AN APPLICATION OF THE GENERALIZED MALIGRANDA-ORLICZ’S LEMMA

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Maligranda-Orlicz’s Lemma René Erlin Castillo and

Eduard Trousselot vol. 9, iss. 3, art. 84, 2008

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AN APPLICATION OF THE GENERALIZED MALIGRANDA-ORLICZ’S LEMMA

RENÉ ERLIN CASTILLO AND EDUARD TROUSSELOT

Departamento de Matemáticas Universidad de Oriente

6101 Cumaná, Edo. Sucre, Venezuela

EMail:rcastill@math.ohiou.edu eddycharles2007@hotmail.com

Received: 17 May, 2008

Accepted: 22 September, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 46F10.

Key words: Banach algebra, Maligranda-Orlicz.

Abstract: Using the generalized Maligranda-Orlicz’s Lemma we will show that BV_(2,α)([a, b])is a Banach algebra.

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1. 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. Later, the concept of bounded variation was generalized in various directions. In his 1908 paper de la Vallée Poussin (see [4]) generalized the Jordan bounded variation concept. De la Vallée Poussin, defined the bounded second variation of a functionf on an interval[a, b]by

V²(f) = V²(f,[a, b]) = sup

Π n−1

X

j=1

f(tj+1)−f(tj)

t_j+1−t_j − f(tj)−f(tj−1) t_j −tj−1

where the supremum is taken over all partitionsΠ :a =t₀ < t₁ <· · ·< t_n =b of [a, b].IfV²(f,[a, b]) <∞, the functionf is said to be of bounded second variation on[a, b].The class of all functions which are of bounded second variation is denoted byBV²([a, b]).

In 1970 the above class of functions was generalized with respect to a strictly increasing continuous functionα(see [3]):

Let f be a real function defined on [a, b]. For a given partition of the form:

Π :a=t₁ <· · ·< t_n=b,we set σ_(2,α)(f,Π) =

n−2

X

j=1

|f_α[t_j, t_j+1]−f_α[t_j+1, t_j+2]|, where

f_α[p, q] = f(q)−f(p) α(q)−α(p), and

V_(2,α)(f,[a, b]) =V_(2,α)(f) = sup

Π

σ_(2,α)(f,Π),

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where the supremum is taken over all partitionsΠof[a, b].

IfV_(2,α)(f)<∞,thenf is said to be of(2, α)-bounded variation.

The set of all these functions will be denoted byBV_(2,α)([a, b]).

A functionf isα-derivable att₀if

t→tlim0

f(t)−f(t0)

α(t)−α(t₀) exists.

If this limit exists, we denote its value byf_α⁰(t₀),which we call theα-derivative off att₀.

The classBV_(2,α)([a, b])is a Banach space equipped with the norm kfk_BV(2,α)([a,b]) =|f(a)|+|f_α⁰(a)|+V_(2,α)(f).

Using the generalized Maligranda-Orlicz’s Lemma (see Theorem3.1of the present paper) we will show thatBV_(2,α)([a, b])is a Banach algebra.

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2. Definition and Notation

We begin this section by giving a definition and several simple lemmas that will be used throughout the paper.

Definition 2.1. A functionf : [a, b] → R is said to beα-Lipschitz if there exists a constantM >0such that

|f(x)−f(y)| ≤M|α(x)−α(y)|,

for allx, y ∈[a, b], x6=y.Byα-Lip[a, b]we will denote the space of functions which areα-Lipschitz. Iff ∈α-Lip[a, b]we define

Lip_α(f) = inf{M >0 :|f(x)−f(y)| ≤M|α(x)−α(y)|, x6=y∈[a, b]}

and

Lip⁰_α(f) = sup

|f(x)−f(y)|

|α(x)−α(y)| :x6=y∈[a, b]

.

It is not hard to prove that

Lip_α(f) = Lip⁰_α(f).

α-Lip[a, b]equipped with the norm

kfk_α-Lip[a,b]=|f(a)|+ Lip_α(f) is a Banach space.

Lemma 2.2. Iff ∈BV_(2,α)([a, b]),then there exists a constantM >0such that

f(x₂)−f(x₁) α(x2)−α(x1)

=|f_α[x₁, x₂]| ≤M for allx1, x2 ∈[a, b].

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Lemma 2.3.

kfk_α-Lip[a,b] ≤ kfk_BV_(2,α)_([a,b]), f ∈BV_(2,α)([a, b]) and

BV_(2,α) ,→α-Lip[a, b].

Lemma 2.4. α-Lip[a, b],→BV[a, b],→B[a, b].

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3. Generalized Maligranda-Orlicz’s Lemma

The following result generalizes the Maligranda-Orlicz Lemma which is due to the authors (see [1]).

Theorem 3.1. Let(X,k · k)be a Banach space whose elements are bounded func- tions, which is closed under pointwise multiplication of functions. Let us assume thatf ·g ∈Xsuch that

kf gk ≤ kfk_∞kgk+kfkkgk_∞+Kkfkkgk, K >0.

Then(X,k · k₁)equipped with the norm

kfk₁ =kfk∞+Kkfk, f ∈X,

is a Banach algebra. IfX ,→B[a, b],thenk · k₁ andk · kare equivalent.

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4. BV

_(2,α)

([a, b]) as a Banach Algebra

The following result shows us thatBV_(2,α)([a, b])is closed under pointwise multiplication of functions.

Theorem 4.1. Iff, g ∈BV_(2,α)([a, b]),thenf·g ∈BV_(2,α)([a, b]).

Proof. LetΠ :a=x₁ < x₂ <·< x_n=bbe a partition of[a, b].Then σ_(2,α)(f·g,Π) =

n−2

X

j=1

(f g)_α[x_j, x_j+1]−(f g)_α[x_j+1, x_j+2]

=

n−2

X

j=1

f(x_j)·g_α[x_j, x_j+1] +g(x_j+1)·f_α[x_j, x_j+1]

−f(x_j+1)·g_α[x_j+1, x_j+2]−g(x_j+2)·f_α[x_j+1, x_j+2]

≤

n−2

X

j=1

f(x_j)·g_α[x_j, x_j+1]−f(x_j)·g_α[x_j+1, x_j+2]

+f(x_j)·g_α[x_j+1, x_j+2]−f(x_j+1)·g_α[x_j+1, x_j+2]

+

n−2

X

j=1

g(x_j+1)·f_α[x_j, x_j+1]−g(x_j+1)·f_α[x_j+1, x_j+2] +g(xj+1)·fα[xj+1, xj+2]−g(xj+2)·fα[xj+1, xj+2]

. Sincef andgare bounded, we have|f(x_j)| ≤ kfk_∞ and |g(x_j+1)| ≤ kgk_∞.

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Hence

σ_(2,α)(f ·g,Π)≤ kfk∞ n−2

X

j=1

g_α[x_j, x_j+1]−g_α[x_j+1, x_j+2]

+

n−2

X

j=1

f(x_j)−f(x_j+1) ·

g_α[x_j+1, x_j+2]

+kgk∞ n−2

X

j=1

fα[xj, xj+1]−fα[xj+1, xj+2]

+

n−2

X

j=1

g(x_j+1)−g(x_j+2) ·

f_α[x_j+1, x_j+2]

=kfk∞·σ_(2,α)(g,Π) +kgk∞·σ_(2,α)(f,Π) +

n−2

X

j=1

|f(x_j)−f(x_j+1)|

|α(x_j)−α(x_j+1)| · |g(x_j+1)−g(x_j+2)|

|α(x_j+1)−α(x_j+2)||α(x_j)−α(x_j+1)|

+

n−2

X

j=1

|g(x_j+1)−g(x_j+2)|

|α(x_j+1)−α(x_j+2)| · |f(x_j+1)−f(x_j+2)|

|α(x_j+1)−α(x_j+2)||α(x_j+1)−α(x_j+2)|.

By Definition2.1and Lemma2.2we obtain

|f(x_j)−f(x_j+1)|

|α(x_j)−α(x_j+1)| ≤Lip_α(f) j = 1,2,· · · , n−1 and |g(x_j)−g(x_j+1)|

|α(x_j)−α(x_j+1)| ≤Lip_α(g) j = 1,2,· · · , n−1.

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Thus

σ_(2,α)(f ·g,Π)≤ kfk∞·σ_(2,α)(g,Π) +kgk∞·σ_(2,α)(f,Π) + (Lip_α(f))(Lip_α(g))

n−2

X

j=1

(α(x_j+1)−α(x_j) +α(x_j+2)−α(x_j+1)).

By Lemma2.3we haveLip_α(f)<+∞andLip_α(g)<+∞. Moreover

n−2

X

j=1

(α(x_j+2)−α(x_j)) =α(b) +

n−2

X

j=2

(α(x_j+1)−α(x_j))−α(a)

≤2(α(b)−α(a)).

Then

σ_(2,α)(f ·g,Π)≤ kfk∞·σ_(2,α)(g,Π) +kgk∞·σ_(2,α)(f,Π)

+ 2(α(b)−α(a))(Lip_α(f))(Lip_α(g)) for all partitionsΠof[a, b].

Hence

V(2,α)(f ·g)≤ kfk∞V(2,α)(g) +kgk∞V(2,α)(f)

+ 2(α(b)−α(a))(Lip_α(f))(Lip_α(g))<+∞.

Thereforef ·g ∈BV_(2,α)([a, b]).

This completes the proof of Theorem4.1.

Corollary 4.2. Iff, g ∈BV_(2,α)([a, b]),then

kf·gk_BV_(2,α)_([a,b]) ≤ kfk∞kgk_BV_(2,α)_([a,b])+kgk∞kfk_BV_(2,α)_([a,b]) + 2(α(b)−α(a))kfk_BV_(2,α)_([a,b])kgk_BV_(2,α)_([a,b]).

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Proof. Note that

Lip_α(f)≤ kfkα-Lip([a,b]) ≤ kfk_BV_(2,α)_([a,b]) and

Lip_α(g)≤ kgkα-Lip([a,b]) ≤ kgk_BV_(2,α)_([a,b]) by Lemma2.3.

From Theorem4.1we have

V_(2,α)(f ·g)≤ kfk∞V_(2,α)(g) +kgk∞V_(2,α)(f)

+ 2(α(b)−α(a))kfk_BV_(2,α)_([a,b])kgk_BV_(2,α)_([a,b]). On the other hand,

|(f g)(a)| ≤2|f(a)| · |g(a)| ≤ kfk∞|g(a)|+kgk∞|f(a)|

|(f g)⁰_α(a)| ≤ |f(a)| · |g_α⁰(a)|+|g(a)| · |f_α⁰(a)|

≤ kfk∞|g_α⁰(a)|+kgk∞|f_α⁰(a)|.

Adding we obtain

|(f g)(a)|+|(f g)⁰_α(a)|+V_(2,α)(f·g)

≤ kfk∞(|g(a)|+|g⁰_α(a)|+V_(2,α)(g)) +kgk∞(|f(a)|+|f_α⁰(a)|+V_(2,α)(f)) + 2(α(b)−α(a))kfk_BV_(2,α)_([a,b])kgk_BV_(2,α)_([a,b]). Therefore

kf·gk_BV_(2,α)_([a,b]) ≤ kfk∞kgk_BV_(2,α)_([a,b])+kgk∞kfk_BV_(2,α)_([a,b])

+ 2(α(b)−α(a))kfk_BV_(2,α)_([a,b])kgk_BV_(2,α)_([a,b]). This completes the proof of Corollary4.2.

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5. Main Result

Theorem 5.1. BV_(2,α)([a, b])equipped with the norm kfk¹_BV

(2,α)([a,b]) =kfk∞+ 2(α(b)−α(a))kfk_BV_(2,α)_([a,b]), f ∈BV_(2,α)([a, b]) is a Banach algebra and the normsk·k_BV_(2,α)_([a,b])andk·k¹_BV

(2,α)([a,b])are equivalent.

Proof. First of all, we need to check the hypotheses from Theorem3.1. Sinceα- Lip[a, b] ,→ B[a, b],by Lemma 2.3 we have BV_(2,α)([a, b]) ⊂ B[a, b]. Next, from Theorem4.1we see that BV_(2,α)([a, b])is closed under pointwise multiplication of functions. Now observe that if we takeK = 2(α(b)−α(a)),the inequality given in Corollary4.2coincides with the one given in Theorem3.1. Also note that

BV_(2,α)([a, b]),→α-Lip[a, b],→B[a, b]

and

kfk∞≤max{1,(α(b)−α(a))}kfk_BV_(2,α)_([a,b]), f ∈BV_(2,α)([a, b]).

Therefore, invoking Theorem3.1we have that(R, BV_(2,α)([a, b]),+,·,k·k¹_BV

(2,α)([a,b])) is a Banach algebra and the normsk · k_BV_(2,α)_([a,b])andk · k¹_BV

(2,α)([a,b])are equivalent.

This completes the proof of Theorem5.1.

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References

[1] R. CASTILLO AND E. TROUSSELOT, A generalization of the Maligranda- Orlicz’s lemma, J. Ineq. Pure and Appli. Math., 8(4) (2007), Art. 115 [ONLINE:

http://jipam.vu.edu.au/article.php?sid=921].

[2] C. JORDAN, Sur la série de Fourier, C.R. Acad. Sci. Paris, 2 (1881), 228–230.

[3] A.M. RUSSEL, Functions of bounded second variation and Stieltjes type inte- grals, J. London Math. Soc. (2), 2 (1970), 193–208.

[4] Ch J. de la VALLÉE POUSSIN, Sur la convergence des formules d’ interpolation entre ordonées équidistantes, Bull. Acad. Sci. Belge, (1908), 314–410.