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

1 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, 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 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 B V φ [ 0 , 1 ] spaces, which we define in Section 2. Each function u ∈ B V φ [ 0 , 1 ] defines a linear and continuous operator M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] given by M u ( f ) = u ⋅ f 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 2 ( μ ) . 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 ∈ B V φ [ 0 , 1 ] 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 ∈ B V φ [ 0 , 1 ] which define Fredholm multiplication operators.

2 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 φ : [ 0 , ∞ ) → [ 0 , ∞ ) 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

where the supremum is taken over all finite partitions P : 0 = t 0 < t 1 < ⋯ < t n = 1 of [ 0 , 1 ] . With this notation, we say that a real-valued function f defined on [ 0 , 1 ] belongs to the space of bounded φ -variation functions B V φ [ 0 , 1 ] (which was introduced by Young in 1937, see [4]) if there exists a λ > 0 such that

For each f ∈ B V φ [ 0 , 1 ] , we can define the quantity

which is a seminorm for B V φ [ 0 , 1 ] (known as the Luxemburg seminorm) and B V φ [ 0 , 1 ] is a Banach space with the norm

where

In fact, for all f , g ∈ B V φ [ 0 , 1 ] we have that f ⋅ g ∈ B V φ [ 0 , 1 ] and

which means that B V φ [ 0 , 1 ] is a Banach algebra. When φ ( t ) = t we have the classical space of bounded variation; while when φ ( t ) = t p , the space B V φ [ 0 , 1 ] becomes into W B V p [ 0 , 1 ] , the space of functions of bounded p -variation, introduced by Wiener in 1924 (see [3]).

We fix a Young function φ and consider the space B V φ [ 0 , 1 ] defined by φ . Next, we have the following:

1. If f ∈ B V φ [ 0 , 1 ] , then there exists λ > 0 such that

   Var φ 1 λ f < ∞ .

   Let 0 < β ≤ 1 λ . Then 0 < β λ ≤ 1 . Since φ is convex and φ ( 0 ) = 0 ,

   Var φ ( β f ) = Var φ λ β f λ + ( 1 − λ β ) 0 ≤ λ β Var φ 1 λ f < ∞ .

   Hence,

   0 ≤ lim β → 0 Var φ ( β f ) ≤ lim β → 0 λ β Var φ 1 λ f = 0

   and so

   lim β → 0 Var φ ( β f ) = 0 .

2. By (a), there exists β > 0 such that

   Var φ ( β f ) ≤ 1 .

   Let λ = 1 β . And so, there exists λ > 0 such that

   Var φ 1 λ f ≤ 1 .

   Hence, we have proved that

   λ > 0 : Var φ 1 λ f ≤ 1 ≠ ∅ .

3. The quantity

   (2.2) ‖ f ‖ φ = inf λ > 0 : Var φ 1 λ f ≤ 1

   satisfies the following property:

   (2.3) Var φ f ‖ f ‖ φ ≤ 1

   whenever f ∈ B V φ [ 0 , 1 ] is not a constant function. In fact, f is a constant function on [ 0 , 1 ] if and only if Var φ ( f ) = 0 which occurs if and only if ‖ f ‖ φ = 0 . In particular, we have

   λ > 0 : Var φ 1 λ f ≤ 1 = ( ‖ f ‖ φ , + ∞ ) .

   Hence, ‖ f ‖ φ ≤ k if and only if Var φ 1 k f ≤ 1 . Furthermore, the relation (2.3) implies that for all f ∈ B V φ [ 0 , 1 ] and t ∈ [ 0 , 1 ] we have

   ∣ f ( t ) ∣ ≤ ∣ f ( 0 ) ∣ + φ − 1 ( 1 ) ‖ f ‖ φ

   and B V φ [ 0 , 1 ] ⊊ B [ 0 , 1 ] , the space of all bounded functions defined on [ 0 , 1 ] , and hence we can write

   ‖ f ‖ ∞ ≤ ∣ f ( 0 ) ∣ + φ − 1 ( 1 ) ‖ f ‖ φ .

   We also have that if f , g ∈ B V φ [ 0 , 1 ] , then f ⋅ g ∈ B V φ [ 0 , 1 ] and

   (2.4) ‖ f ⋅ g ‖ φ ≤ ‖ f ‖ ∞ ‖ g ‖ φ + ‖ g ‖ ∞ ‖ f ‖ φ .

   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 ‖ f ‖ φ > 0 and ‖ g ‖ φ > 0 . We set

   M = ‖ f ‖ ∞ ‖ g ‖ φ + ‖ g ‖ ∞ ‖ f ‖ φ ,

   then for any partition P : 0 = t 0 < t 1 < ⋯ < t n = 1 we can write

   ∑ k = 1 n φ 1 M ∣ f ( t k ) g ( t k ) − f ( t k − 1 ) g ( t k − 1 ) ∣ = ∑ k = 1 n φ 1 M ∣ ( f ( t k ) − f ( t k − 1 ) ) g ( t k ) + f ( t k − 1 ) ( g ( t k ) − g ( t k − 1 ) ) ∣ ≤ ∑ k = 1 n φ ‖ g ‖ ∞ M ∣ f ( t k ) − f ( t k − 1 ) ∣ + ‖ f ‖ ∞ M ∣ g ( t k ) − g ( t k − 1 ) ∣ = ∑ k = 1 n φ ‖ g ‖ ∞ ‖ f ‖ φ M ∣ f ( t k ) − f ( t k − 1 ) ∣ ‖ f ‖ φ + ‖ f ‖ ∞ ‖ g ‖ φ M ∣ g ( t k ) − g ( t k − 1 ) ∣ ‖ g ‖ φ ≤ 1 M ‖ g ‖ ∞ ‖ f ‖ φ Var φ f ‖ f ‖ φ + 1 M ‖ f ‖ ∞ ‖ g ‖ φ Var φ g ‖ g ‖ φ ≤ 1 M ‖ g ‖ ∞ ‖ f ‖ φ + 1 M ‖ f ‖ ∞ ‖ g ‖ φ = 1 ,

   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

   Var φ f ⋅ g M ≤ 1

   which proves that ‖ f ⋅ g ‖ φ ≤ M as affirmed.

This is known as an indicatrix function. In particular, we have the following very useful property:

3 Multiplication operators with closed range on B V φ [ 0 , 1 ]

In this section, we characterize all functions u ∈ B V φ [ 0 , 1 ] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] with closed range. It is convenient, from now, to define the following set:

hence, Z u c = [ 0 , 1 ] ⧹ Z u = { t ∈ [ 0 , 1 ] : u ( t ) ≠ 0 } . Also, for ε > 0 fixed, we set

with these notations, we have the following result.

4 On the compactness

In this section, we analyze when a function u ∈ B V φ [ 0 , 1 ] defines compact multiplication operators M u acting on B V φ [ 0 , 1 ] . 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 n } ⊂ X . It is known that the limit of compact operators is also a compact operator and that the identity operator I : X → X defined by I f = f is compact if and only if dim ( X ) < ∞ . We recall that for ε > 0 we set

then we have the following result.

For the converse, we need to impose a reasonable condition to the Young function φ . The Young function φ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfies the global Δ 2 condition if there exists a constant K Δ > 0 such that

for all t ≥ 0 . For instance, the Young function φ 1 ( t ) = t log ( 1 + t ) satisfies the global Δ 2 condition but the Young function φ 2 ( t ) = e t − 1 does not satisfy the global Δ 2 condition. We also recall that a continuous operator S : X → X has finite range if dim ( Ran ( S ) ) < ∞ . It is well known that each operator having finite range is a compact operator. With this notation and definition, we have the following result: