotes the Gamma function. The above operator actually corresponds to the integral operator (8) with the kernel This is the Riemann-Liouville left-sided fractional integral of order α, denoted in [23] by I α 0 + . The fractional integral operators, along with their applications, have been extensively studied through decades (see [18,23] and the references therein). Among other properties of the map I α 0 + first of all we should mention that the mapping is well-defined for any α > 0 and q > 1 as a mapping from L q to L q (see [23,Theorem 2.6]).
In [5,7] the Authors considered the following problem where x (α) denotes the fractional derivative of the function x of the order α ∈ (0, 1). The problem (14) corresponds to the operator equation (cf. [7, Proposition 4]) where y 0 (t) = x 0 (α) t α−1 . Let us observe that, by Example 3, the function y 0 belongs to I BV q 1 for α ∈ (1 − 1 q , 1). As we will see below, Theorem 4 may be applied also to the problems of the form (14). (12) and α ∈ (1 − 1 q , 1), then for each x ∈ L q , K x ∈ I BV q 1 and the mapping K : L q → I BV q 1 is continuous.

Theorem 5 If K is given by
Proof We are going to show that k : I × I → R given by (13) satisfies the assumptions of Theorem 1. It is-obviously-measurable and the function t → k(t, s) is in L q for a.e. s ∈ I . In order to check the assumption (A2) we should fix s ∈ I and calculate k(·, s) L q = 1 (α)