Key words and phrases: Hardy’s integral inequality, Commutator, Hardy operator, Lipschitz function

http://jipam.vu.edu.au/

Volume 7, Issue 5, Article 183, 2006

HARDY’S INTEGRAL INEQUALITY FOR COMMUTATORS OF HARDY OPERATORS

QING-YU ZHENG AND ZUN-WEI FU DEPARTMENT OFMATHEMATICS

LINYINORMALUNIVERSITY

LINYISHANDONG, 276005 PEOPLE’SREPUBLIC OFCHINA

zqy7336@163.com lyfzw@tom.com

Received 26 March, 2006; accepted 09 October, 2006 Communicated by B. Yang

ABSTRACT. The authors establish the Hardy integral inequality for commutators generated by Hardy operators and Lipschitz functions.

Key words and phrases: Hardy’s integral inequality, Commutator, Hardy operator, Lipschitz function.

2000 Mathematics Subject Classification. 26D15, 42B25.

1. INTRODUCTION ANDMAIN RESULTS

Letf be a non-negative and integral function onR⁺, Hardy operators are defined by (Hf)(x) = 1

x Z x

0

f(t)dt, x >0, and

(Vf)(x) = Z ∞

x

f(t)dt, x >0.

The Hardy integral inequality results are well known [9, 10, 11]; in particular (1.1)

Z ∞ 0

(Hf(x))^pdx ¹_p

≤ p p−1

Z ∞ 0

(f(x))^pdx ¹_p

,

and (1.2)

Z ∞ 0

(Vf(x))^pdx ¹_p

≤p Z ∞

0

(xf(x))^pdx _p¹

.

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

This paper was supported by the National Natural Science Foundation (No.10371080) of People’s Republic of China.

083-06

In (1.1), the constant _p−1^p is the best possible. In (1.2), the constantpis 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

(H^αf)(x) = 1 x^1−α

Z x 0

f(t)dt, x >0,

for ¹_p − ¹_q = α, 0 < α < 1. There are fractional order Hardy integral inequalities which correspond to (1.1) and (1.2):

(1.3)

Z ∞ 0

(H^αf(x))^qdx ¹_q

≤C Z ∞

0

(f(x))^pdx ¹_p

,

and (1.4)

Z ∞ 0

(Vf(x))^qdx ¹_q

≤C Z ∞

0

(x^1−αf(x))^pdx ¹_p

. 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 impor- tant 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. Let0≤ α <1andb(x)be a measurable, locally integrable function. Then the commutator of the Hardy operatorU_b^α is defined by

H^α_bf(x) = 1 x^1−α

Z x 0

f(t)(b(x)−b(t))dt, x >0.

Fu [7] obtained the following results.

Theorem 1.1. Forb∈∧˙_β(R⁺),0< β <1,H_b^αis a bounded operator fromL^p(R⁺)toL^q(R⁺), 1< p < q <∞,0≤α <1,0< α+β <1, ¹_p − ¹_q =α+β.

Remark 1.2. In Theorem 1.1, If we letα = 0, Then the result corresponds to Hardy inequality (1.3).

Definition 1.2. Letb(x)be a measurable, locally integrable function. Then the commutator of the Hardy operatorV_b is defined by

V_bf(x) = Z ∞

x

f(t)(b(x)−b(t))dt, x >0.

In Definition 1.1, if we letα= 0, then we denoteH^α_b byH_b.

It can be seen that ifb ∈ ∧˙_β(Rⁿ), 0 < β < 1, then H_b has a similar boundedness property to H^β. A natural question regarding the boundedness property of V_b, can be answered in the affirmative by the following inequality (1.5).

Theorem 1.3. Ifb∈∧˙_β(R⁺), ¹_p −¹_q =β,0< β <1, p >1. Then (1.5) kVbfkq ≤Ckbk∧˙_β(R⁺)k(·)f(·)kp.

Theorem 1.4. Ifb ∈∧˙_β(R⁺),0< β <1, ¹_p − ¹_q =α+β,0< α+β < 1,0≤α < 1, p > 1.

Then

(1.6) kV_bfk_q ≤Ckbk∧˙β(R⁺)k(·)^1−αf(·)k_p.

If we letα= 0in Theorem 1.4, then Theorem 1.3 can be obtained without difficulty. Thus we just need to prove Theorem 1.4. Before we prove the main theorem, let us state some lemmas and notations.

The Besov-Lipschitz space∧˙_β(R⁺)is the space of functionsf satisfying kfk∧˙_β(R⁺) = sup

x,h∈R⁺,h6=0

|f(x+h)−f(x)|

|h|^β <∞.

Lemma 1.5 ([8, 15]). For anyx, y ∈R⁺, iff ∈∧˙_β(R⁺),0< β <1, then (1.7) |f(x)−f(y)| ≤ |x−y|^βkfk∧˙_β(R⁺),

and given any intervalI inR⁺ sup

x∈I

|f(x)−f_I| ≤C|I|^βkfk∧˙_β(R⁺),

where

f_I = 1

|I|

Z

I

f.

Lemma 1.6 ([7, 14]). Lets >0, q≥p > 1, then

∞

X

i=−∞

∞

X

k=i

2^(i−k)/s

Z 2^k+1 2^k

|f(t)|^pdt

!¹_p

q

≤C Z ∞

0

|f(t)|^pdt ^q_p

,

where

C =

2^p/2s 2^p/2s−1

2^q0/2s 2^q0/2s−1

q q0

, 1

q⁰ + 1 q = 1.

There are two different methods to prove Theorem 1.4.

2. PROOF OFTHEOREM1.4 First Proof.

Z ∞ 0

|Vbf(x)|^qdx=

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

Z ∞ x

(b(x)−b(t))f(t)dt

q

dx

≤

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

Z ∞ 2ⁱ

|(b(x)−b(t))f(t)|dt q

dx

=

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

|(b(x)−b(t))f(t)|dt

!q

dx

≤2^q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

(b(x)−b₍₂ⁱ_,2ⁱ⁺¹_])f(t) dt

!q

dx

+ 2^q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

(b(t)−b₍₂ⁱ_,2ⁱ⁺¹_])f(t) dt

!q

dx :=I+J.

By Lemma 1.5 and the Hölder inequality, I = 2^q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

b(x)−b₍₂ⁱ_,2ⁱ⁺¹_]

qdx

∞

X

k=i

Z 2^k+1 2^k

|f(t)|dt

!q

≤2^q/q0

∞

X

i=−∞

2ⁱ sup

x∈(2ⁱ,2ⁱ⁺¹]

b(x)−b₍₂ⁱ_,2ⁱ⁺¹_]

!q

×







∞

X

k=i

Z 2^k+1 2^k

|f(t)|^pdt

!¹_p

Z 2^k+1 2^k

dt

!¹_p

0





q

≤C2^q/q0

∞

X

i=−∞

2^i(qβ+1) kbk∧˙_β(R⁺)

q







∞

X

k=i

Z 2^k+1 2^k

|f(t)|^pdt

!¹_p

Z 2^k+1 2^k

dt

!_p¹0





q

≤C2^q/q0

∞

X

i=−∞

2^i(qβ+1) kbk∧˙_β(R⁺)

q





∞

X

k=i

2^k/p0

Z 2^k+1 2^k

2^{−k(1−α)p}|t^1−αf(t)|^pdt

!¹_p



q

.

Notice that ¹_p −¹_q =α+β,

I ≤C2^q/q0 kbk∧˙_β(R⁺)

q

∞

X

i=−∞





∞

X

k=i

2^{(i−k)(1q+β)}

Z 2^k+1 2^k

|t^1−αf(t)|^pdt

!¹_p



q

.

By Lemma 1.6

s = _1+qβ^q ,

(2.1) I ≤C kbk∧˙_β(R⁺)

qZ ∞ 0

|t^1−αf(t)|^pdt ^q_p

.

Now estimateJ, J = 2^q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

(b(t)−b₍₂ⁱ_,2ⁱ⁺¹_])f(t) dt

!q

dx

≤2^2q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

(b(t)−b₍₂^k_,2^k+1_])f(t) dt

!q

dx

+ 2^2q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

(b₍₂k,2^k+1]−b₍₂ⁱ_,2ⁱ⁺¹_])f(t) dt

!q

dx

:=J₁+J₂.

Notice that ¹_p −¹_q =α+β, ¹_p +_p¹0 = 1, by Lemma 1.5, it can be inferred that J1 ≤2^2q/q0

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

sup

t∈(2^k,2^k+1]

b(t)−b₍₂^k_,2^k+1_]

!Z 2^k+1 2^k

|f(t)|dt

!q

dx

≤C2^2q/q0 kbk∧˙_β(R⁺)

q

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

2^kβ Z 2^k+1

2^k

|f(t)|dt

!q

dx

≤C2^2q/q0 kbk∧˙_β(R⁺)

q

∞

X

i=−∞

2ⁱ





∞

X

k=i

2^k(β+p10)

Z 2^k+1 2^k

|f(t)|^pdt

!¹_p



q

≤C2^2q/q0 kbk∧˙_β(R⁺)

q

∞

X

i=−∞

2ⁱ





∞

X

k=i

2^k(β+p10)

Z 2^k+1 2^k

2^{−k(1−α)p}|t^1−αf(t)|^pdt

!¹_p



q

=C2^2q/q0 kbk∧˙_β(R⁺)

q

∞

X

i=−∞





∞

X

k=i

2^(i−k)/q

Z 2^k+1 2^k

|t^1−αf(t)|^pdt

!¹_p



q

. In the third inequality, the Hölder inequality is applied.

By Lemma 1.6(s =q),

(2.2) J₁ ≤C kbk∧˙_β(R⁺)

qZ ∞ 0

|t^1−αf(t)|^pdt _p^q

.

To estimateJ₂, fori > k, by Lemma 1.5, the following result is obtained.

b₍₂k,2^k+1]−b₍₂ⁱ_,2ⁱ⁺¹_] ≤ 1

2ⁱ Z 2ⁱ⁺¹

2ⁱ

b(y)−b₍₂k,2^k+1]

dy

≤ 1 2ⁱ

1 2^k

Z 2^k+1 2^k

Z 2ⁱ⁺¹ 2ⁱ

|b(y)−b(z)|dydz

≤ kbk∧˙_β(R⁺)

1 2ⁱ

1 2^k

Z 2^k+1 2^k

Z 2ⁱ⁺¹ 2ⁱ

|y−z|^βdydz

≤ kbk∧˙_β(R⁺)

1 2ⁱ

Z 2ⁱ⁺¹ 2ⁱ

y^βdy+ 1 2^k

Z 2^k+1 2^k

z^βdz

!

≤C2^iβ+1kbk∧˙β(R⁺).

Notice that ¹_p −¹_q =α+β, ¹_p +_p¹0 = 1; by Lemma 1.5 and the Hölder inequality, we have J₂ ≤C2^2q/q0+q kbk∧˙β(R⁺)

q

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

2^iβ

∞

X

k=i

Z 2^k+1 2^k

|f(t)|dt

!q

dx

≤C2^2q/q0+q kbk∧˙_β(R⁺)

q

∞

X

i=−∞



2^i(β+1q)

∞

X

k=i

2^k/p0

Z 2^k+1 2^k

|f(t)|^pdt

!¹_p



q

≤C2^2q/q0+q kbk∧˙_β(R⁺)

q

∞

X

i=−∞



2^i(β+1q)

∞

X

k=i

2^k/p0

Z 2^k+1 2^k

2^{−k(1−α)p}|t^1−αf(t)|^pdt

!¹_p



q

=C2^2q/q0+q kbk∧˙_β(R⁺)

q

∞

X

i=−∞





∞

X

k=i

2^{(i−k)(β+1q)}

Z 2^k+1 2^k

|t^1−αf(t)|^pdt

!¹_p



q

. In the second inequality, the Hölder inequality is applied.

By Lemma 1.6

s= _qβ+1^q ,

(2.3) J₂ ≤C kbk∧˙β(R⁺)

qZ ∞ 0

|t^1−αf(t)|^pdt _p^q

.

Combining (2.1), (2.2) and (2.3), we complete the proof of Theorem 1.4.

Second Proof. By inequality (1.7) and comparing the size ofxandt, we have Z ∞

0

|V_bf(x)|^qdx=

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

Z ∞ x

f(t)|b(x)−b(x)|dt

q

dx

≤

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

|t−x|^βkbk∧˙_β(R⁺)|f(t)|dt

!q

dx

≤C kbk∧˙_β(R⁺)

q

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

t^β|f(t)|dt

!q

dx

=C kbk∧˙_β(R⁺)

q

∞

X

i=−∞

Z 2ⁱ⁺¹ 2ⁱ

∞

X

k=i

Z 2^k+1 2^k

t^1−α|f(t)|

t^1−α−β dt

!q

dx

=C kbk∧˙_β(R⁺)

q

∞

X

i=−∞

2ⁱ

∞

X

k=i

Z 2^k+1 2^k

t^1−α|f(t)|

t^1−α−β dt

!q

.

By the Hölder inequality, ¹_p +_p¹0 = 1,the following estimate is obtained.

Z 2^k+1 2^k

t^1−α|f(t)|

t^1−α−β dt≤

Z 2^k+1 2^k

|t^1−αf(t)|^pdt

!¹_p

Z 2^k+1 2^k

t^{(α+β−1)p0}dt

!_p¹0

.

Notice that ¹_p −¹_q =α+β, Z ∞

0

|V_bf(x)|^qdx≤C kbk∧˙_β(R⁺)

q

∞

X

i=−∞







∞

X

k=i

2^(i−k)/q

Z 2^k+1 2^k

|t^1−αf(t)|^pdt

!¹_p





q

.

By Lemma 1.6, the desired result is obtained.

REFERENCES

[1] C. BENNET, R.A. DEVOREANDR. SHARPLEY, WeakL^∞and BMO, Ann. of Math., 113 (1981), 601–611.

[2] G.A. BLISS, An integral inequality, J. London Math. Soc., 5(1930), 40-46.

[3] T.A.A. BROADBENT, A proof of Hardy’s convergence theorem, J. London Math. Soc., 3 (1928), 242–243

[4] A.P. CALDERÓN, Space betweenL¹andL^∞and the theorem of Marcikiewiez, Studia Math., 26 (1966), 273–299.

[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 (Nat- ural Science), 42 (2006), 342–345.

[8] Z.G. LIUANDZ.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.

[11] G.H. HARDY, J.E. LITTLEWOODANDG. 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. LONGANDJ. WANG, Commutators of Hardy operators, J. of Math. Anal. Appl., 274 (2002), 626–644.

[15] S.Z. LU, Q. WUAND D.C. YANG, Boundedness of commutators on Hardy type spaces, Science in China, 8 (2002), 984–997.

[16] B.G. PACHPATTE, A note on a certain inequality related to Hardy’s inequality, Indiana J. Pure Appl., 23 (1992), 773–776.

[17] B.G. PACHPATTE, On some integral similar to Hardy’s integral inequality, J. Math. Anal. Appl., 129 (1988), 596–606.

[18] B.G. PACHPATTE, On a new Hardy type inequality, J. Math. Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 265–274.

[19] B.G. PACHPATTE, On some new generalizations of Hardy’s integral inequality, J. Math. Anal.

Appl., 1 (1999), 15–30.

[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.

[21] B.C. YANG, Z.H. ZENGANDL. DEBNATH, Generalizations of Hardy integral inequality, Inter- nat. J. Math. Math. Sci, 3 (1999), 535–542.

[22] J. XIAO,L^p and BMO bounds of weighted Hardy–Littlewood averages, J. Math. Anal. Appl., 262 (2001), 660–666.