On Bounded Second Variation

In this paper, we discuss various aspects of the problem of space-invariance, under compositions, of certain subclasses of the space of all continuously differentiable functions on an interval   , a b   · · k . We present a result about integrability of products of the form f f f   under suitable mild conditions and, finally, we prove that a Nemytskij operator g g S maps   , , BV a b   loc g BV    a distinguished subspace of the space of all functions of second bounded variation, into itself if, and only if . A similar result is obtained for the space of all functions of bounded  -variation   , 2   1 1 , p    p  2 , . p A a b


Introduction
Throughout this paper we use the following notations: if g, f are given functions, the expression g f  stands for the composite function  ,  g f t whenever it is welldefined;   , a b denotes a compact interval in (the field of all real numbers) and   denotes the Lebesgue measure on .As usual, the set of all natural numbers will be denoted by . diferentiable.It is well known that the composition of two functions of bounded variation, say g and f, in general, need not be of bounded variation; in fact, not even if we choose the inner function well-enough behaved guarantees that the composition g f    is of bounded variation.For instance, if g However, the multiplication of f    k by a derivative of f, which is of bounded variation, improves the properties of that composite function.Indeed, a proof of the following theorem can be found in [3, Theorem 5].
Let be a subspace of  .Given a function , the autonomous Nemytskij (or Superposition, see [4]) opera Given two linear spaces and a function , a primary objective of research is to investigate under what conditions on the generating function the associated Nemytskij operator maps into  .This problem is known as the Superposition Operator Problem.
Recently (see e.g.[5]), the Superposition Operator Problem have been studied extensively in various spaces of differentiable functions related to the spaces .We prove a version of Theorem 1.1 about the integrability of products of the form is continuous, and obtain an estimation of the norm we give necessary

Some Function Spaces
finitions and state some Finally, we prove two results in which and sufficient conditions for the autonomous Nemytskij Operator to map the space of all functions of second bounded variation into itself and the class of functions of bounded   , 2 p -variation into itself.
In this section we recall some de results which will be needed for the further development of this work. We will use the notation

  
; , M a b to denote functi BV the space of all bounded ons f such that The class of all Lipschitz continuous functions in The class of all absolutely con us functions on tinuo   , a b , which is actually an algebra, is denoted as The equivalence (a) In this case, the relation In this case the relation A a b .Clearly a continuously differentiable function is Lipschitz continuous and any Lipschitz continuous function is > 1, p the following ch absolutely continuous.In fact, ain of strict inclusions holds (see e.g., [5,10]):

Main Results
We begin this section by stating some fundamentals facts ions of functions on BV and AC.In sic properties of the inner function concerning composit these cases the intrin (in the composition) will show to play also an important role.We recall that if D and E are given sets, X is a linear subspace of E  and φ is a map from D to E, the linear composition operator

instances of a very remarkable phe n that of nonlinear
In what follows we will observe more nomeno ten occurs in functional analysis: is the case in which given two functions, say g and f, the multiplication of g f  by a continuous derivative integrability of products of the form a b similar considerations as those discussed above apply with respect to the g f f   or even  p g f f    .Now, not even the fact that the function g is integrable and the function f is absolutely continuous guarantees that the product and, for 0 x  , let   : if, and only if, the function , where Notice that G bsolutely continuous function, which bring us back to the same situatio above.

bility) properties of the product
is an a ns considered It turns out that, if g is an integrable function, multiplication by a continuous derivative of f improves the (integra g f f   .By an lemma alogy with an useful notion originated from the theory of partial differential equations, we might call this derivative an integrating factor.The following provides a version of Theorem 1.1 when the outer function in the composition is an integrable function.The proposition might be of some interest in itself.
Lemma 3.4.Suppose that , defined by el nite integral function G by (3.1), is absolut y continuous, thus Remark 3.1, the monotonicity of f on ,  is integrable on this interval.Hence, since f  does not change sign on , wher tion e the nota , is continuous, (generalized) for i ies th the mean value theorem ntegrals impl at, on each , The proof is complete.□ For convenience we state the next result as a single proposition.The proof of it is based in three separate results of M. Josephy [6] (see also [12]), N. Merentes [13] and N. Merentes and S. Rivas [14].

The Autonomous Nemytskij Operator on the Spaces
Proposition 3.5.Suppose   is said to be of bounded variation on   , a b if the (total) variation of f   is of bounded variation on   , a b if, and only if, it is the diference of two monotone functions.In particular, every function in  and right limit   + f x at every point  ,  x a b  ; also, by the celebrated Lebesgue's Theorem (see e.g.[2, Theorem 1.2.8]) every function in   , BV a b is -. .a e paper, we discuss various aspects of the the Superposition Operator Problem when the spaces 

1 .
(Luzin N property).A real-valued de Def function fined on a compact interval I   is said to satisfy the Luzin N property (or simply, property N) if it carries sets of -  measure zero into set s of  measure zero.It is easy to see that the composition of tw unctions that ha o f ve property N also has property N.The class of all continuous functions that satisfy property N on an interval   , a b will be denoted as   , N a b .The following characterization of absolutely continuous fun ons is well known (c ap cti f. [2, Ch ter 7]).Proposition 2.2.The following statements on a function is an algebra with respect to pointwise multiplica continuo tion (see[3]), and the Banach-Zareckii Theorem.Proposition 3.2.(Burenkov).If f has an absolutely us - , then the function For any given pair of r β eal numbers α, with α < β denote by ab f  the linear difeomorphism Séances de l'Académie des Sciences, Vol. 2, 1881, pp.228-230.
one given for the the necessity of the condition in th Th □