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
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
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Received:
17 May, 2008
Accepted:
22 September, 2008
Communicated by:
JJ
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S.S. Dragomir
J
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2000 AMS Sub. Class.:
46F10.
Page 1 of 13
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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Contents
1
Introduction
3
2
Definition and Notation
5
3
Generalized Maligranda-Orlicz’s Lemma
7
4
BV(2,α) ([a, b]) as a Banach Algebra
5
Main Result
8
12
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
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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 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
|fα [tj , tj+1 ] − fα [tj+1 , tj+2 ]|,
j=1
where
fα [p, q] =
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
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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.
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
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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 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
vol. 9, iss. 3, art. 84, 2008
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|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 2.2. 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].
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
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Lemma 2.3.
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 2.4. α-Lip[a, b] ,→ BV [a, b] ,→ B[a, b].
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
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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 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.
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
Then (X, k · k1 ) equipped with the norm
kf k1 = kf k∞ + Kkf k, f ∈ X,
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is a Banach algebra. If X ,→ B[a, b], then k · k1 and k · k are equivalent.
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4.
BV(2,α) ([a, b]) as a Banach Algebra
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
n−2
X
(f g)α [xj , xj+1 ] − (f g)α [xj+1 , xj+2 ]
σ(2,α) (f · g, Π) =
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
j=1
=
n−2
X
f (xj ) · gα [xj , xj+1 ] + g(xj+1 ) · fα [xj , xj+1 ]
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j=1
Contents
− 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 ]
+
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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∞ .
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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 ]
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
j=1
n−2
X
g(xj+1 ) − g(xj+2 ) · fα [xj+1 , xj+2 ]
+
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j=1
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= 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(x
)
−
α(x
)|
|α(x
)
−
α(x
)|
j+1
j+2
j+1
j+2
j=1
By Definition 2.1 and Lemma 2.2 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 )|
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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 2.3 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)
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
j=2
≤ 2(α(b) − α(a)).
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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
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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)) < +∞.
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]) .
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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 2.3.
From Theorem 4.1 we have
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
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]) .
vol. 9, iss. 3, art. 84, 2008
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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
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|(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]) .
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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.
Main Result
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 αLip[a, b] ,→ B[a, b], by Lemma 2.3 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
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
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Contents
BV(2,α) ([a, b]) ,→ α-Lip[a, b] ,→ B[a, b]
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.
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References
[1] R. CASTILLO AND E. TROUSSELOT, A generalization of the MaligrandaOrlicz’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 integrals, J. London Math. Soc. (2), 2 (1970), 193–208.
Maligranda-Orlicz’s Lemma
René Erlin Castillo and
Eduard Trousselot
vol. 9, iss. 3, art. 84, 2008
[4] Ch J. de la VALLÉE POUSSIN, Sur la convergence des formules d’ interpolation
entre ordonées équidistantes, Bull. Acad. Sci. Belge, (1908), 314–410.
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