ACTA UNIVERSITATIS APULENSIS No 20/2009 BOUNDEDNESS FOR MULTILINEAR COMMUTATOR OF INTEGRAL OPERATOR ON HARDY AND HERZ-HARDY SPACES Ren Sheng and Lanzhe Liu Abstract. In this paper,the (H

BOUNDEDNESS FOR MULTILINEAR COMMUTATOR OF INTEGRAL OPERATOR ON HARDY AND HERZ-HARDY SPACES

Ren Sheng and Lanzhe Liu

Abstract. In this paper,the (H^p

~b, L^p) and (HK˙^α,p

q,~b,K˙_q^α,p) type boundedness for the multilinear commutator associated with some integral operator are obtained.

2000Mathematics Subject Classification: 42B20, 42B25.

1. Introduction

Letb∈BM O(Rⁿ), andT be the Calder´on-Zygmund operator. The commutator [b, T] generated byb and T is defined by

[b, T]f(x) =b(x)T f(x)−T(bf)(x).

A classical result of Coifman, Rochberg and Weiss (see [3]) proved that the commu- tator [b, T] is bounded on L^p(Rⁿ) (1< p <∞). However, it was observed that the [b, T] is not bounded, in general, fromH^p(Rⁿ) toL^p(Rⁿ). But ifH^p(Rⁿ) is replaced by a suitable atomic space H_~b^P(Rⁿ) andHK˙^α,p

q,~b(Rⁿ), then [b, T] maps continuously H_~b^P(Rⁿ) into L^p(Rⁿ) andHK˙^α,p

q,~b(Rⁿ) into ˙K_q^α,p. In addition we easily known that H_~b^p(Rⁿ)⊂H^p(Rⁿ), K˙^α,p

q,~b(Rⁿ)⊂HK˙_q^α,p(Rⁿ). The main purpose of this paper is to consider the continuity of the multilinear commutators related to the Littlewood- Paley operators andBM O(Rⁿ) functions on certain Hardy and Herz-Hardy spaces.

Let us first introduce some definitions (see [1][4-16][18][19]).

Given a positive integer m and 1≤ j ≤ m, we denote by C_j^m the family of all finite subsets σ={σ(1),· · ·, σ(j)}of{1,· · ·, m}ofj different elements. Forσ ∈C_j^m, set σ^c = {1,· · ·, m} \σ. For~b = (b1,· · ·, b_m) and σ = {σ(1),· · ·, σ(j)} ∈ C_j^m, set

~bσ = (b_σ(1),· · ·, b_σ(j)),bσ =b_σ(1)· · ·b_σ(j)and||~bσ||_{BM O}=||b_σ(1)||_{BM O}· · · ||b_σ(j)||_{BM O}.

Definition 1. Let bi (i= 1,· · ·, m) be a locally integrable function and0< p≤ 1. A bounded measurable functiona onRⁿ is said a (p,~b) atom, if

(1) suppa⊂B =B(x0, r) (2) ||a||_L^∞ ≤ |B|^−1/p

(3) ^R_Ba(y)dy=^R_Ba(y)^Q_l∈σb_l(y)dy= 0 for anyσ ∈C_j^m ,1≤j ≤m .

A temperate distribution f is said to belong toH_~b^p(Rⁿ), if, in the Schwartz dis- tribution sense, it can be written as

f(x) =

∞

X

j=1

λ_ja_j(x).

where a⁰_js are (p,~b) atoms, λ ∈ C and ^P∞_j=1|λ|^p < ∞. Moreover, ||f||_H^p

~b

≈ (^P∞_j=1|λ_j|^p)^1/p.

Definition 2.Let 0< p, q <∞, α∈R.Fork∈Z, setB_k={x∈Rⁿ:|x| ≤2^k} and C_k = B_k\Bk−1. Denote by χ_k the characteristic function of C_k and χ₀ the characteristic function of B0.

(1) The homogeneous Herz space is defined by

K˙_q^α,p(Rⁿ) =ⁿf ∈L^q_loc(Rⁿ\ {0}) :||f||_K˙α,p

q <∞^o, where

||f||_K˙α,p

q =





∞

X

k=−∞

2^kαp||f χ_Bk||^p_Lq





1/p

.

(2) The nonhomogeneous Herz space is defined by K_q^α,p(Rⁿ) =ⁿf ∈L^q_loc(Rⁿ) :||f||_K^α,p

q <∞^o, where

||f||_K^α,p

q =

"_∞ X

k=1

2^kαp||f χ_k||^p_Lq+||f χ_B0||^p_Lq

#1/p

.

Definition 3.Let α∈Rⁿ, 1< q <∞, α≥n(1−¹_q), bi ∈BM O(Rⁿ), 1≤i≤ m. A function a(x) is called a central (α, q,~b) -atom (or a central (α, q,~b)-atom of restrict type ), if

(1) suppa∈B =B(x0, r)(or for some r≥1), (2) ||a||_L^q ≤ |B|^−α/n

(3) ^R_Ba(x)x^βdx=^R_Ba(x)x^βQ_i∈σb_i(x)dx= 0 for any σ∈C_j^m ,1≤j ≤m.

A temperate distributionf is said to belong to HK˙^α,

q,~b(Rⁿ)(or HK^α,p

q,~b(Rⁿ)), if it can be written as f =^P∞_j=−∞λ_ja_j (or f =^P∞_j=0λ_ja_j), in the S⁰(Rⁿ) sense, where

aj is a central (α, q,~b)-atom(or a central (α, q,~b)-atom of restrict type ) supported on B(0,2^j) and ^P∞_−∞|λ_j|^p<∞(or ^P∞_j=0|λ_j|<∞). Moreover,

||f||_HK˙^α,p

q,~b

( or ||f||_HK^α,p

q,~b

) = inf(^X

j

|λ_j|^p)^1/p,

where the infimum are taken over all the decompositions of f as above.

Definition 4.Suppose bj (j= 1,· · ·, m) are the fixed locally integrable functions on Rⁿ. Let F_t(x, y) be the function defined on Rⁿ×Rⁿ×[0,+∞). Set

St(f)(x) = Z

Rⁿ

Ft(x, y)f(y)dy and

S^~b_t(f)(x) = Z

Rⁿ m

Y

j=1

(b_j(x)−b_j(y))F_t(x, y)f(y)dy

for every bounded and compactly supported function f. Let H be the Banach space H ={h :||h|| <∞} such that, for each fixed x ∈Rⁿ, S_t(f)(x) and S^~b_t(f)(x) may be viewed as the mappings from [0,+∞) to H. The multilinear commutator related to St is defined by

T_δ^~b(f)(x) =||S^~b_t(f)(x)||, where Ft satisfies: for fixed ε >0 and 0< δ < n,

||F_t(x, y)|| ≤C|x−y|^−n+δ and

||F_t(x, y)−Ft(x, z)||+||F_t(y, x)−Ft(z, x)|| ≤C|y−z||x−z|^−n−ε+δ if 2|y−z| ≤ |x−z|. We also define T_δ(f)(x) =||S_t(f)(x)||.

2. Theorems and Proofs

Lemma.(see [18])Let 1< r <∞, bj ∈BM O(Rⁿ) for j = 1,· · ·, k and k∈N. Then, we have

1

|Q|

Z

Q k

Y

j=1

|b_j(y)−(bj)_Q|dy≤C

k

Y

j=1

||b_j||_{BM O}

and



 1

|Q|

Z

Q k

Y

j=1

|b_j(y)−(bj)Q|^rdy





1/r

≤C

k

Y

j=1

||b_j||_{BM O}.

Theorem 1.Let bi ∈ BM O(Rⁿ), 1 ≤ i ≤ m, ~b = (b1,· · ·, bm), 0 < δ < n, n/(n+ε−δ)< q≤1,1/q = 1/p−δ/n.Suppose thatT_δ^~bis the multilinear commutator as in Definition 4 such thatT is bounded fromL^s(Rⁿ)toL^r(Rⁿ)for any1< s < n/δ and 1/r= 1/s−δ/n. Then T_δ^~b is bounded from H_~b^p(Rⁿ) to L^q(Rⁿ).

Proof. It suffices to show that there exist a constantC >0, such that for every (p,~b) atom a,

||T_δ^~b(a)||_L^q ≤C.

Let abe a (p,~b) atom supported on a ballB =B(x0, l). We write Z

Rⁿ

|T^~b_δ(a)(x)|^qdx= Z

|x−x₀|≤2l

|T_δ^~b(a)(x)|^qdx+ Z

|x−x₀|>2l

|T_δ^~b(a)(x)|^qdx=I+II.

For I, takingr, s >1 withq < s < n/δ and 1/r= 1/s−δ/n, by H¨older’s inequality and the (L^s, L^r)- boundedness ofT_δ^~b,we see that

I ≤ Z

|x−x0|≤2l

|T_δ^~b(a)(x)|^rdx

!q/r

· |B(x₀,2l)|^1−q/r

≤ C||T^~b_δ(a)(x)||^q_Ls · |B(x0,2l)|^1−q/r

≤ C||a||^q_Ls|B|^1−q/r

≤ C|B|−q/p+q/s+1−q/r

≤ C.

For II, denoting λ = (λ1,· · ·, λm) with λi = (bi)_B, 1 ≤ i ≤ m, where (bi)_B =

1

|B(x₀,l)|

R

B(x0,l)b_i(x)dx, by H¨older’s inequality and the vanishing moment of a, we get

II =

∞

X

k=1

Z

2^k+1B\2^kB

|T_δ^~b(a)(x)|^qdx

≤ C

∞

X

k=1

|2^k+1B|^1−q Z

2^k+1B\2^kB

|T_δ^~b(a)(x)|dx

!q

≤ C

∞

X

k=1

|2^k+1B|^1−q

×



 Z

2^k+1B\2^kB

||

Z

B m

Y

j=1

(b_j(x)−b_j(y))F_t(x, y)a(y)dy||dx





q

≤ C

∞

X

k=1

|2^k+1B|^1−q

×



 Z

2^k+1B\2^kB

Z

B

||F_t(x, y)−Ft(x,0)||

m

Y

j=1

|(b_j(x)−bj(y))||a(y)|dydx





q

; noting that y∈B, x∈2^k+1B\2^kB, then

Z

B

||F_t(x, y)−Ft(x,0)||

m

Y

j=1

|(b_j(x)−bj(y))||a(y)|dy

≤ C

Z

B m

Y

j=1

|(b_j(x)−b_j(y))|||F_t(x, y)−F_t(x,0)|||a(y)|dy

≤ C

Z

B m

Y

j=1

|(b_j(x)−b_j(y))| |y|^ε

|x|^n+ε−δ|a(y)|dy,

thus II ≤ C

∞

X

k=1

|2^k+1B|^1−q



 Z

2^k+1B\2^kB

|x|^{−(n+ε−δ)}



 Z

B m

Y

j=1

|b_j(x)−bj(y)||y|^ε|a(y)|dy



dx





q

≤ C

∞

X

k=1

|2^k+1B|^1−q

×







m

X

j=0

X

σ∈C_j^m

Z

2^k+1B\2^kB

|x|^{−(n+ε−δ)}|(~b(x)−λ)_σ|dx Z

B

|(~b(y)−λ)_σ^c||y|^ε|a(y)|dy







q

≤ C

m

X

j=0

X

σ∈C^m_j

Z

B

|(~b(y)−λ)_σ^c||y|^ε|a(y)|dy q

×

∞

X

k=1

|2^k+1B|^1−q

"

Z

2^k+1B\2^kB

|x|^{−(n+ε−δ)}|(~b(x)−λ)_σ|dx

#q

≤ C

m

X

j=0

X

σ∈C^m_j

||~bσ^c||^q_{BM O}· ||~bσ||^q_{BM O}

∞

X

k=1

|2^k+1B|1−q(n+ε−δ)/nk^q|B|(1+ε/n−1/p)q

≤ C||~b||^q_{BM O}

∞

X

k=1

k^q·2−knq(1+ε/n−δ/n−1/q)

≤ C||~b||^q_{BM O}.

This finish the proof of Theorem 1.

Theorem 2.Let 0 < p < ∞, 0 < δ < n, 1 < q1, q2 <∞, 1/q1 −1/q2 = δ/n, n(1−1/q₁) +δ ≤ α < n(1−1/q₁) +ε+δ and b_i ∈ BM O(Rⁿ),1 ≤ i ≤ m, ~b =

(b₁,· · ·, b_m).Suppose that T^~b_δ is the multilinear commutator as in Definition 4 such thatT is bounded from L^s(Rⁿ) toL^r(Rⁿ) for any1< s < n/δand 1/r= 1/s−δ/n.

Then T_δ^~b is bounded from HK˙^α,p

q1,~b(Rⁿ) to K˙_q^α,p₂ (Rⁿ).

Proof. Letf ∈HK˙^α,p

q1,~b(Rⁿ) andf(x) =^P∞_j=−∞λ_ja_j(x) be the atomic decom- position for f as in Definition 3, we write

||T_δ^~b(f)(x)||_K˙α,p q2

≤ C





∞

X

k=−∞

2^kαp(

∞

X

j=−∞

|λ_j|||T_δ^~b(aj)χ_k||_L^q2)^p





1/p

≤ C





∞

X

k=−∞

2^kαp(

k−3

X

j=−∞

|λ_j|||T_δ^~b(aj)χ_k||_L^q2)^p





1/p

+C





∞

X

k=−∞

2^kαp(

∞

X

j=k−2

|λ_j|||T_δ^~b(aj)χ_k||_L^q2)^p





1/p

= I+II.

ForII, noting that suppaj ⊆B(0,2^j),||a_j||_L^q1 ≤ |B(0,2^j)|^−α/n, by the boundedness of T_δ^~b on (L^q1(Rⁿ), L^q2(Rⁿ)) and the H¨older’s inequality, we get

II = C





∞

X

k=−∞

2^kαp(

∞

X

j=k−2

|λ_j|||T_δ^~b(a_j)χ_k||_L^q2)^p





1/p

≤ C





∞

X

k=−∞

2^kαp





∞

X

j=k−2

|λ_j|||a_j||_L^q1





p



1/p

≤ C





∞

X

k=−∞

2^kαp





∞

X

j=k−2

|λ_j| ·2^−jα





p



1/p

≤ C





 hP∞

j=−∞|λ_j|^pPj+2_k=−∞2^{(k−j)αpi1/p}, 0< p≤1 hP∞

j=−∞|λ_j|^p(^Pj+2_k=−∞2^(k−j)αp/2)(^Pj+2_k=−∞2^{(k−j)αp0/2})^p/p0i1/p, 1< p <∞

≤ C





∞

X

j=−∞

|λ_j|^p





1/p

≤ C||f||_HK˙α,p q1,~b

.

For I, when m=1, let C_k = B_k\B_k−1, χ_k = χ_Ck, bⁱ_j = |B_j|^−1R_B

jbi(x)dx, 1 ≤i ≤ m, ~b⁰ = (b¹_j,· · ·, b^m_j ).We have

T_δ^b1(a_j)(x) ≤ Z

Bj

||F_t(x, y)−F_t(x,0)|||b₁(x)−b₁(y)||a_j(y)|dy

≤ Z

Bj

|a_j(y)||b₁(x)−b₁(y)| |y|^ε

|x|^n+ε−δdy

≤ C|x|^{−(n+ε−δ)} Z

Bj

|y|^ε|a_j(y)||b₁(x)−b¹_j|dy +C|x|^{−(n+ε−δ)}

Z

Bj

|y|^ε|a_j(y)||b₁(y)−b¹_j|dy

≤ C|x|^{−(n+ε−δ)}|b₁(x)−b¹_j|2j(ε+n(1−1/q₁)−α)

+ 2j(ε+n(1−1/q₁)−α)||b₁||_{BM O}; Then

||T_δ^b1(a_j)χ_k||_Lq

2

≤ C2j(ε+n(1−1/q1)−α)[ Z

B_k

|x|^{−q2(n+ε−δ)}|b₁(x)−b¹_j|^q2dx 1/q2

+ Z

Bk

|x|^{−q2(n+ε−δ)}dx 1/q2

||b₁||_{BM O}]

≤ C2j(ε+n(1−1/q₁)−α)h

2^{−k(n+ε−δ)}· |B_k|^1/q2||b₁||_{BM O}+ 2^{−k(n+ε−δ)}· |B_k|^1/q2||b₁||_{BM O}ⁱ

≤ C||b₁||_{BM O}2j(ε+n(1−1/q₁)−α)·2−k(ε+n(1−1/q₁)), thus

I = C





∞

X

k=−∞

2^kαp





k−3

X

j=−∞

|λ_j|||T_δ^b1(a_j)χ_k||_L^q2





p



1/p

≤ C||b₁||_{BM O}





∞

X

k=−∞

2^kαp





k−3

X

j=−∞

|λ_j|2j(ε+n(1−1/q₁)−α)−k(ε+n(1−1/q₁))





p



1/p

≤ C||b₁||_{BM O}













 hP∞

k=−∞2^kαpPk−3_j=−∞|λ_j|^p2p[j(ε+n(1−1/q1)−α)−k(ε+n(1−1/q1))]i1/p

, 0< p≤1 hP∞

k=−∞2^kαpPk−3_j=−∞|λ_j|^p2[j(ε+n(1−1/q1)−α)−k(ε+n(1−1/q1))]p/2

×^Pk−3_j=−∞2[j(ε+n(1−1/q₁)−α)−k(ε+n(1−1/q₁))]p⁰/2p/p⁰1/p

, 1< p <∞

≤ C||b₁||_{BM O}













 hP∞

j=−∞|λ_j|^pP∞_k=j+32p(j−k)(ε+n(1−1/q1)−α)i1/p

, 0< p≤1 hP∞

j=−∞|λ_j|^pP∞_k=j+32(j−k)(ε+n(1−1/q1)−α)p/2

×^P∞_k=j+32(j−k)(ε+n(1−1/q1)−α)p⁰/2p/p⁰1/p

, 1< p <∞

≤ C||b₁||_{BM O}





∞

X

j=−∞

|λ_j|^p





1/p

≤ C||f||_HK˙^α,p

q1,~b

.

When m >1, similar to the proof of T_δ^b1(a_j)(x),we have T_δ^~b(aj)(x) ≤ C

Z

Bj

m

Y

i=1

|b_i(x)−bi(y)|||F_t(x, y)−Ft(x,0)|||a_j(y)|dy

≤ C|x|^{−(n+ε−δ)} Z

Bj

|y|^ε|a_j(y)|

m

Y

i=1

|b_i(x)−b_i(y)|dy

≤ C|x|^{−(n+ε−δ)}

m

X

i=0

X

σ∈C_i^m

|(~b(x)−~b⁰)σ| Z

Bj

|y|^ε|a_j(y)||(~b(x)−~b)_σ^c|dy

≤ C|x|^{−(n+ε−δ)}

m

X

i=0

X

σ∈C_i^m

|(~b(x)−~b⁰)σ|2^jε·2^−jε·2^jn(1−1/q1)||~b_σ^c||_{BM O}

≤ C|x|^{−(n+ε−δ)}·2j(ε+n(1−1/q1)−α) m

X

i=0

X

σ∈C^m_i

|(~b(x)−~b⁰)_σ|||~b_σ^c||_{BM O};

So

||T_δ^~b(aj)χk||_L^q2

≤ C2j(ε+n(1−1/q)−α)||~bσ^c||_{BM O}



 Z

Bk



|x|^{−(n+ε−δ)}

m

X

i=0

X

σ∈C_i^m

|(~b(x)−b~⁰)σ|





q2

dx





1/q2

≤ C||~b_σ^c||_{BM O}2j(ε+n(1−1/q1)−α)·2−k(n+ε−δ)+kn/q2||~b_σ||_{BM O}

≤ C||~b||_{BM O}2j(ε+n(1−1/q1)−α)−k(ε+n(1−1/q1))

and

I = C





∞

X

k=−∞

2^kαp





k−3

X

j=−∞

|λ_j|||T_δ^~b(a_j)χ_k||_L^q2





p



1/p

≤ C||~b||_{BM O}





∞

X

k=−∞

2^kαp





k−3

X

j=−∞

|λ_j|2[j(ε+n(1−1/q1)−α)−k(ε+n(1−1/q1))]





p



1/p

≤ C||~b||_{BM O}













 hP∞

k=−∞2^kαpPk−3_j=−∞|λ_j|^p2p[j(ε+n(1−1/q₁)−α)−k(ε+n(1−1/q₁))]i1/p

, 0< p≤1 hP∞

k=−∞2^kαpPk−3_j=−∞|λ_j|^p2[j(ε+n(1−1/q₁)−α)−k(ε+n(1−1/q₁))]p/2

×^Pk−3_j=−∞2[j(ε+n(1−1/q1)−α)−k(ε+n(1−1/q1))]p⁰/2p/p⁰1/p

, 1< p <∞

≤ C||~b||_{BM O}













 hP∞

j=−∞|λ_j|^pP∞_k=j+32(j−k)(ε+n(1−1/q1)−α)pi1/p

, 0< p≤1 hP∞

j=−∞|λ_j|^pP∞_k=j+32(j−k)(ε+n(1−1/q1)−α)p/2

×^P∞_k=j+32(j−k)(ε+n(1−1/q1)−α)p⁰/2p/p⁰1/p

, 1< p <∞

≤ C||~b||_{BM O}





∞

X

j=−∞

|λ_j|^p





1/p

≤ C||f||_HK˙^α,p

q1,~b

.

Remark. Theorem 2 also hold for nonhomogeneous Herz-type spaces, we omit the details.

4. Applications

Now we give some applications of Theorems in this paper.

Application 1. Littlewood-Paley operator.

Fixed 0< δ < nandε >0. Letψbe a fixed function which satisfies the following properties:

(1) ^R_Rnψ(x)dx= 0,

(2) |ψ(x)| ≤C(1 +|x|)^{−(n+1−δ)},

(3) |ψ(x+y)−ψ(x)| ≤C|y|^ε(1 +|x|)^{−(n+1+ε−δ)} when 2|y|<|x|.

The Littlewood-Paley multilinear operators are defined by g^~b_ψ,δ(f)(x) =

Z _∞

0

|F_t^~b(f)(x)|²dt t

1/2

, where

F_t^~b(f)(x) = Z

Rⁿ m

Y

j=1

(b_j(x)−b_j(y))ψ_t(x−y)f(y)dy

and ψ_t(x) =t^−n+δψ(x/t) for t >0. SetF_t(f)(y) =f∗ψ_t(y). We also define g_ψ,δ(f)(x) =

Z ∞ 0

|F_t(f)(x)|²dt t

1/2

,

which is the Littlewood-Paley operator(see [18]). LetH be the space H=

(

h:||h||= Z ∞

0

|h(t)|²dt/t 1/2

<∞ )

,

then, for each fixedx∈Rⁿ,F^~b_t(f)(x) andF_t^~b(f)(x, y) may be viewed as the mappings from [0,+∞) to H, and it is clear that

g^~b_ψ,δ(f)(x) =||F_t^~b(f)(x)||, gψ,δ(f)(x) =||F_t(f)(x)||.

It is easily to see that g_ψ,δ satisfies the conditions of Theorem 1 and 2 (see [5-9]), thus Theorem 1 and 2 hold for g^~b_ψ,δ.

Application 2. Marcinkiewicz operator.

Fixed 0 < δ < n and 0 < γ ≤ 1. Let Ω be homogeneous of degree zero on Rⁿ with ^R_Sn−1Ω(x⁰)dσ(x⁰) = 0. Assume that Ω∈ Lip_γ(Sⁿ⁻¹). The Marcinkiewicz multilinear operators are defined by

µ^~b_Ω,δ(f)(x) = Z ∞

0

|F_t^~b(f)(x)|²dt t³

1/2

, where

F_t^~b(f)(x) = Z

|x−y|≤t m

Y

j=1

(b_j(x)−b_j(y)) Ω(x−y)

|x−y|^n−1−δf(y)dy.

Set

F_t(f)(x) = Z

|x−y|≤t

Ω(x−y)

|x−y|^n−1−δf(y)dy.

We also define

µ_Ω,δ(f)(x) = Z ∞

0

|F_t(f)(x)|²dt t³

1/2

,

which is the Marcinkiewicz operator(see [8][20]). Let H be the space H =

(

h:||h||= Z ∞

0

|h(t)|²dt/t³ 1/2

<∞ )

.

Then, it is clear that

µ^~b_Ω,δ(f)(x) =||F_t^~b(f)(x)||, µΩ,δ(f)(x) =||F_t(f)(x)||,

It is easily to see that µ_Ω,δ satisfies the conditions of Theorem 1 and 2 (see [8][20]), thus Theorem 1 and 2 hold for µ^~b_Ω,δ.

Application 3. Bochner-Riesz operator .

Let η > (n−1)/2, B_t^η(f)(ξ) = (1ˆ −t²|ξ|²)^η₊fˆ(ξ) and B_t^η(z) = t⁻ⁿB^η(z/t) for t >0. Set

F_η,t^~b (f)(x) = Z

Rⁿ m

Y

j=1

(b_j(x)−b_j(y))B_t^η(x−y)f(y)dy.

The maximal Bochner-Riesz multilinear commutator are defined by B^~b_η,∗(f)(x) = sup

t>0

|B^~b_η,t(f)(x)|.

We also define that

Bη,∗(f)(x) = sup

t>0

|B_t^η(f)(x)|,

which is the maximal Bochner-Riesz operator(see [10]). Let H be the space H = {h:||h||= sup

t>0

|h(t)|<∞}, then

B^~b_η,∗(f)(x) =||B^~b_η,t(f)(x)||, B_∗^η(f)(x) =||B_t^η(f)(x)||.

It is easily to see thatB^~b_η,∗satisfies the conditions of Theorem 1 and 2 withδ = 0(see [9]), thus Theorem 1 and 2 hold for B^~b_η,∗.

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Sheng Ren and Liu Lanzhe Department of Mathematics Hunan University

Changsha, 410082, P.R.of China E-mail: [email protected]