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

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1 Vol. XIX, N o 1, Junio (211) Matemáticas: Matemáticas: Enseñanza Universitaria c Escuela Regional de Matemáticas Universidad del Valle - Colombia The Nemytskii oerator on bounded -variation in the mean saces René Erlín Castillo Universidad Nacional de Colombia Received Aug. 2, 21 Acceted Nov. 22, 21 Abstract We introduce the notion of bounded -variation in the sense of L -norm. We obtain a Riesz tye result for functions of bounded -variation in the mean. We show that if the Nemytskii oerator ma the bounded -variation in the mean saces into itself and satisfy some Lischitz condition there exist two functions g and h belonging to the bounded -variation in the mean sace such that f(t, y) = g(t)y + h(t), t [, 2π], y R. Keywords: Nemytskii s Oerator. MSC(2): Primary 26A45, 26B3. Secondary 26A16, 26A24. 1 Introduction Two centuries ago, around 188, 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 saces with same tye of variation (see [1]). The circle grou T is defined as the quotient R/2πZ, where, as indicated by the notation, 2πZ is the grou of integral multiles of 2π. There is a natural identification between functions on T and 2π-eriodic functions on R, which allows an imlicit 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 receding identification: a function f is integrable on T if the corresonding 2π-eriodic function, which we denote again by f, is integrable on [, 2π], and we set f(t)dt = f(x)dx. T Let f be a real-value function in L 1 on the circle grou T. We define the corresonding interval function by f(i) = f(b) f(a), where I denotes the interval [a, b]. Let = t < t 1 < < t n = 2π be a artition of [, 2π] and I kx = [x + t k 1, x + t k ], if { } V m (f, T ) = su f (I kx ) dx < T

2 32 R. Castillo where the suremum is taken over all artition of [, 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 concet was introduce by Móricz and Siddiqi [6], who investigated the convergence in the mean of the artial sums of S[f], the Fourier series of f. If f is of bounded variation (f BV ) with variation V (f, T ), then f (I kx ) dx 2πV (f, T ), T and so it is clear that BV BV M. A straightforward calculation shows that BV M is a Banach sace with norm f BV M = f 1 + V m (f, T ). In the resent aer we introduce the concet of bounded -variation in the mean in the sense of L [, 2π] norm (see Definition 2.1) and rove a characterization of the class BV M in terms of this concet. In 191 in [7], F. Riesz defined the concet of bounded -variation (1 < ) and roved that for 1 < < this class coincides with the class of functions f, absolutely continuous with derivative f L [a, b]. Moreover, the -variation of a function f on [a, b] is given by f L[a,b], that is V (f; [a, b]) = f L[a,b] (1) In this aer we obtain an analogous result for the class BV M. More recisely we show that if f BV M is such that f is continuous on [, 2π], then f L[, 2π] and m (f) = 2π f L (See theorem 2.9). 2 Bounded -variation in the mean Definition 2.1. let f L [, 2π] with 1 < <. Let P : = t < t 1 < < t n = 2π be a artition of [, 2π] if { m (f, T ) = su T } f (x + t k ) f (x + t k 1 ) t k t k 1 1 dx <, (2) where the suremum is taken over all artitions P of [, 2π], then f is said to be of bounded -variation in the mean.

3 Nemytskii oerator on bounded -variation in the mean saces 33 We denote the class of all functions which are of bounded -variation in the mean by BV M, that is BV M = { f L [, 2π] : V m (f, T ) < } (3) Remark 2.2. For 1 < <, it is not hard to rove that f BM = f L + { V m (f, T ) } 1/ (4) defines a norm on BV M. Proosition 2.3. Let f and g be two functions in BV M, then i) f + g BV M, ii) kf BV M, for any k R. In order words, BV M is a vector sace. Moreover V m (f + g, T ) 2 1 [ m (f, T ) + m (g, T ) ], and m (kf, T ) = k m (f, T ). Theorem 2.4. For 1 < <, BV M BV M and V m (f, T ) (2π) 2 2 [ V m (f, T ) ] 1. (5) Demostración. Let P : = t < t 1 < < t n = 2π be a artition of [, 2π] and consider f BV M, then by Hölder s inequality we obtain f (x + t k ) f (x + t k 1 ) dx ( (2π) 2 2 t k t k 1 dx ) 1 q ( ) f (x + t k ) f (x + t k 1 ) 1 t k t k 1 1 dx [ V m (f, T ) ] 1. (6) Thus f BV M, therefore BV M BV M. By (6) we obtain (5). This comletes the roof of Theorem 2.4. Theorem 2.5. Li [, 2π] BV M, where Li [, 2π] denotes the class of all functions which are Lischitz on [, 2π].

4 34 R. Castillo Demostración. Let f Li [, 2π], then there exists a ositive constant M > such that f(x) f(y) M x y, for all x, y [, 2π]. Let P : = t < t 1 < < t n = 2π be a artition of [, 2π], thus from (7) we have f (x + t k ) f (x + t k 1 ) M t k t k 1, (7) f (x + t k ) f (x + t k 1 ) t k t k 1 1 dx 4πM (8) by (8) we get f BV M. This comletes the roof of the Theorem 2.5. Remark 2.6. By Theorem 2.4 and 2.5, we can observe the following embedding: Li [, 2π] BV M BV M. Theorem 2.7. Let f Li [, 2π] and g BV M. Then f g BV M. Demostración. Let P : = t < t 1 < < t n = 2π be a artition of [, 2π] then Thus M f (g (x + t k )) f (g (x + t k 1 )) t k t k 1 1 dx g (x + t k ) g (x + t k 1 ) t k t k 1 1 dx. f (g (x + t k )) f (g (x + t k 1 )) t k t k 1 1 dx M V m (g, T ) (9) for all artitions of [, 2π]. By (9) we obtain Hence f g BV M. m (f g, T ) M m (g, T ). Theorem 2.8. BV M, Banach sace. equied with the norm defined in Remark 2.1. is a Demostración. Let {f n } n be a Cauchy sequence in BV M. Then for any ɛ > there exists a ositive integer no such that f n f m BM < ɛ whenever n, m n, (1)

5 Nemytskii oerator on bounded -variation in the mean saces 35 From (3) and (9) we have f n f m L f n f m BM < ɛ Whenever n, m n, this imlies that {f n } n N is Cauchy sequence in L since this sace is comlete, thus lím n f n exits, call it f. By Fatou s lemma and (3) we obtain f n f m BM lím inf f n f m m L + lím inf m { V m (f n f m, T ) } 1/ < ɛ whenever n n. Finally we need to rove that f BV M. In other to do that we invoke Fatou s lemma again. f BM lím inf n f n L + lím inf { V m (f n, T ) } 1/ <. Thus f BV M. This comletes the roof of Theorem 2.8 Theorem 2.9. Let f BV M such that f is continuous on [, 2π], then f L [, 2π] and Vm(f) = 2π f L (11) Demostración. Let P : = t < t 1 <... < t n = 2π be a artition of [, 2π]. By the Mean value theorem there exists ɛ k (x + t k 1, x + t k ) for any x [, 2π] such that for 1 < < by (12) we obtain 2π lím from (13) we have f(x + t k ) f(x + t k 1 ) t k t k 1 1 = f (ɛ k ) (t k t k 1 ) (12) f (ɛ k ) (t k t k 1 ) f(x + t k ) f(x + t k 1 ) t k t k 1 1 dx (13) 2π f (x) dx m (f). (14) Thus (14) imlies that f L [, 2π] and also we have on the other hand 2π f L V m (f) (15) f(x + t k ) f(x + t k 1 ) = x+tk 1 x+t k 1 f (t)dt (16)

6 36 R. Castillo by Hölder s inequality we obtain x+tk 1 ( ) x+tk 1 f (t)dt f (t) dt t k t k 1 1, x+t k 1 hence by (16) we get x+t k 1 then f(x + t k ) f(x + t k 1 ) x+tk 1 t k t k 1 1 f (t), dt x+t k 1 From (17) we finally have Combining (15) and (16) we obtain (11) f(x + t k ) f(x + t k 1 ) t k t k 1 1 2π f (x) dx. (17) V m (f) 2π f L (18) 3 Substitution Oerators Let Ω R be a bounded oen set. A function f : Ω R R is said to satisfy the Carathéodory conditions if i) For every t R, the function f(, t) : Ω R is Lebesgue measurable ii) For a.e. x Ω, the fuction f(x, ) : R R is continuous. Set M = {ϕ : Ω R : ϕ for each ϕ M define the oerator is Lebesgue measurable} (N f ϕ)(t) = f(t, ϕ(t)) The oerator N f is said to be the substitution or Nemytskii oerator generated by the function f. The urose of this section is to resent one condition under which the oerator N f mas BV M into itself. Also if N f satisfy the hyothesis condition from Lemma 3.1 below we will show that these exist two functions g and h which belong to the bounded variation in the mean sace such that f(t, y) = g(t)y + h(t), t [, 2π], y R.

7 Nemytskii oerator on bounded -variation in the mean saces 37 Lemma 3.1. N f : BV M BV M if there exits a constant L > such that f(s, ϕ(s) f(t, ϕ(t)) L ϕ(s) ϕ(t) for every ϕ M. Demostración. Let ϕ BV M, then { } (N f ϕ)(x + t k ) (N f ϕ)(x + t k 1 ) su Γ t k t k 1 1 dx { } f(x + t k, ϕ(x + t k )) f(x + t k 1, ϕ(x + t k 1 )) = su Γ t k t k 1 1 dx { } ϕ(x + t k ) ϕ(x + t k 1 ) L su t k t k 1 1 dx <. Thus N f BV M. Γ Theorem 3.2. Let f : [, 2π] R R and the Nemytskii oerator N f generated by f and defined by N f : BV M BV M u N f u, with (N f u) (t) = f (t, u(t)), t [, 2π]. If there exists a constant K > such that N f u 1 N f u 2 BM K u 1 u 2 BM (19) for u 1, u 2 BV M. Then there exist g, h BV M such that f(t, y) = g(t)y + h(t), t [, 2π], y R. Demostración. Let y R, define and u : [, 2π] R t u (t) = y, (a constant function) N f : BV M BV M u N f u with (N f u ) (t) = f (t, u (t)) = f(t, y). Note that f(t, y) BV M, y R, by hyothesis.

8 38 R. Castillo Next, let t, t [, 2π], t < t ; y 1, y 2, y 1, y 2 R. Now, we define u 1, u 2 by u i : [, 2π] R, i = 1, 2; y i, s < t, y s u i (s) = i y i t t (s t), t s t, y i, t < s 2π. Note that each u i belong to Li[, 2π], thus u 1 u 2 Li[, 2π]. Then y 1 y 2, s < t, y (u 1 u 2 ) (s) = 1 y 1 (y 2 y 2) t t (s t) + y 1 y 2, t s t, y 1 y 2, t < s 2π. Observe that, s < t, (u 1 u 2 ) y (s) = 1 y 1 (y 2 y 2) t t, t s t,, t < s 2π. and also that (u 1 u 2 ) is a continuous function on [, 2π]. Now, we can aly Theorem 2.5, obtaining 2π (u1 u 2 ) L = 2π (u1 u 2 ) (s) ds = 2π t t y 1 y 2 + y 2 y 1 t t = 2π y 1 y 2 + y 2 y 1 t t 1 <, ds hence Therefore m (u 1 u 2 ) = 2π y 1 y 2 + y 2 y 1 t t 1. u 1 u 2 BM = u 1 u 2 L + ( V m (u 1 u 2 ) ) 1 = u 1 u 2 L + (2π y 1 y 2 + y 2 y 1 t t 1 ) 1. By hyothesis N f u 1, N f u 2 belong to BV M and thus N f u 1 N f u 2 BV M with N f u i : [, 2π] R s (N f u i ) (s) = f (s, u i (s)),

9 Nemytskii oerator on bounded -variation in the mean saces 39 where f ( (s, y i ), ) s < t, f (s, u i (s)) = f s, y i y i t t (s t) + y i, t s t, f (s, y i ), t < s 2π. Next, let us consider the artition Π : < t < t < 2π, then ( (Nf u 1 N f u 2 ) (t ) (N f u 1 N f u 2 ) (t) t t 1 ( m (N f u 1 N f u 2, T ) ) 1 ) 1 N f u 1 N f u 2 L + ( m (N f u 1 N f u 2 ) ) 1 = N f u 1 N f u 2 BM K u 1 u 2 BM. Hence, by hyothesis we have ( f (t, y 1 ) f (t, y 2 ) f (t, y 1) + f (t, y 2 ) ) 1 t t 1 [ K u 1 u 2 L + (2π y 1 y 2 + y 2 y 1 ) 1 ] t t 1. (2) Then f (t, y 1 ) f (t, y 2 ) f (t, y 1) + f (t, y 2 ) t t 1 K [ u 1 u 2 L + (2π y 1 y 2 + y 2 y 1 ) 1 ] t t 1, since (A + B) 2 (A + B ) for A, B, we obtain f (t, y 1 ) f (t, y 2 ) f (t, y 1) + f (t, y 2 ) t t 1 [ 2 K u 1 u 2 L + 2π y 1 y 2 + y 2 y 1 ] t t 1, and f ( t, y 1) ( f t, y 2 ) f (t, y1 ) + f (t, y 2 ) (2K) [ u 1 u 2 L t t 1 + 2π y 1 y 2 + y 2 y 1 ]. Since f(, y) is continuous, if we let t t, then we have f ( t, y 1) f ( t, y 2 ) f (t, y1 ) + f (t, y 2 ) 2πK y 1 y 2 + y 2 y 1. (21)

10 4 R. Castillo Next, we make the following substitution: y 1 = w + z, y 2 = w, y 1 = z, y 2 =. (22) Putting (22) into (21) we get f (t, w + z) f (t, w) + f (t, ) f (t, z) 2πK w + z w z =, thus f (t, w + z) f (t, w) + f (t, ) f (t, z) =, from this latter equation we have f (t, w + z) f (t, ) = (f (t, w) f (t, )) + (f (t, z) f (t, )). Writing P t ( ) = f(t, ) f(t, ), then P t (w + z) = P t (w) + P t (z), which means that P t is additive and also P t ( ) = f(t, ) f(t, ) is a continuous function thus P t ( ) satisfy the functional Cauchy equation and its unique solution is given by P t (y) = g(t)y, with g : [, 2π] R, y R. Let h : [, 2π] R t h(t) = f(t, ), then h BV M and P t (y) = f(t, y) f(t, ) can be reduce to where f(t, y) = g(t)y + h(t). Finally, since g(t)y = f(t, y) h(t), f(t, 1) f(t, ) = (P t (1) + f(t, )) f(t, ) = g(t) for t [, 2π], we conclude that g BV M. Now the roof of Theorem 3.2 is comlete. Remark 3.3. The converse of Theorem 3.2 does not hold because L is not a Banach algebra.

11 Nemytskii oerator on bounded -variation in the mean saces 41 References [1] Castillo, R. E. and Trousselot, E.: On functions of (,α)-bounded variation. Real Anal. Exchange, 34, (29), n.1,49-6. [2] Jordan, C.: Sur la Série de Fourier, C. R. Math. acad. Sci. Paris, 2, (1881), [3] Merentes, N.: Functions of bounded (φ,2) Variation Annals Univ Sci Budaest, XXXIV, (1991), [4] Merentes, N.: On Functions of bounded (,2)-variation. Collect Math, 43, 2, (1992), [5] Medvedév, Y.: A generalization of certain Theorem of Riesz, Usekhi. Mat. Nauk, 6, (1953), [6] Móricz, F. and Siddiqi, A. H.: A quatified version of the Dirichlet-Jordan test in L 1 -norm, Rend. Circ. Mat. Palermo, 45, (1996), [7] Riesz, F.: Untersuchungen über system intergrierbarer function, Mathematische Annalen, 69, (191), [8] Schramm, M.: Functions of Φ-Bounded variation and Riemann-Stieltjes Integration. Trans of the Amer Math. Soc., 287, 1, (1985), [9] Waterman, D.: On Λ-Bounded Variation, Studia Mathematicae, LVII, (1976), Author s address René Erlín Castillo Deartamento de Matemáticas, Universidad Nacional de Colombia, Bogotá-Colombia recastillo@unal.edu.co