Some Classes of Function Spaces, Their Properties, and Their Applications 2014View this Special Issue
Locally Lipschitz Composition Operators in Space of the Functions of Bounded -Variation
We give a necessary and sufficient condition on a function under which the nonlinear composition operator , associated with the function , , acts in the space and satisfies a local Lipschitz condition.
Given a function , the composition operator associated with the function maps each function into the composition function defined by
More generally, given , we consider the operator , defined by
This operator is also called superposition operator or substitution operator or Nemytskij operator associated with . In what follows, we will refer to (1) as the autonomous case and to (2) as the nonautonomous case. For an extensive treatment of composition operator and function spaces we refer to the monographs Appell et al. , Appell and Zabrejko , and Runst and Sickel .
In 1984, Sobolevskij  proved the following statement: “the autonomous composition operator associated with is locally Lipschitz in the space Lip if and only if the derivative exists and is locally Lipschitz.” In recent articles Appell et al.  and Merentes et al.  obtained several results of the Sobolevskij type. As the authors explain in the introduction, the significance of these results lies in the fact that in most applications to many nonlinear problems it is sufficient to impose a local Lipschitz condition, instead of a global Lipschitz condition. In fact they proved that Sobolevskij’s result is valid in the spaces , , , and .
Although the composition operator (or Nemytskij operator) is very simple, it turns out to be one of the most interesting and important operators studied in nonlinear functional analysis; the behavior of this operator exhibits many surprising and even pathological features in various function spaces. For example, about 35 years ago Dahlberg  proved the following: for and integer, if maps the Sobolev space into itself, then is a linear function. Among these pathologies there is one called degeneracy phenomenon, which states that the global Lipschitz condition necessarily leads to affine functions in various functions spaces. This property was first proved in  for the space Lip. Additional information about the degeneracy phenomena can be found in [9, 10].
This paper is organized as follows: Section 2 contains definitions, notations, and necessary background about the class of functions of bounded -variation in the sense of Schramm-Korenblum; Section 3 contains the main theorem. Also in this section we state and prove a Helly-type theorem, which plays a crucial role in the demonstration of our Sobolevskij-type result.
2. Some Function Spaces
The concept of functions of bounded variation has been well known since C. Jordan gave the complete characterization of functions of bounded variation as a difference of two increasing functions in 1881. This class of functions exhibits so many interesting properties that it makes a suitable class of functions in a variety of contexts with wide applications in pure and applied mathematics [1, 11].
Definition 1. Let be a function. For a given partition of the interval ,
is called the variation of on with respect to .
The (possibly infinite) number, where the supremum is taken over all partitions of the interval is called the total variation of on . If , we say that has bounded variation. The collection of all functions of bounded variation on is denoted by .
This notion of a function of bounded variation has been generalized by several authors. One of these generalized versions was given by Korenblum in 1975 . He considered a new kind of variation, called -variation, and introduced a function for distorting the expression in the partition itself, rather than the expression in the range. One advantage of this alternative approach is that a function of bounded -variation may be decomposed into the difference of two simpler functions called -decreasing functions.
Definition 2. A function is called a distortion function (-function) if satisfies the following properties: (1)is continuous with and ;(2) is concave and increasing;(3).
Korenblum (see ) introduced the definition of bounded -variation as follows.
Definition 3. Let be a distortion function, a real function , and a partition of the interval . Let one consider where the supremum is taken over all partitions of the interval . In the case one says that has bounded -variation on and one will denote by the space of functions of bounded -variation on .
Schramm in 1985  considered a -sequence as follows.
Definition 4 (-sequence). Let be a sequence of increasing convex functions, defined on such that (1), ;(2) for .
We will say that is a -sequence if for all and and a -sequence if in addition diverges for .
From now on, all sequences considered in this work will be -sequences. We will consider a nonoverlapping family of subintervals of the interval , ; it means that either is empty or contains a single point for , .
Definition 5. If is a -sequence, one says that a function is of bounded -variation if the -sums for any nonoverlapping collection of the interval .
Definition 6 (condition generalized for small values ). The -sequence satisfies condition if and only if there exist and such that
We may define, for of bounded -variation, the total -variation of by where the supremum is taken over all . Hernández and Rivas (see ) showed that if is a -sequence and satisfies condition , then is a linear space. We denote by the collection of all functions such that is of bounded -variation for some .
S. K. Kim and J. Kim in 1986  considered a bounded -variation as follows.
Definition 7. Let be a distortion function and a -sequence and let . One defines
If , we say that has bounded -variation in the interval and this number denotes the -variation of in Schramm-Korenblum’s sense in . The class of functions that has bounded -variation in the interval is denoted by . The vector space generated by this class is denoted by .
Let us consider as a function of variable . If is a sequence of increasing convex functions, , , we have , . Let and let . Then as . With this in mind, we define a norm in the space as follows: We will consider the following norm in the space : where and denotes the supremum norm.
By the above definition, we have the following.
Theorem 8 (see ). Let be a sequence such that converges to almost everywhere with . Then that is, the Luxemburg norm is lower semicontinuous on .
Theorem 9 (see ). is a Banach space.
Definition 10 (see ). Let be a -sequence. A real function is said to be -decreasing on if there exists a positive constant such that for each subinterval of
Lemma 11 (see ). For any -function and any -sequence , one has the following: (1), ,(2)if , then , .
Lemma 12 (see ). Let be a distortion function and a -sequence and let and . Then if and only if .
Theorem 14 (see ). Let be a distortion function, let be a -sequence, let , and let be the composition operator associated with . maps the space into the space or if and only if is locally Lipschitz. Furthermore, the operator is bounded.
The following lemma is basic for our main result.
Lemma 15 (invariance principle). Let be a function. Then the composition operator (1) maps the space into itself if and only if it maps, for any other choice of , the space into itself.
Proof. Suppose that the composition operator defined by maps the space into itself. The function defined by
is a strictly increasing homeomorphism between and with inverse
which satisfies and . Let denote the family of all partitions of . Thus, with
defines a one-to-one correspondence between all partitions of and all partitions of .
Given , the function belongs to , by the definition of functions of bounded -variation, and so belongs to , by assumption. But for and as above we have
Passing to the supremum with respect to and we conclude that .
3. Main Results
In the proof of the main result of this paper, we will employ a compactness result, for instance, Helly’s selection principle or second Helly’s theorem. Helly’s theorem for functions of generalized variation has been of some importance for a long time. Helly’s selection principle has been the subject of intensive research, and many applications, generalizations, and improvements of them can be found in the literature (see, e.g., [19–21] and the references therein).
In this part we will state and prove our main results. In the proof of our main result we make use of a Helly-type selection theorem for a -decreasing function.
In the paper  Cyphert and Kelingos proved the same result for an arbitrary infinite family of functions defined on which is both uniformly bounded and uniformly -decreasing.
Theorem 16 (Helly-type selection theorem). An arbitrary infinite family of functions defined on which is both uniformly bounded and uniformly -decreasing contains a subsequence which converges at every point of to a -decreasing function.
Proof. Let us denote by an arbitrary infinite family of functions defined on , which is both uniformly bounded and uniformly -decreasing. Then, there exists a constant such that for every and every pair
Using (17) we can, by means of the standard Cantor diagonalization technique, find a sequence of functions in which converges pointwise at each rational point of , to a function . Since each satisfies (18), so does , for all rational numbers .
Define at irrational points by
The existence of this limit can be seen as follows:
Let and be two sequences of rational points converging to , arranged so that and such that and as . Then
Then , and hence .
From (19) we obtain, by taking limits of rational points in inequality (18), that satisfies (18) for all pairs of positive real numbers; that is, is -decreasing with constant on . By Theorem 13 is of bounded -variation and is continuous. Hence, by another Cantor diagonalization process, a convergent subsequence of the functions can be found.
Now, let us consider and . Then, we fix two rational numbers and with such that Since the sequence , , converges to in the rational numbers, there exists such that
Now, from (22) and (23) we obtain
We are now in a position to formulate and prove our main result.
Theorem 17. Let us suppose that the composition operator associated with maps the space into itself. Then is locally Lipschitz if and only if exists and is locally Lipschitz in .
Proof. First let us assume that is locally Lipschitz in . Given , for , we denote by the minimal Lipschitz constant of and by the supremum of on the bounded set
The finiteness of implies that satisfies a local Lipschitz condition with respect to the norm , so we only have to prove a local Lipschitz condition for with respect to the -variation norm. We will prove this by applying twice the mean value theorem.
In fact, let us fix with and , . Given a partition of , we split the index set into a union of disjoint sets and by defining the following:
if and if By the classical mean value theorem we find between and such that Now, by definition of we have
A straightforward calculation shows then that Since for , we obtain that and dividing by and adding on we get that
Again, by the mean value theorem, we find between and and between and such that By definition of we have
A straightforward calculation shows that Since for , we obtain that and dividing by and adding on we get that
Summing up both partial sums and observing that and do not depend on the partition we conclude that which proves the assertion.
Conversely, suppose that satisfies a Lipschitz condition. By assumption, the constant is finite for each . Considering, in particular, both functions and in (40) constant, we see that This shows that is locally Lipschitz, and so the derivative exists almost everywhere in . It remains to prove that exists everywhere in and is locally Lipschitz. For the proof of the first claim we show that exists in any closed interval .
Given , we consider with . Let be a decreasing sequence of positive real numbers converging to ; without loss of generality, we may assume that for all . We define a sequence of functions by
Since the composition operator associated with acts in the space , by assumption, the functions given by (42) belong to .
Now, we show that the sequence has uniformly bounded -variation for all with . In fact, let be a partition of the interval . For each we define functions and by
Then and . Furthermore, from Lemma 11, (42), and (43), we obtain the estimates
Since the partition was arbitrary, the inequality holds for every and each with . From Lemma 11, the definition of the function in (42), and the definition of the functions and in (43), we further get and hence . By Lemma 11, we conclude that which shows that the sequence satisfies the hypotheses of Theorem 16.
Theorem 16 ensures the existence of a pointwise convergent subsequence of ; without loss of generality we assume that the whole sequence converges pointwise on to some function .
Now we define , where is so small that . By (43) we see that for almost all . Since the primitive of and the function are both absolutely continuous and have the same derivative on , we conclude that they differ only by some constant on , and so exists everywhere on . From the invariance principle (Lemma 15) we deduce that the derivative of exists on any interval and so everywhere in .
It remains to prove that satisfies a local Lipschitz condition. Denoting by the composition operator associated with the function from (48), we claim that, for with , we have where is the Lipschitz constant from (40). In fact, we conclude that whenever the sequence of functions converges pointwise on to a function . Combining this with (47) and the observation that the sequence converges as , we obtain (49). We conclude that the composition operator maps the space into itself, and so the corresponding function is locally Lipschitz on . By (48), the same is true for the function .
Conflict of Interests
The authors declare that there is no conflict of interests.
This research has been partially supported by the Central Bank of Venezuela. The authors want to give thanks to the library staff of B.C.V. for compiling the references.
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