Volume 9 (2008), Issue 3, Article 84, 6 pp.
AN APPLICATION OF THE GENERALIZED MALIGRANDA-ORLICZ’S LEMMA
RENÉ ERLIN CASTILLO AND EDUARD TROUSSELOT
D EPARTAMENTO DE M ATEMÁTICAS
U NIVERSIDAD DE O RIENTE
6101 C UMANÁ , E DO . S UCRE , V ENEZUELA
rcastill@math.ohiou.edu
eddycharles2007@hotmail.com
Received 17 May, 2008; accepted 22 September, 2008
Communicated by S.S. Dragomir
A BSTRACT. Using the generalized Maligranda-Orlicz’s Lemma we will show that BV(2,α) ([a, b])
is a Banach algebra.
Key words and phrases: Banach algebra, Maligranda-Orlicz.
2000 Mathematics Subject Classification. 46F10.
1. I NTRODUCTION
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 function f on an interval [a, b] by
n−1 X
f (tj+1 ) − f (tj ) f (tj ) − f (tj−1 ) 2
2
V (f ) = V (f, [a, b]) = sup
−
tj+1 − tj
t
−
t
Π
j
j−1
j=1
where the supremum is taken over all partitions Π : a = t0 < t1 < · · · < tn = b of [a, b]. If
V 2 (f, [a, b]) < ∞, the function f is said to be of bounded second variation on [a, b]. The class
of all functions which are of bounded second variation is denoted by BV 2 ([a, b]).
In 1970 the above class of functions was generalized with respect to a strictly increasing
continuous function α (see [3]):
Let f be a real function defined on [a, b]. For a given partition of the form: Π : a = t1 <
· · · < tn = b, we set
σ(2,α) (f, Π) =
n−2
X
j=1
152-08
|fα [tj , tj+1 ] − fα [tj+1 , tj+2 ]|,
2
R ENÉ E RLIN C ASTILLO AND E DUARD T ROUSSELOT
where
fα [p, q] =
f (q) − f (p)
,
α(q) − α(p)
and
V(2,α) (f, [a, b]) = V(2,α) (f ) = sup σ(2,α) (f, Π),
Π
where the supremum is taken over all partitions Π of [a, b].
If V(2,α) (f ) < ∞, then f is said to be of (2, α)-bounded variation.
The set of all these functions will be denoted by BV(2,α) ([a, b]).
A function f is α-derivable at t0 if
lim
t→t0
f (t) − f (t0 )
exists.
α(t) − α(t0 )
If this limit exists, we denote its value by fα0 (t0 ), which we call the α-derivative of f at t0 .
The class BV(2,α) ([a, b]) is a Banach space equipped with the norm
kf kBV(2,α)([a,b]) = |f (a)| + |fα0 (a)| + V(2,α) (f ).
Using the generalized Maligranda-Orlicz’s Lemma (see Theorem 3.1 of the present paper)
we will show that BV(2,α) ([a, b]) is a Banach algebra.
2. D EFINITION AND N OTATION
We begin this section by giving a definition and several simple lemmas that will be used
throughout the paper.
Definition 2.1. A function f : [a, b] → R is said to be α-Lipschitz if there exists a constant
M > 0 such that
|f (x) − f (y)| ≤ M |α(x) − α(y)|,
for all x, y ∈ [a, b], x 6= y. By α-Lip[a, b] we will denote the space of functions which are
α-Lipschitz. If f ∈ α-Lip[a, b] we define
Lipα (f ) = inf{M > 0 : |f (x) − f (y)| ≤ M |α(x) − α(y)|, x 6= y ∈ [a, b]}
and
Lip0α (f )
= sup
|f (x) − f (y)|
: x 6= y ∈ [a, b] .
|α(x) − α(y)|
It is not hard to prove that
Lipα (f ) = Lip0α (f ).
α-Lip[a, b] equipped with the norm
kf kα-Lip[a,b] = |f (a)| + Lipα (f )
is a Banach space.
Lemma 2.1. If f ∈ BV(2,α) ([a, b]), then there exists a constant M > 0 such that
f (x2 ) − f (x1 ) α(x2 ) − α(x1 ) = |fα [x1 , x2 ]| ≤ M
for all x1 , x2 ∈ [a, b].
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 84, 6 pp.
http://jipam.vu.edu.au/
G ENERALIZED M ALIGRANDA -O RLICZ ’ S L EMMA
3
Lemma 2.2.
kf kα-Lip[a,b] ≤ kf kBV(2,α) ([a,b]) , f ∈ BV(2,α) ([a, b])
and
BV(2,α) ,→ α-Lip[a, b].
Lemma 2.3. α-Lip[a, b] ,→ BV [a, b] ,→ B[a, b].
3. G ENERALIZED M ALIGRANDA -O RLICZ ’ S L EMMA
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 functions, which
is closed under pointwise multiplication of functions. Let us assume that f · g ∈ X such that
kf gk ≤ kf k∞ kgk + kf kkgk∞ + Kkf kkgk, K > 0.
Then (X, k · k1 ) equipped with the norm
kf k1 = kf k∞ + Kkf k, f ∈ X,
is a Banach algebra. If X ,→ B[a, b], then k · k1 and k · k are equivalent.
4. BV(2,α) ([a, b]) AS A BANACH A LGEBRA
The following result shows us that BV(2,α) ([a, b]) is closed under pointwise multiplication of
functions.
Theorem 4.1. If f, g ∈ BV(2,α) ([a, b]), then f · g ∈ BV(2,α) ([a, b]).
Proof. Let Π : a = x1 < x2 < · < xn = b be a partition of [a, b]. Then
σ(2,α) (f · g, Π) =
n−2
X
(f g)α [xj , xj+1 ] − (f g)α [xj+1 , xj+2 ]
j=1
=
n−2
X
f (xj ) · gα [xj , xj+1 ] + g(xj+1 ) · fα [xj , xj+1 ]
j=1
− f (xj+1 ) · gα [xj+1 , xj+2 ] − g(xj+2 ) · fα [xj+1 , xj+2 ]
≤
n−2
X
f (xj ) · gα [xj , xj+1 ] − f (xj ) · gα [xj+1 , xj+2 ]
j=1
+ f (xj ) · gα [xj+1 , xj+2 ] − f (xj+1 ) · gα [xj+1 , xj+2 ]
n−2
X
g(xj+1 ) · fα [xj , xj+1 ] − g(xj+1 ) · fα [xj+1 , xj+2 ]
+
j=1
+ g(xj+1 ) · fα [xj+1 , xj+2 ] − g(xj+2 ) · fα [xj+1 , xj+2 ].
Since f and g are bounded, we have |f (xj )| ≤ kf k∞ and |g(xj+1 )| ≤ kgk∞ .
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 84, 6 pp.
http://jipam.vu.edu.au/
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R ENÉ E RLIN C ASTILLO AND E DUARD T ROUSSELOT
Hence
n−2
X
gα [xj , xj+1 ] − gα [xj+1 , xj+2 ]
σ(2,α) (f · g, Π) ≤ kf k∞
j=1
+
n−2
X
f (xj ) − f (xj+1 ) · gα [xj+1 , xj+2 ]
j=1
+ kgk∞
n−2
X
fα [xj , xj+1 ] − fα [xj+1 , xj+2 ]
j=1
+
n−2
X
g(xj+1 ) − g(xj+2 ) · fα [xj+1 , xj+2 ]
j=1
= kf k∞ · σ(2,α) (g, Π) + kgk∞ · σ(2,α) (f, Π)
n−2
X
|f (xj ) − f (xj+1 )| |g(xj+1 ) − g(xj+2 )|
+
·
|α(xj ) − α(xj+1 )|
|α(xj ) − α(xj+1 )| |α(xj+1 ) − α(xj+2 )|
j=1
n−2
X
|g(xj+1 ) − g(xj+2 )| |f (xj+1 ) − f (xj+2 )|
+
·
|α(xj+1 ) − α(xj+2 )|.
|α(xj+1 ) − α(xj+2 )| |α(xj+1 ) − α(xj+2 )|
j=1
By Definition 2.1 and Lemma 2.1 we obtain
|f (xj ) − f (xj+1 )|
≤ Lipα (f ) j = 1, 2, · · · , n − 1
|α(xj ) − α(xj+1 )|
and
|g(xj ) − g(xj+1 )|
≤ Lipα (g) j = 1, 2, · · · , n − 1.
|α(xj ) − α(xj+1 )|
Thus
σ(2,α) (f · g, Π) ≤ kf k∞ · σ(2,α) (g, Π) + kgk∞ · σ(2,α) (f, Π)
+ (Lipα (f ))(Lipα (g))
n−2
X
(α(xj+1 ) − α(xj ) + α(xj+2 ) − α(xj+1 )).
j=1
By Lemma 2.2 we have Lipα (f ) < +∞ and Lipα (g) < +∞. Moreover
n−2
X
(α(xj+2 ) − α(xj )) = α(b) +
j=1
n−2
X
(α(xj+1 ) − α(xj )) − α(a)
j=2
≤ 2(α(b) − α(a)).
Then
σ(2,α) (f · g, Π) ≤ kf k∞ · σ(2,α) (g, Π) + kgk∞ · σ(2,α) (f, Π)
+ 2(α(b) − α(a))(Lipα (f ))(Lipα (g))
for all partitions Π of [a, b].
Hence
V(2,α) (f · g) ≤ kf k∞ V(2,α) (g) + kgk∞ V(2,α) (f )
+ 2(α(b) − α(a))(Lipα (f ))(Lipα (g)) < +∞.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 84, 6 pp.
http://jipam.vu.edu.au/
G ENERALIZED M ALIGRANDA -O RLICZ ’ S L EMMA
5
Therefore f · g ∈ BV(2,α) ([a, b]).
This completes the proof of Theorem 4.1.
Corollary 4.2. If f, g ∈ BV(2,α) ([a, b]), then
kf · gkBV(2,α) ([a,b]) ≤ kf k∞ kgkBV(2,α) ([a,b]) + kgk∞ kf kBV(2,α) ([a,b])
+ 2(α(b) − α(a))kf kBV(2,α) ([a,b]) kgkBV(2,α) ([a,b]) .
Proof. Note that
Lipα (f ) ≤ kf kα-Lip([a,b]) ≤ kf kBV(2,α) ([a,b])
and
Lipα (g) ≤ kgkα-Lip([a,b]) ≤ kgkBV(2,α) ([a,b])
by Lemma 2.2.
From Theorem 4.1 we have
V(2,α) (f · g) ≤ kf k∞ V(2,α) (g) + kgk∞ V(2,α) (f )
+ 2(α(b) − α(a))kf kBV(2,α) ([a,b]) kgkBV(2,α) ([a,b]) .
On the other hand,
|(f g)(a)| ≤ 2|f (a)| · |g(a)| ≤ kf k∞ |g(a)| + kgk∞ |f (a)|
|(f g)0α (a)| ≤ |f (a)| · |gα0 (a)| + |g(a)| · |fα0 (a)|
≤ kf k∞ |gα0 (a)| + kgk∞ |fα0 (a)|.
Adding we obtain
|(f g)(a)| + |(f g)0α (a)| + V(2,α) (f · g)
≤ kf k∞ (|g(a)| + |gα0 (a)| + V(2,α) (g)) + kgk∞ (|f (a)| + |fα0 (a)| + V(2,α) (f ))
+ 2(α(b) − α(a))kf kBV(2,α) ([a,b]) kgkBV(2,α) ([a,b]) .
Therefore
kf · gkBV(2,α) ([a,b]) ≤ kf k∞ kgkBV(2,α) ([a,b]) + kgk∞ kf kBV(2,α) ([a,b])
+ 2(α(b) − α(a))kf kBV(2,α) ([a,b]) kgkBV(2,α) ([a,b]) .
This completes the proof of Corollary 4.2.
5. M AIN R ESULT
Theorem 5.1. BV(2,α) ([a, b]) equipped with the norm
kf k1BV(2,α) ([a,b]) = kf k∞ + 2(α(b) − α(a))kf kBV(2,α) ([a,b]) , f ∈ BV(2,α) ([a, b])
is a Banach algebra and the norms k · kBV(2,α) ([a,b]) and k · k1BV(2,α) ([a,b]) are equivalent.
Proof. First of all, we need to check the hypotheses from Theorem 3.1. Since α-Lip[a, b] ,→
B[a, b], by Lemma 2.2 we have BV(2,α) ([a, b]) ⊂ B[a, b]. Next, from Theorem 4.1 we see that
BV(2,α) ([a, b]) is closed under pointwise multiplication of functions. Now observe that if we
take K = 2(α(b) − α(a)), the inequality given in Corollary 4.2 coincides with the one given in
Theorem 3.1. Also note that
BV(2,α) ([a, b]) ,→ α-Lip[a, b] ,→ B[a, b]
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 84, 6 pp.
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R ENÉ E RLIN C ASTILLO AND E DUARD T ROUSSELOT
and
kf k∞ ≤ max{1, (α(b) − α(a))}kf kBV(2,α) ([a,b]) , f ∈ BV(2,α) ([a, b]).
Therefore, invoking Theorem 3.1 we have that (R, BV(2,α) ([a, b]), +, ·, k · k1BV(2,α) ([a,b]) ) is a
Banach algebra and the norms k · kBV(2,α) ([a,b]) and k · k1BV(2,α) ([a,b]) are equivalent.
This completes the proof of Theorem 5.1.
R EFERENCES
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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 integrals, 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.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 84, 6 pp.
http://jipam.vu.edu.au/