λ-CENTRAL BMO ESTIMATES FOR COMMUTATORS OF N-DIMENSIONAL HARDY OPERATORS

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λ-Central BMO Estimates Zun-Wei Fu vol. 9, iss. 4, art. 111, 2008

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λ-CENTRAL BMO ESTIMATES FOR

COMMUTATORS OF N -DIMENSIONAL HARDY OPERATORS

ZUN-WEI FU

Department of Mathematics Linyi Normal University

Linyi Shandong, 276005, P.R. of China EMail:lyfzw@tom.com

Received: 12 April, 2008

Accepted: 13 October, 2008 Communicated by: R.N. Mohapatra

2000 AMS Sub. Class.: 26D15, 42B25, 42B99.

Key words: Commutator, N-dimensional Hardy operator, λ-central BMO space, Central Morrey space.

Abstract: This paper gives theλ-central BMO estimates for commutators ofn-dimensional Hardy operators on central Morrey spaces.

Acknowledgements: The research is supported by the NNSF (Grant No. 10571014; 10871024) of People’s Republic of China.

The author would like to express his thanks to Prof. Shanzhen Lu for his constant encourage. This paper is dedicated to him for his70^thbirthday. The author also would like to express his gratitude to the referee for his valuable comments.

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1. Introduction and Main Results

Let f be a locally integrable function on Rⁿ. The n-dimensional Hardy operators are defined by

Hf(x) := 1

|x|ⁿ Z

|t|≤|x|

f(t)dt, H^∗f(x) :=

Z

|t|>|x|

f(t)

|t|ⁿ dt, x∈Rⁿ\ {0}.

In [4], Christ and Grafakos obtained results for the boundedness ofHonL^p(Rⁿ) spaces. They also found the exact operator norms ofH on L^p(Rⁿ)spaces, where 1< p <∞.

It is easy to see thatHandH^∗ satisfy (1.1)

Z

Rⁿ

g(x)Hf(x)dx= Z

Rⁿ

f(x)H^∗g(x)dx.

We have

|Hf(x)| ≤C_nM f(x),

whereM is the Hardy-Littlewood maximal operator which is defined by

(1.2) M f(x) = sup

Q3x

1

|Q|

Z

Q

|f(t)|dt, where the supremum is taken over all balls containingx.

Recently, Fu et al. [2] gave the definition of commutators ofn-dimensional Hardy operators.

Definition 1.1. Letbbe a locally integrable function onRⁿ. We define the commu- tators ofn-dimensional Hardy operators as follows:

H_bf :=bHf− H(f b), H^∗_bf :=bH^∗f − H^∗(f b).

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In [2], Fu et al. gave the central BMO estimates for commutators ofn-dimensional Hardy operators. In 2000, Alvarez, Guzmán-Partida and Lakey [1] studied the rela- tionship between central BMO spaces and Morrey spaces. Furthermore, they intro- ducedλ-central bounded mean oscillation spaces and central Morrey spaces, respec- tively.

Definition 1.2 (λ-central BMO space). Let 1 < q < ∞ and −¹_q < λ < ¹_n. A functionf ∈ L^q_loc(Rⁿ) is said to belong to theλ-central bounded mean oscillation spaceCM O˙ ^{q, λ}(Rⁿ)if

(1.3) kfk_CM O˙ ^{q, λ}(Rⁿ) = sup

R>0

1

|B(0,R)|^1+λq Z

B(0,R)

|f(x)−f_B(0,R)|^qdx ¹_q

<∞.

Remark 1. If two functions which differ by a constant are regarded as a function in the spaceCM O˙ ^{q, λ}(Rⁿ), thenCM O˙ ^{q, λ}(Rⁿ)becomes a Banach space. Apparently, (1.3) is equivalent to the following condition (see [1]):

sup

R>0 c∈infC

1

|B(0,R)|^1+λq Z

B(0,R)

|f(x)−c|^qdx ¹_q

<∞.

Definition 1.3 (Central Morrey spaces, see [1]). Let1< q <∞and−¹_q < λ <0.

The central Morrey spaceB˙^{q, λ}(Rⁿ)is defined by

(1.4) kfkB˙^{q, λ}(Rⁿ)= sup

R>0

1

|B(0,R)|^1+λq Z

B(0,R)

|f(x)|^qdx ¹_q

<∞.

Remark 2. It follows from (1.3) and (1.4) thatB˙^{q, λ}(Rⁿ)is a Banach space continu- ously included inCM O˙ ^{q, λ}(Rⁿ).

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Inspired by [2], [3] and [5], we will establish the λ-central BMO estimates for commutators ofn-dimensional Hardy operators on central Morrey spaces.

Theorem 1.4. LetH_b be defined as above. Suppose1 < p₁ < ∞, p⁰₁ < p₂ < ∞,

1 q = _p¹

1 + _p¹

2, −¹_q < λ < 0,0 ≤ λ₂ <_n¹ andλ = λ₁ +λ₂. If b ∈ CM O˙ ^p²^{, λ}²(Rⁿ), then the commutatorHb is bounded from B˙^p¹^{, λ}(Rⁿ) to B˙^{q, λ}(Rⁿ) and satisfies the following inequality:

kH_bfkB˙^{q, λ}(Rⁿ) ≤Ckbk_CM O˙ ^p²^{, λ}²(Rⁿ)kfkB˙^p¹^{, λ}¹(Rⁿ).

Letλ₂ = 0in Theorem1.4. We can obtain the central BMO estimates for commutators ofn-dimensional Hardy operators,H_b, on central Morrey spaces.

Corollary 1.5. LetH_b be defined as above. Suppose1 < p₁ < ∞, p⁰₁ < p₂ < ∞,

1 q = _p¹

1 + _p¹

2 and −¹_q < λ < 0. If b ∈ CM O˙ ^p²(Rⁿ), then the commutator H_b is bounded fromB˙^p¹^{, λ}(Rⁿ)toB˙^{q, λ}(Rⁿ)and satisfies the following inequality:

kH_bfkB˙^{q, λ}(Rⁿ) ≤Ckbk_CM O˙ ^p²(Rⁿ)kfkB˙^p¹^{, λ}(Rⁿ). Similar to Theorem1.4, we have:

Theorem 1.6. LetH^∗_b be defined as above. Suppose1 < p₁ < ∞, p⁰₁ < p₂ < ∞,

1 q = _p¹

1 + _p¹

2, −¹_q < λ < 0,0≤ λ2 < _n¹ andλ = λ1 +λ2. Ifb ∈ CM O˙ ^p²^{, λ}²(Rⁿ), then the commutatorH^∗_b is bounded fromB˙^p¹^{, λ}¹(Rⁿ)toB˙^{q, λ}(Rⁿ)and satisfies the following inequality:

kH_b^∗fkB˙^{q, λ}(Rⁿ)≤Ckbk_CM O˙ ^p²^{, λ}²(Rⁿ)kfkB˙^p¹^{, λ}¹(Rⁿ).

Letλ2 = 0in Theorem1.6. We can get the central BMO estimates for commutators ofn-dimensional Hardy operators,H_b^∗, on central Morrey spaces.

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Corollary 1.7. LetH_b^∗ be defined as above. Suppose1 < p₁ < ∞, p⁰₁ < p₂ < ∞,

1 q = _p¹

1 + _p¹

2 and −¹_q < λ < 0. If b ∈ CM O˙ ^p²(Rⁿ), then the commutatorH^∗_b is bounded fromB˙^p¹^{, λ}(Rⁿ)toB˙^{q, λ}(Rⁿ)and satisfies the following inequality:

kH^∗_bfkB˙^{q, λ}(Rⁿ) ≤Ckbk_CM O˙ ^p²(Rⁿ)kfkB˙^p¹^{, λ}(Rⁿ).

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2. Proofs of Theorems

Proof of Theorem1.4. Letf be a function inB˙^p¹^{, λ}¹(Rⁿ). For fixedR > 0, denote B(0, R)byB. Write

1

|B|

Z

B

|H_bf(x)|^qdx ¹_q

= 1

|B|

Z

B

1

|x|ⁿ Z

B(0,|x|)

f(y)(b(x)−b(y))dy

q

dx ¹_q

≤ 1

|B|

Z

B

1

|x|ⁿ Z

B(0,|x|)

f(y)(b(x)−b_B)dy

q

dx ¹_q

+ 1

|B|

Z

B

1

|x|ⁿ Z

B(0,|x|)

f(y)(b(y)−b_B)dy

q

dx ¹_q

:=I+J.

For ¹_q = _p¹

1 +_p¹

2, by Hölder’s inequality and the boundedness ofHfromL^p¹ toL^p¹, we have

I ≤ |B|⁻¹^q Z

B

|b(x)−b_B|^p²dx _p¹

2 Z

B

|H(f χ_B)(x)|^p¹dx _p¹

1

≤C|B|⁻¹^qkbk_CM O˙ ^p²^{, λ}²(Rⁿ)|B|^p¹²^+λ² Z

B

|f(x)|^p¹dx _p¹

1

=C|B|^λkbk_CM O˙ ^p²^{, λ}²(Rⁿ)

1

|B|^1+p¹^λ¹ Z

B

|f(x)|^p¹dx _p¹

1

≤C|B|^λkbk_CM O˙ ^p²^{, λ}²(Rⁿ)kfkB˙^p¹^{, λ}¹(Rⁿ).

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ForJ, we have J^q = 1

|B| Z

B

1

|x|ⁿ Z

B(0,|x|)

f(y)(b(y)−bB)dy

q

dx

= 1

|B|

0

X

k=−∞

Z

2^kB\2^k−1B

1

|x|ⁿ Z

B(0,|x|)

f(y)(b(y)−bB)dy

q

dx

≤ C

|B|

0

X

k=−∞

1

|2^kB|^q Z

2^kB\2^k−1B

k

X

i=−∞

Z

2ⁱB\2ⁱ⁻¹B

f(y)(b(y)−b_B)dy

q

dx

≤ C

|B|

0

X

k=−∞

1

|2^kB|^q Z

2^kB\2^k−1B

k

X

i=−∞

Z

2ⁱB\2ⁱ⁻¹B

f(y)(b(y)−b₂ⁱ_B)dy

q

dx

+ C

|B|

0

X

k=−∞

1

|2^kB|^q Z

2^kB\2^k−1B

k

X

i=−∞

Z

2ⁱB\2ⁱ⁻¹B

f(y)(b₂ⁱ_B−b_B)dy

q

dx

:=J₁+J₂

By Hölder’s inequality (_p¹

1 +_p¹

2 = ¹_q), we have J₁ ≤ C

|B|

0

X

k=−∞

|2^kB|

|2^kB|^q ( _k

X

i=−∞

|2ⁱB|^q¹⁰ Z

2ⁱB

|f(y)|^p¹dy _p¹

1

× Z

2ⁱB

|b(y)−b₂ⁱ_B|^p²dy _p¹

2

)q

≤ C

|B|kbk^q

CM O˙ ^p²^{, λ}²(Rⁿ)kfk^q_˙

B^p¹^{, λ}¹(Rⁿ) 0

X

k=−∞

|2^kB|

|2^kB|^q ( _k

X

i=−∞

|2ⁱB|^λ+1 )^q

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≤C|B|^qλkbk^q

B^p¹^{, λ}¹(Rⁿ). To estimateJ₂, the following fact is applied.

Forλ₂ ≥0,

|b₂ⁱ_B−b_B| ≤

−1

X

j=i

|b₂^j+1_B−b₂^j_B|

≤

−1

X

j=i

1

|2^jB|

Z

2^jB

|b(y)−b₂^j+1_B|dy

≤C

−1

X

j=i

1

|2^j+1B|

Z

2^j+1B

|b(y)−b₂^j+1_B|^p²dy _p¹

2

≤Ckbk_CM O˙ ^p²^{, λ}²(Rⁿ)|B|^λ²

−1

X

j=i

2^(j+1)nλ²

≤Ckbk_CM O˙ ^p²^{, λ}²(Rⁿ)|i||B|^λ². By Hölder’s inequality (_p¹

1 +_p¹0

1 = 1), we have J₂ = C

|B|

0

X

k=−∞

1

|2^kB|^q Z

2^kB\2^k−1B

k

X

i=−∞

Z

2ⁱB\2ⁱ⁻¹B

f(y)(b₂ⁱ_B−b_B)dy

q

dx

≤ C

|B|kbk^q

B^p¹^{, λ}¹(Rⁿ) 0

X

k=−∞

|2^kB||B|^qλ²

|2^kB|^q

( _k X

i=−∞

|i||2ⁱB|^λ¹⁺¹ )^q

≤ C

|B|kbk^q

B^p¹^{, λ}¹(Rⁿ) 0

X

k=−∞

|2^kB||B|^qλ²|k|^q|2^kB|^(λ¹^+1)q

|2^kB|^q

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≤C|B|^qλkbk^q

B^p¹^{, λ}¹(Rⁿ).

Combining the estimates ofI,J1 andJ2, we get the required estimate for Theorem 1.4.

Proof of Theorem1.6. We omit the details here.

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References

[1] J. ALVARAREZ, M. GUZMAN-PARTIDAANDJ. LAKEY, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math., 51 (2000), 1–47.

[2] Z.W. FU, Z.G. LIU, S.Z. LUANDH.B. WANG, Characterization for commuta- tors of N-dimensional fractional Hardy operators. Science in China (Ser. A), 10 (2007), 1418–1426.

[3] Z.W. FU, Y. LINAND S.Z. LU, λ-Central BMO estimates for commutators of singular integral operators with rough kernels. Acta Math. Sinica (English Ser.), 3 (2008), 373–386.

[4] M. CHRIST AND L. GRAFAKOS, Best constants for two non-convolution in- equalities, Proc. Amer. Math. Soc., 123 (1995), 1687–1693.

[5] S.C. LONG AND J. WANG, Commutators of Hardy operators, J. Math. Anal.

Appl., 274 (2002), 626–644.