The Nemytskii operator on bounded p-variation in the mean spaces

We introduce the notion of bounded p-variation in the sense of Lp-norm. We obtain a Riesz type result for functions of bounded p-variation in the mean. We show that if the Nemytskii operator map the bounded p-variation in the mean spaces into itself and satisfy some Lipschitz condition there exist two functions g and h belonging to the bounded p-variation in the mean space such that f(t, y) = g(t)y + h(t), t ∈ [0, 2π], y ∈ R.


Introduction
Two centuries ago, around 1880, C. Jordan (see [2]) introduced the notion of a function of bounded variation and established the relation between these functions and monotonic ones; since then a number of authors such as, Yu Medvedév (see [5]), N Merentes (see [3] and [4]), D Waterman (see [9]), M Schramm (see [8]) and recently Castillo and Trousselot had been study different spaces with same type of variation (see [1]).The circle group T is defined as the quotient R/2πZ, where, as indicated by the notation, 2πZ is the group of integral multiples of 2π.There is a natural identification between functions on T and 2π-periodic functions on R, which allows an implicit introduction of notions such as continuity, differentiability, etc for functions on T .The Lebesgue measure on T also can be defined by means of the preceding identification: a function f is integrable on T if the corresponding 2π-periodic function, which we denote again by f , is integrable on [0, 2π], and we set Let f be a real-value function in L 1 on the circle group T .We define the corresponding interval function by where the supremum is taken over all partition of [0, 2π], then f is said to be of variation in the mean (or bounded variation in L 1 -norm).
We denote the class of all functions which are of bounded variation in the mean by BV M .This concept was introduce by Móricz and Siddiqi [6], who investigated the convergence in the mean of the partial sums of S[f ], the Fourier series of f .
If f is of bounded variation (f ∈ BV ) with variation V (f, T ), then and so it is clear that BV ⊂ BV M .A straightforward calculation shows that BV M is a Banach space with norm In the present paper we introduce the concept of bounded p-variation in the mean in the sense of L p [0, 2π] norm (see Definition 2.1) and prove a characterization of the class BV p M in terms of this concept.
In 1910 in [7], F. Riesz defined the concept of bounded p-variation (1 ≤ p < ∞) and proved that for 1 < p < ∞ this class coincides with the class of functions f , absolutely continuous with derivative f ∈ L p [a, b].Moreover, the p-variation of a function f on [a, b] is given by f Lp[a,b] , that is In this paper we obtain an analogous result for the class BV p M. More precisely we show that if f ∈ BV p M is such that f is continuous on [0, 2π], then f ∈ Lp[0, 2π] and 2 Bounded p-variation in the mean Definition 2.1.
where the supremum is taken over all partitions P of [0, 2π], then f is said to be of bounded p-variation in the mean.
We denote the class of all functions which are of bounded p-variation in the mean by BV p M , that is Remark 2.2.For 1 < p < ∞, it is not hard to prove that defines a norm on BV p M .
Proposition 2.3.Let f and g be two functions in BV p M , then In order words, BV p M is a vector space.Moreover Demostración.Let P : 0 = t 0 < t 1 < • • • < t n = 2π be a partition of [0, 2π] and consider f ∈ BV p M , then by Hölder's inequality we obtain Thus f ∈ BV M , therefore BV p M ⊂ BV M .By (6) we obtain (5).This completes the proof of Theorem 2.4.
from (7) we have by ( 8) we get f ∈ BV p M .This completes the proof of the Theorem 2.5.
Demostración.Let {f n } n be a Cauchy sequence in BV p M. Then for any > 0 there exists a positive integer no such that From ( 3) and ( 9) we have Whenever n, m ≥ n 0 , this implies that {f n } n∈N is Cauchy sequence in Lp since this space is complete, thus lím n→∞ f n exits, call it f .By Fatou's lemma and (3) we obtain Finally we need to prove that f ∈ BV p M. In other to do that we invoke Fatou's lemma again.
Thus f ∈ BV p M .This completes the proof of Theorem 2.8 Theorem 2.9.
Demostración.Let P : 0 = t 0 < t 1 < . . .< t n = 2π be a partition of [0, 2π].By the Mean value theorem there exists k ∈ (x + t k−1 , x + t k ) for any x ∈ [0, 2π] such that for 1 by ( 12) we obtain 2π lím Thus (14) implies that f ∈ L p [0, 2π] and also we have on the other hand by Hölder's inequality we obtain hence by ( 16) we get From ( 17) we finally have Combining ( 15) and ( 16) we obtain (11) The operator N f is said to be the substitution or Nemytskii operator generated by the function f .The purpose of this section is to present one condition under which the operator N f maps BV p M into itself.Also if N f satisfy the hypothesis condition from Lemma 3.1 below we will show that these exist two functions g and h which belong to the bounded p − variation in the mean space such that and the Nemytskii operator N f generated by f and defined by If there exists a constant K > 0 such that for u 1 , u 2 ∈ BV p M .Then there exist g, h ∈ BV p M such that Demostración.Let y ∈ R, define and Note that each u i belong to Lip[0, 2π], thus Observe that and also that (u 1 − u 2 ) is a continuous function on [0, 2π].Now, we can apply Theorem 2.5, obtaining Therefore Next, let us consider the partition Π : 0 < t < t < 2π, then Hence