Properties of multiplication operators on the space of functions of bounded φ-variation

In 1881, Camille Jordan [1] introduced the notion of functions of bounded variation and established the relation between those functions and monotonic ones when he was studying convergence of Fourier series. Later on the concept of bounded variation was generalized in various directions by many mathematicians such as F. Riesz, N. Wiener, R. E. Love, H. Ursell, L. C. Young, W. Orlicz, J. Musielak, L. Tonelli, L. Cesari, R. Caccioppoli, E. de Giorgi, O. Olcinik, E. Conway, J. Smoller, A. Volpert, S. Hudjacv, L. Ambrosio, G. Dal Maso, among many others. It is noteworthy to mention that many of these generalizations were motivated by problems in such areas as calculus of variation, convergence of Fourier series, geometric measure theory, mathematical physics, etc. For many applications of functions of bounded variation see, e.g., the monograph [2]. On the other side, the space of functions of bounded p-variation, introduced byWiener in 1924 (see [3]), led in 1937 to Young [4] to give a generalization of the concept of function of bounded variation by introducing the notion of φ-variation of a function; this concept, in turn, was generalized by Chistyakov, [5], for functions which take values in a linear normed vector space. We recommend the excellent book of Appell et al. [6] for the study of the properties of functions of bounded φ-variation. The aim of this article is to study the properties of multiplication operator acting on [ ] BV 0, 1 φ spaces, which we define in Section 2. Each function ∈ [ ] u BV 0, 1 φ defines a linear and continuous operator


Introduction
In 1881, Camille Jordan [1] introduced the notion of functions of bounded variation and established the relation between those functions and monotonic ones when he was studying convergence of Fourier series. Later on the concept of bounded variation was generalized in various directions by many mathematicians such as F. Riesz Maso, among many others. It is noteworthy to mention that many of these generalizations were motivated by problems in such areas as calculus of variation, convergence of Fourier series, geometric measure theory, mathematical physics, etc. For many applications of functions of bounded variation see, e.g., the monograph [2].
On the other side, the space of functions of bounded p-variation, introduced by Wiener in 1924 (see [3]), led in 1937 to Young [4] to give a generalization of the concept of function of bounded variation by introducing the notion of φ-variation of a function; this concept, in turn, was generalized by Chistyakov, [5], for functions which take values in a linear normed vector space. We recommend the excellent book of Appell et al. [6] for the study of the properties of functions of bounded φ-variation.
The aim of this article is to study the properties of multiplication operator acting on which is known as multiplication operator with symbol u. This operator has been widely studied in the context of spaces of measurable functions. We mention here the pioneering work by Singh and Kumar in [7,8] on properties of multiplication operators acting on spaces of measurable functions. These authors studied the compactness and closedness of the range of multiplication operators on ( ) L μ 2 . Additionally, we note that the work of Arora et al. in [9][10][11] examined properties of M u on Lorentz and Lorentz-Bochner spaces. Further significant results regarding multiplication operators were obtained by Castillo et al. in [12][13][14], in which these authors showed that the techniques used by previously mentioned authors can be modified to study multiplication operators on weak L p spaces, Orlicz-Lorentz spaces, and variable Lebesgue spaces. The most complete study of the properties of multiplication operators acting on measurable function spaces was carried out by Hudzik et al. [15] (see also [16] and [17]). However, in the context of spaces of functions with certain type of bounded variation, it is a recent topic, being the first work due to Astudillo-Villalba and Ramos-Fernández [18] (see also [19]) where the authors made an exhaustive study of the properties of multiplication operator acting on the space of functions of bounded variation.
Consequently, the main objective of this article is to extend the results achieved in [18] to more general setting of space of functions of bounded φ-variation. With the above end, in Section 2, we give some properties of functions of bounded φ-variation. Section 3 is dedicated to characterize all symbols ∈ [ ] u BV 0, 1 φ which define multiplication operators having closed range; while in Section 4 we study the compactness of M u making special emphasis on the properties of the symbol u which define multiplication operators with finite dimensional range. Finally, in Section 5, we use the results obtained in Sections 4 and 5 to characterize the symbols ∈ [ ] u BV 0, 1 φ which define Fredholm multiplication operators.

Some remarks on functions of bounded φ-variation
For the reader's ease, in this section we collect some properties of φ-functions which will be of great utility in the proof of our results. A function is said to be a Young function (or φ-function) if it is convex, strictly increasing and such that ( ) = φ 0 0. In particular, according to this definition we also have that ( ) → ∞ φ t as → ∞ t . Thus, if φ is a fixed Young function, then we can define the φ-variation (also known as the Young variation) of a real-valued function f defined on [ ] 0, 1 as   [3]).
We fix a Young function φ and consider the space [ ] BV 0, 1 φ defined by φ. Next, we have the following: and so satisfies the following property: , the space of all bounded functions defined on [ ] 0, 1 , and hence we can write We also have that if We include a proof of the above inequality to illustrate the use of the property (2.3) and the definition of Luxemburg seminorm (2.2). We can suppose that where we have used the convexity of φ in the second inequality and the property (2.3) in the last inequality. Thus, since the partition P was arbitrary, we obtain that In this kind of spaces, the following functions will play an important role. For a subset B of [ ] 0, 1 , the characteristic function of B is denoted by 1 B and it is defined by This is known as an indicatrix function. In particular, we have the following very useful property: and by definition of Luxemburg seminorm, we conclude that And so the proposition is proved. □

Remark.
It is important to remark that from the proof of the above proposition, we can see that gives us a more precise information. with closed range. It is convenient, from now, to define the following set: with these notations, we have the following result.
and in this case we have where we have used that φ is an increasing function. The above estimation is also valid for the case ∈ t Z Next, taking into account the sum in each of the four cases, we obtain  Thus, { } h n is a Cauchy sequence in the closed set ( ) M Ran u and because of this fact, there exists a function as → ∞ n . Thus, if we fix ∈ m and we consider ∈ + t A m 2 1 , then for all ≥ n m we have that ∈ + t A n 2 1 and therefore for all ∈ n . We will see that a function f with this property cannot belong to . This implies that f does not belong to [ ] BV 0, 1 φ and we have a contradiction. Therefore, there exists a > δ 0 such that | ( )| ≥ u t δ for all ∈ t Z u c . □

Remark.
As an important application of Theorem 3.1 we can see that when ( ) = φ t t, we have Theorem 8 in [18], while if ( ) = φ t t p with > p 1 we have Theorem 7 in [19].

On the compactness
In this section, we analyze when a function ∈ [ ] u BV 0, 1 φ defines compact multiplication operators M u acting on [ ] BV 0, 1 φ . We recall that an operator → T X X : is said to be compact if { ( )} T x n has a convergent subsequence in X for all bounded sequences { } ⊂ x X n . It is known that the limit of compact operators is also a compact operator and that the identity operator → I X X : defined by = If f is compact if and only if ( ) < ∞ X dim . We recall that for > ε 0 we set then we have the following result. is a compact operator and that there exists an > ε 0 such that E ε is an infinite set. Then E ε has a sequence { } t n such that ≠ t t n m for ≠ n mand the sequence of functions { } f n defined by are linearly independent and belong to the space To see that, we consider, as in the proof of Theorem 3.1, four cases for the function is a bijective and compact operator. Hence, we have arrived to the following conclusion: X ε is a finite dimensional space, which leads a contradiction with the conclusion of the first part of the proof. □ For the converse, we need to impose a reasonable condition to the Young function φ. The Young function does not satisfy the global Δ 2 condition. We also recall that a continuous operator → S X X : has finite range if ( ( )) < ∞ S dim Ran . It is well known that each operator having finite range is a compact operator. With this notation and definition, we have the following result:   . Hence, we can build the partition Then we have    and we shall estimate is any partition of [ ] 0, 1 . As in the proof of Theorem 3.1, we consider four cases but now we will use the Δ 2 -condition: