Journal of Inequalities in Pure and
Applied Mathematics
HARDY’S INTEGRAL INEQUALITY FOR COMMUTATORS OF
HARDY OPERATORS
volume 7, issue 5, article 183,
2006.
QING-YU ZHENG AND ZUN-WEI FU
Department of Mathematics
Linyi Normal University
Linyi Shandong, 276005
People’s Republic of China
EMail: zqy7336@163.com
EMail: lyfzw@tom.com
Received 26 March, 2006;
accepted 09 October, 2006.
Communicated by: B. Yang
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
083-06
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Abstract
The authors establish the Hardy integral inequality for commutators generated
by Hardy operators and Lipschitz functions.
2000 Mathematics Subject Classification: 26D15, 42B25.
Key words: Hardy’s integral inequality, Commutator, Hardy operator, Lipschitz function.
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
This paper was supported by the National Natural Science Foundation
(No.10371080) of People’s Republic of China.
Qing-Yu Zheng and Zun-Wei Fu
Contents
Title Page
1
Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
7
Contents
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J. Ineq. Pure and Appl. Math. 7(5) Art. 183, 2006
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1.
Introduction and Main Results
Let f be a non-negative and integral function on R+ , Hardy operators are defined by
Z
1 x
(Hf )(x) =
f (t) dt, x > 0,
x 0
and
Z ∞
(Vf )(x) =
f (t) dt, x > 0.
x
The Hardy integral inequality results are well known [9, 10, 11]; in particular
Z
∞
p
p1
(Hf (x)) dx
(1.1)
0
p
Qing-Yu Zheng and Zun-Wei Fu
p1
∞
Z
p
≤
p−1
(f (x)) dx
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
,
0
Title Page
and
Contents
Z
(1.2)
0
∞
(Vf (x))p dx
p1
Z
≤p
∞
(xf (x))p dx
p1
.
0
p
In (1.1), the constant p−1
is the best possible. In (1.2), the constant p is also
the best possible. The inequality (1.1) was first proved by Hardy in an attempt
to give a simple proof of Hilbert’s double series theorem [12].
Let f be a non-negative and integral function on R+ , and the fractional Hardy
operator be defined by
Z x
1
α
(H f )(x) = 1−α
f (t) dt, x > 0,
x
0
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for p1 − 1q = α, 0 < α < 1. There are fractional order Hardy integral inequalities
which correspond to (1.1) and (1.2):
Z
∞
q
α
(H f (x)) dx
(1.3)
1q
p1
∞
Z
p
≤C
(f (x)) dx
0
,
0
and
Z
∞
q
(Vf (x)) dx
(1.4)
1q
Z
≤C
0
∞
(x
1−α
p
f (x)) dx
p1
.
0
The Hardy inequality (1.3) can be found in [2], (1.4) in [13].
The Hardy integral inequalities have received considerable attention and a
large number of papers have appeared which deal with its alternative proofs,
various generalizations, numerous variants and applications. For earlier developments of this kind of inequality and many important applications in analysis,
see [11]. Among numerous papers dealing with such inequalities, we choose
to refer to the papers [3], [5], [9], [10], [16] – [21] and some of the references
cited therein.
Definition 1.1. Let 0 ≤ α < 1 and b(x) be a measurable, locally integrable
function. Then the commutator of the Hardy operator Ubα is defined by
Z x
1
α
Hb f (x) = 1−α
f (t)(b(x) − b(t))dt, x > 0.
x
0
Fu [7] obtained the following results.
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
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Theorem 1.1. For b ∈ ∧˙ β (R+ ), 0 < β < 1, Hbα is a bounded operator from
Lp (R+ ) to Lq (R+ ), 1 < p < q < ∞, 0 ≤ α < 1, 0 < α+β < 1, p1 − 1q = α+β.
Remark 1. In Theorem 1.1, If we let α = 0, Then the result corresponds to
Hardy inequality (1.3).
Definition 1.2. Let b(x) be a measurable, locally integrable function. Then the
commutator of the Hardy operator Vb is defined by
Z ∞
Vb f (x) =
f (t)(b(x) − b(t))dt, x > 0.
x
In Definition 1.1, if we let α = 0, then we denote Hbα by Hb .
It can be seen that if b ∈ ∧˙ β (Rn ), 0 < β < 1, then Hb has a similar boundedness property to Hβ . A natural question regarding the boundedness property
of Vb , can be answered in the affirmative by the following inequality (1.5).
Theorem 1.2. If b ∈ ∧˙ β (R+ ),
(1.5)
1
p
−
1
q
= β, 0 < β < 1, p > 1. Then
kVb f kq ≤ Ckbk∧˙ β (R+ ) k(·)f (·)kp .
Theorem 1.3. If b ∈ ∧˙ β (R+ ), 0 < β < 1,
0 ≤ α < 1, p > 1. Then
1
p
−
1
q
= α + β, 0 < α + β < 1,
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
Title Page
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(1.6)
kVb f kq ≤ Ckbk∧˙ β (R+ ) k(·)1−α f (·)kp .
If we let α = 0 in Theorem 1.3, then Theorem 1.2 can be obtained without
difficulty. Thus we just need to prove Theorem 1.3. Before we prove the main
theorem, let us state some lemmas and notations.
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The Besov-Lipschitz space ∧˙ β (R+ ) is the space of functions f satisfying
kf k∧˙ β (R+ ) =
|f (x + h) − f (x)|
< ∞.
|h|β
x,h∈R+ ,h6=0
sup
Lemma 1.4 ([8, 15]). For any x, y ∈ R+ , if f ∈ ∧˙ β (R+ ), 0 < β < 1, then
|f (x) − f (y)| ≤ |x − y|β kf k∧˙ β (R+ ) ,
(1.7)
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
and given any interval I in R+
sup |f (x) − fI | ≤ C|I|β kf k∧˙ β (R+ ) ,
x∈I
where
1
fI =
|I|
Z
Qing-Yu Zheng and Zun-Wei Fu
Title Page
f.
Contents
I
Lemma 1.5 ([7, 14]). Let s > 0, q ≥ p > 1 , then
! p1 q
pq
Z ∞
Z 2k+1
∞ X
X
∞ (i−k)/s
2
|f (t)|p dt ≤ C
|f (t)|p dt ,
k
2
0
i=−∞ k=i
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where
C=
2p/2s
2p/2s − 1
0
2q /2s
2q0 /2s − 1
qq0
,
1
1
+
= 1.
q0 q
There are two different methods to prove Theorem 1.3.
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2.
Proof of Theorem 1.3
First Proof.
Z ∞
|Vb f (x)|q dx
0
∞ Z 2i+1 Z
X
=
≤
i
i=−∞ 2
∞ Z 2i+1
X
∞ Z
X
≤2
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
q
|(b(x) − b(t))f (t)| dt dx
2i+1
i=−∞
|(b(x) − b(t))f (t)| dt
∞ Z
X
2i
∞
X
k=i
Z
2i+1
2i
!q
2k+1
Qing-Yu Zheng and Zun-Wei Fu
dx
2k
k=i
i=−∞
0
∞
∞ Z
X
2i+1
∞ Z
X
+ 2q/q
q
(b(x) − b(t))f (t)dt dx
2i
2i
i=−∞
q/q 0
x
Z
2i
i=−∞
=
∞
2k+1
!q
(b(x) − b(2i ,2i+1 ] )f (t) dt dx
2k
∞ Z
X
k=i
2k+1
!q
(b(t) − b(2i ,2i+1 ] )f (t) dt dx
Title Page
Contents
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2k
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:= I + J.
Close
By Lemma 1.4 and the Hölder inequality,
q/q 0
I=2
∞ Z
X
i=−∞
2i+1
2i
Quit
∞ Z
X
b(x) − b(2i ,2i+1 ] q dx
k=i
2k+1
2k
!q
Page 7 of 16
|f (t)| dt
J. Ineq. Pure and Appl. Math. 7(5) Art. 183, 2006
http://jipam.vu.edu.au
∞
X
q/q 0
≤2
!q
2i
sup
x∈(2i ,2i+1 ]
i=−∞
×

Z
∞
X

q/q 0
≤ C2
∞
X
b(x) − b(2i ,2i+1 ] ! p1
2k+1
|f (t)|p dt
2k
k=i
2i(qβ+1) kbk∧˙ β (R+ )
Z
2k+1
2k
! p1 0 q

dt

q
i=−∞
×

Z
∞
X

q/q 0
≤ C2
∞
X
! p1
2k+1
|f (t)|p dt
2i(qβ+1) kbk∧˙ β (R+ )
2k+1
2k
2k
k=i
Z
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
! 10 q
p 
dt

Qing-Yu Zheng and Zun-Wei Fu
q
Title Page
i=−∞

×
∞
X
Z
k/p0
2
2k
k=i
Notice that
q/q 0
I ≤ C2
1
p
−
1
q
2k+1
! p1 q
2−k(1−α)p |t1−α f (t)|p dt  .
= α + β,
kbk∧˙ β (R+ )
∞
q X
i=−∞
Contents
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
∞
X

k=i
(i−k)( 1q +β)
Z
2k+1
2
2k
! p1 q
|t1−α f (t)|p dt
 .
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By Lemma 1.5 s =
q
1+qβ
,
I ≤ C kbk∧˙ β (R+ )
(2.1)
q
∞
Z
1−α
|t
pq
f (t)| dt .
p
0
Now estimate J,
∞ Z
X
q/q 0
J =2
2i
i=−∞
2q/q 0
≤2
∞ Z
X
2i+1
∞ Z
X
2q/q 0
+2
∞ Z
X
2i+1
k=i
∞ Z
X
i=−∞
!q
(b(t) − b(2i ,2i+1 ] )f (t) dt dx
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
2k
k=i
2i
i=−∞
2k+1
2i+1
2i
2k+1
!q
(b(t) − b(2k ,2k+1 ] )f (t) dt dx
Qing-Yu Zheng and Zun-Wei Fu
2k
∞ Z
X
2k+1
!q
(b(2k ,2k+1 ] − b(2i ,2i+1 ] )f (t) dt dx
Title Page
2k
k=i
Contents
:= J1 + J2 .
Notice that
2q/q 0
J1 ≤ 2
1
p
−
1
q
∞ Z
X
i=−∞
2q/q 0
≤ C2
= α + β,
2i+1
2i
kbk∧˙ β (R+ )
1
p
+
1
p0
∞
X
k=i
= 1, by Lemma 1.4, it can be inferred that
sup
b(t) − b(2k ,2k+1 ] t∈(2k ,2k+1 ]
∞ Z
q X
i=−∞
JJ
J
2i+1
2i
!Z
|f (t)|dt
2k
∞
X
k=i
kβ
Z
2k+1
dx
Close
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!q
|f (t)|dt
2
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!q
2k+1
II
I
dx
Page 9 of 16
2k
J. Ineq. Pure and Appl. Math. 7(5) Art. 183, 2006
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0
≤ C22q/q kbk∧˙ β (R+ )

∞
q X
2i 
∞
X
i=−∞
i=−∞
= C2

2k
k=i
kbk∧˙ β (R+ )

∞
∞
X
X
k(β+ p10 )
i
×
2
2
2q/q 0
|f (t)|p dt
2
q
2q/q 0
≤ C2
k(β+ p10 )
! p1 q
2k+1
Z
kbk∧˙ β (R+ )

∞
X

i=−∞
2−k(1−α)p |t1−α f (t)|p dt
2k
k=i
∞
q X
! p1 q
2k+1
Z
Z
(i−k)/q
! p1 q
2k+1
1−α
|t
2
p
f (t)| dt
2k
k=i

 .
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
In the third inequality, the Hölder inequality is applied.
By Lemma 1.5 (s = q),
(2.2)
J1 ≤ C kbk∧˙ β (R+ )
q
Title Page
pq
∞
1−α
p
|t f (t)| dt .
Z
0
To estimate J2 , for i > k, by Lemma 1.4, the following result is obtained.
b(2k ,2k+1 ] − b(2i ,2i+1 ] ≤ 1
2i
≤
Z
b(y) − b(2k ,2k+1 ] dy
2i
1 1
2i 2k
JJ
J
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2i+1
Z
Contents
2k+1
Z
2i+1
|b(y) − b(z)| dydz
2k
≤ kbk∧˙ β (R+ )
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Page 10 of 16
2i
1 1
2i 2k
Close
Z
2k+1
2k
Z
2i+1
2i
|y − z|β dydz
J. Ineq. Pure and Appl. Math. 7(5) Art. 183, 2006
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1
2i
≤ kbk∧˙ β (R+ )
Z
2i+1
1
y β dy + k
2
2i
Z
!
2k+1
z β dz
2k
≤ C2iβ+1 kbk∧˙ β (R+ ) .
Notice that p1 − 1q = α + β, p1 + p10 = 1; by Lemma 1.4 and the Hölder inequality,
we have
!q
∞ Z 2i+1
∞ Z 2k+1
X
X
0
q
J2 ≤ C22q/q +q kbk∧˙ β (R+ )
2iβ
|f (t)|dt dx
2i
i=−∞
2q/q 0 +q
≤ C2
kbk∧˙ β (R+ )
∞
q X
k=i

i(β+ 1q )
2
∞
X
i=−∞
2q/q 0 +q
= C2
kbk∧˙ β (R+ )
Z

∞
X

i=−∞
! 1 q
2k+1
2
p
|f (t)|p dt
! p1 q
2k+1
2−k(1−α)p |t1−α f (t)|p dt
(i−k)(β+ 1q )
Z
! 1 q
2k+1
p
1−α
|t
2
2k
In the second inequality,
the Hölder
inequality is applied.
q
By Lemma 1.5 s = qβ+1 ,
(2.3)
J2 ≤ C kbk∧˙ β (R+ )
Z
0
∞
Contents

2k
k=i
q
Qing-Yu Zheng and Zun-Wei Fu

2k
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Title Page
Z
k=i
∞
q X
k/p0
k=i
q
0
≤ C22q/q +q kbk∧˙ β (R+ )

∞
∞
X
1 X
0
)
i(β+
2
q
2k/p
×
i=−∞
2k
pq
1−α
p
|t f (t)| dt .
p
f (t)| dt
 .
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Combining (2.1), (2.2) and (2.3), we complete the proof of Theorem 1.3.
Second Proof. By inequality (1.7) and comparing the size of x and t, we have
∞
Z
|Vb f (x)|q dx =
0
∞ Z
X
2i
i=−∞
≤
2i+1
∞ Z
X
x
q
f (t)|b(x) − b(x)|dt dx
∞ Z
X
2i+1
2i
i=−∞
∞
Z
k=i
≤ C kbk∧˙ β (R+ )
|t − x|β kbk∧˙ β (R+ ) |f (t)| dt
2k
∞ Z
q X
q
∞
X
Z
= C kbk∧˙ β (R+ )
i=−∞
By the Hölder inequality,
Z
2k+1
2k
t1−α |f (t)|
dt ≤
t1−α−β
1
p
1
p0
+
Z
2
tβ |f (t)| dt
dx
2k
2k
k=i
k=i
!q
2k+1
2k+1
2k
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
∞ Z 2k+1
X
∞ Z
X
dx
1−α
t
|f (t)|
t1−α−β
!q
dt
dx
!q
t1−α |f (t)|
dt
t1−α−β
Title Page
Contents
.
JJ
J
= 1, the following estimate is obtained.
2k+1
2k
i
∞ Z
X
k=i
2i+1
2i
i=−∞
∞
q X
2i+1
2i
i=−∞
= C kbk∧˙ β (R+ )
!q
2k+1
II
I
Go Back
! p1
|t1−α f (t)|p dt
Z
2k+1
2k
! 10
p
0
t(α+β−1)p dt
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.
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Notice that
Z
1
p
−
1
q
= α + β,
∞
|Vb f (x)|q dx
0
≤ C kbk∧˙ β (R+ )
q
∞
X
i=−∞

∞
X

2(i−k)/q
k=i
By Lemma 1.5, the desired result is obtained.
Z
2k+1
2k
! p1 q

1−α
p
|t f (t)| dt
.

Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
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References
[1] C. BENNET, R.A. DEVORE AND R. SHARPLEY, Weak L∞ and BMO,
Ann. of Math., 113 (1981), 601–611.
[2] G.A. BLISS, An integral inequality, J. London Math. Soc., 5(1930), 4046.
[3] T.A.A. BROADBENT, A proof of Hardy’s convergence theorem, J. London Math. Soc., 3 (1928), 242–243
1
[4] A.P. CALDERÓN, Space between L and L
cikiewiez, Studia Math., 26 (1966), 273–299.
∞
and the theorem of Mar-
[5] E. COPSON, Some integral inequality, Proc. Roy. Soc. Edinburgh Sect. A.
75 (1975-76), 157–164.
[6] R.A. DEVORE AND R.C. SHARPLY, Maximal functions measuring
smoothness, Mem. Amer. Math. Soc., 47 (1984).
[7] Z.W. FU, Commutators of Hardy–Littlewood average operators, J. Beijing
Normal University (Natural Science), 42 (2006), 342–345.
[8] Z.G. LIU AND Z.W. FU, Weighted Hardy–Littlewood average operators
on Herz spaces, Acta Math. Sinica (Chinese Series), 49 (2006), 1085–
1090.
[9] G.H. HARDY, Note on a theorem of Hilbert, Math. Z., 6 (1920), 314–317.
[10] G.H. HARDY, Note on some points in the integral calculus, Messenger
Math., 57 (1928), 12–16.
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
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[11] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, London/New York: Cambridge Univ. Press, 1934.
[12] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, 2nd ed.,
London/New York: Cambridge Univ. Press, 1952.
[13] J.C. KUANG, Applied Inequalities, 3rd ed., Shandong: Shandong Sci.
Tech. Press, 2004.
[14] S.C. LONG AND J. WANG, Commutators of Hardy operators, J. of Math.
Anal. Appl., 274 (2002), 626–644.
Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
[15] S.Z. LU, Q. WU AND D.C. YANG, Boundedness of commutators on
Hardy type spaces, Science in China, 8 (2002), 984–997.
Qing-Yu Zheng and Zun-Wei Fu
[16] B.G. PACHPATTE, A note on a certain inequality related to Hardy’s inequality, Indiana J. Pure Appl., 23 (1992), 773–776.
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[17] B.G. PACHPATTE, On some integral similar to Hardy’s integral inequality, J. Math. Anal. Appl., 129 (1988), 596–606.
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[19] B.G. PACHPATTE, On some new generalizations of Hardy’s integral inequality, J. Math. Anal. Appl., 1 (1999), 15–30.
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[20] B.C. YANG, Z.H. ZENG AND L. DEBNATH, Note on new generalizations
of Hardy’s integral inequality. J. Math. Anal. Appl., 2 (1998), 321–327.
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[21] B.C. YANG, Z.H. ZENG AND L. DEBNATH, Generalizations of Hardy
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Hardy’s Integral Inequality For
Commutators Of Hardy
Operators
Qing-Yu Zheng and Zun-Wei Fu
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