Acta Universitatis Apulensis ISSN: 1582-5329 No. 24/2010 pp. 95-107

BOUNDEDNESS FOR MULTILINEAR COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR ON HARDY AND

HERZ-HARDY SPACES

Jiasheng Zeng

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 the Littlewood-Paley operator are obtained.

2000Mathematics Subject Classification: 42B20, 42B25.

1. Introduction and definition

Let 0 < q < ∞ and L^q_loc(Rⁿ) = {f^q is locally integrable on Rⁿ}. Suppose f ∈ L¹_loc(Rⁿ), B = B(x0, r) = {x ∈ Rⁿ : |x−x0| < r} denotes a ball of Rⁿ centered at x₀ and having radius r, write f_B = |B|⁻¹R

Bf(x)dx and f^#(x) = sup_x∈B|B|⁻¹R

B|f(x)−fB|dx < ∞. f is said to belong to BM O(Rⁿ), if f^# ∈ L^∞(Rⁿ) and define ||f||_{BM O} =||f^#||_L^∞.

LetT be the Calder´on-Zygmund singular integral operator andb∈BM O(Rⁿ).

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[2]) proved that the commu- tator [b, T] is bounded on L^r(Rⁿ) for any 1 < r < ∞. However, it was observed that the [b, T] is not bounded, in general, fromH^p(Rⁿ) to L^p(Rⁿ) when 0< p≤1.

But if H^p(Rⁿ) is replaced by a suitable atomic space H^p

~b(Rⁿ)(see [1][7][12]), then [b, T] maps continuously H^p

~b(Rⁿ) intoL^p(Rⁿ), and a similar results holds for Herz- type spaces. In addition we have easily known that H_~b^p(Rⁿ) ⊂H^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][3-10][12][13]).

Definition 1.Let ε >0,n > δ >0and ψbe 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 commutator is defined by

S^~b_δ(f)(x) =

"

Z Z

Γ(x)

|F_t^~b(f)(x, y)|²dydt tⁿ⁺¹

#1/2

,

where

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

Rⁿ





m

Y

j=1

(b_j(x)−b_j(z))



ψ_t(y−z)f(z)dz,

Γ(x) = {(y, t) ∈ Rⁿ⁺¹₊ : |x −y| < t} and ψ_t(x) = t^−n+δψ(x/t) for t > 0. Set F_t(f)(y) =R

Rⁿψ_t(y−x)f(x)dx. We also define that S_δ(f)(x) =

Z Z

Γ(x)

|F_t(f)(x, y)|²dydt tⁿ⁺¹

!1/2

, which is the Littlewood-Paley operator with δ = 0(see [15]).

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

~b_σ = (b_σ(1),· · ·, b_σ(j)),bσ =b_σ(1)· · ·b_σ(j)and||~b_σ||_{BM O}=||b_σ(1)||_{BM O}· · · ||b_σ(j)||_{BM O}. Definition 2.Let b_i (i= 1,· · · , m) be a locally integrable functions and0< p≤ 1. A bounded measurable functiona onRⁿ is called a (p,~b) atom, if

(1) supp a⊂B =B(x0, r) (2) ||a||_L^∞ ≤ |B(x₀, r)|^−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(see [14][15]) f is said to belong to H_~b^p(Rⁿ), if, in the Schwartz distribution sense, it can be written as

f(x) =

∞

X

j=1

λjaj(x).

where a⁰_js are (p,~b) atoms, λj ∈ C and P∞

j=1|λ_j|^p < ∞. Moreover, ||f||_H^p

~b

≈ (P∞

j=1|λ_j|^p)^1/p.

Definition 3.Let0< p, q <∞, α∈R.Fork∈Z, setBk={x∈Rⁿ:|x| ≤2^k}

(1) The homogeneous Herz space is defined by K˙_q^α,p(Rⁿ) =

n

f ∈L^q_loc(Rⁿ\ {0}) :||f||_K˙^α,p

q <∞o .

where

||f||_K˙α,p

q =

" _∞ X

k=−∞

2^kαp||f χ_k||^p_Lq

#1/p

.

(2) The nonhomogeneous Herz space is defined by K_q^α,p(Rⁿ) =n

f ∈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 χ₀||^p_Lq

#1/p

.

Definition 4.Let α∈Rⁿ, 1< q <∞, α≥n(1−1/q), b_i ∈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) supp a⊂B =B(x₀, r)(or for somer ≥1), (2) ||a||_L^q ≤ |B(x₀, r)|^−α/n,

(3) R

Ba(x)x^βdx =R

Ba(x)x^βQ

i∈σbi(x)dx= 0 for any σ ∈C_j^m ,1 ≤ j ≤ m, 0 ≤ |β| ≤ α, where β = (β₁, ..., β_n) is the multi-indices with β_i ∈ N for 1 ≤i ≤n and |β|=Pn

i=1βi.

A temperate distributionf is said to belong toHK˙^α,p

q,~b(Rⁿ)(orHK^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.

2. Theorems and Proofs To prove the theorems, we need the following lemmas.

Lemma 1.(see [12])Let1< p < q < n/α, 1/q= 1/p−α/n. Then Sδis bounded from L^p(Rⁿ) to L^q(Rⁿ).

Lemma 2.(see [12])Let1< p < q < n/α,1/q= 1/p−α/n. ThenS^~b_δ is bounded from L^p(Rⁿ) to L^q(Rⁿ).

Theorem 1.Letε >0,n/(n+ε−δ)< p≤1, 1/q= 1/p−δ/n,bi ∈BM O, 1≤ i ≤ m, ~b = (b₁,· · ·, b_m). Then the multilinear commutator S^~b_δ 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,

||S^~b_δ(a)||_Lq ≤C.

Let abe a (p,~b) atom supported on a ball B =B(x₀, d). When m = 1 see [7], and now we prove m >1. Write

Z

Rⁿ

|S^~b_δ(a)(x)|^qdx= Z

|x−x₀|≤2d

|S^~b_δ(a)(x)|^qdx+ Z

|x−x₀|>2d

|S^~b_δ(a)(x)|^qdx=I+II.

For I, taking r, s > 1 with q < s < n/δ and 1/r = 1/s−δ/n, by H¨older’s inequality and the (L^s, L^r)-boundedness ofS^~b_δ,we have

I ≤ Z

|x−x₀|≤2d

|S^~b_δ(a)(x)|^rdx

!q/r

· |B(x₀,2d)|^1−q/r

≤ C||S^~b_δ(a)(x)||^q_Lr · |B(x₀, d)|^1−q/r

≤ C||a||^q_Ls|B|^1−q/r

≤ C.

For II, denoting λ= (λ₁,· · · , λ_m) with λ_i = (b_i)_B, 1 ≤i≤m, where (b_i)_B =

|B(x₀, r)|⁻¹R

B(x0,r)bi(x)dx, by the vanishing moment ofa, we get II =

∞

X

k=1

Z

2^k+1r≥|x−x₀|>2^kd

|S^~b_δ(a)(x)|^qdx

≤ C

∞

X

k=1

|B(x₀,2^k+1d)|^1−q Z

2^k+1d≥|x−x0|>2^kd

|S^~b_δ(a)(x)|dx

!q

≤ C

∞

X

k=1

|B(x₀,2^k+1d)|^1−q

×





 Z

2^k+1d≥|x−x₀|>2^kd



 Z Z

Γ(x)

| Z

B m

Y

j=1

(bj(x)−bj(z))ψt(y−z)a(z)dz|²dydt tⁿ⁺¹





1/2

dx







q

≤ C

∞

X

k=1

|B(x₀,2^k+1d)|^1−q[ Z

2^k+1d≥|x−x0|>2^kd

×



 Z Z

Γ(x)



 Z

B

|ψ_t(y−z)−ψt(y−x0)|

m

Y

j=1

|(b_j(x)−bj(z))||a(z)|dz





2

dydt tⁿ⁺¹





1/2

dx]^q;

noting that z∈B, x∈B(x₀,2^k+1d)\B(x₀,2^kd), then S^~b_δ(a)(x)

=



 Z Z

Γ(x)



 Z

B

|ψ_t(y−z)−ψt(y−x0)|

m

Y

j=1

|b_j(x)−bj(z)||a(z)|dz





2

dydt tⁿ⁺¹





1/2

≤ C



 Z Z

Γ(x)



 Z

B

t^−n+δ|a(z)|

m

Y

j=1

|b_j(x)−bj(z)| (|x₀−z|/t)^ε

(1 +|x₀−y|/t)^n+1+ε−δdz





2

dydt tⁿ⁺¹





1/2

≤ C

Z Z

Γ(x)

t¹⁻ⁿ

(t+|x₀−y|)^{2(n+1+ε−δ)}dydt

!1/2

Z

B m

Y

j=1

|b_j(x)−b_j(z)||x₀−z|^ε|a(z)|dz

≤ C

Z Z

Γ(x)

t¹⁻ⁿ2^{2(n+1+ε−δ)}

(2t+ 2|x₀−y|)^{2(n+1+ε−δ)}dydt

!1/2

Z

B m

Y

j=1

|b_j(x)−bj(z)||x₀−z|^ε|a(z)|dz;

Notice that 2t+|x₀−y|>2t+|x₀−x| − |x−y|> t+|x₀−x|when |x−y|< t,

and it is easy to calculate that Z ∞

0

tdt

(t+|x−x0|)^{2(n+1+ε−δ)} =C|x−x0|^{−2(n+ε−δ)}; then, we deduce

S^~b_δ(a)(x)

≤ C

Z Z

Γ(x)

t¹⁻ⁿ

(2t+|x₀−y|)^{2(n+1+ε−δ)}dydt

!1/2

Z

B m

Y

j=1

|b_j(x)−bj(z)||x₀−z|^ε|a(z)|dz

≤ C

Z Z

Γ(x)

t¹⁻ⁿ

(t+|x−x₀|)^{2(n+1+ε−δ)}dydt

!1/2

Z

B m

Y

j=1

|b_j(x)−b_j(z)||x₀−z|^ε|a(z)|dz

≤ C

Z ∞ 0

tdt

(t+|x−x0|)^{2(n+1+ε−δ)} 1/2Z

B m

Y

j=1

|b_j(x)−bj(z)||x₀−z|^ε|a(z)|dz

≤ C|B|^ε/n−1/p|x−x₀|^{−(n+ε−δ)} Z

B m

Y

j=1

|b_j(x)−b_j(z)|dz.

So

II ≤ C|B|^{(ε/n−1/p)q}

∞

X

k=1

|B(x₀,2^k+1r)|^1−q

×



 Z

2^k+1d≥|x−x0|>2^kd

|x−x0|^{−(n+ε−δ)}



 Z

B m

Y

j=1

|b_j(x)−bj(z)|dz



dx





q

≤ C|B|^{(ε/n−1/p)q}

∞

X

k=1

|B(x₀,2^k+1r)|^1−q

×





m

X

j=0

X

σ∈C_j^m

Z

2^k+1d≥|x−x₀|>2^kd

|x−x0|^{−(n+ε−δ)}|(~b(x)−λ)σ|dx Z

B

|(~b(z)−λ)σ^c|dz





q

≤ C|B|^{(ε/n−1/p)q}

m

X

j=0

X

σ∈C_j^m

Z

B

|(~b(z)−λ)_σ^c|dz q

×

∞

X

k=1

|B(x0,2^k+1d)|^1−q

"

Z

2^k+1d≥|x−x0|>2^kd

|x−x0|^{−(n+ε−δ)}|(~b(x)−λ)σ|dx

#q

≤ C

m

X

j=0

X

σ∈C_j^m

||~b_σ^c||^q_{BM O}· ||~b_σ||^q_{BM O}

∞

X

k=1

|B(x₀,2^k+1d)|1−(n+ε−δ)q/n

k^q|B|(1+ε/n−1/p)q

≤ C||~b||^q_{BM O}

∞

X

k=1

k^q·2k(n−q−qn+qδ)

≤ C.

This finishes the proof of Theorem 1.

Theorem 2. Let ε > 0, 0 < p < ∞, 1 < q1, q2 < ∞, 1/q1 −1/q2 = δ/n, n(1−1/q₁)≤α < n(1−1/q₁) +ε andb_i ∈BM O(Rⁿ),1≤i≤m,~b= (b₁,· · · , b_m).

Then S^~b_δ is bounded from HK˙_q^α,p_1,DmA(Rⁿ) to K˙q^α,p2 (Rⁿ).

Proof. Let f ∈HK˙^α,p

q1,~b(Rⁿ) and f(x) =P∞

j=−∞λjaj(x) be the atomic decom-

position for f as in Definition 4, we write

||S^~b_δ(f)(x)||_K˙α,p q2 =

∞

X

k=−∞

2^kαp||S^~b_δ(f)χ_k||^p_Lq2

!1/p

≤ C





∞

X

k=−∞

2^kαp(

∞

X

j=−∞

|λ_j|||S^~b_δ(aj)χk||_L^q2)^p





1/p

≤ C





∞

X

k=−∞

2^kαp(

k−3

X

j=−∞

|λ_j|||S^~b_δ(aj)χk||_L^q2)^p





1/p

+C





∞

X

k=−∞

2^kαp(

∞

X

j=k−2

|λ_j|||S^~b_δ(aj)χ_k||_L^q2)^p





1/p

= I+II.

For II, by the (L^q1, L^q2)-boundedness ofS^~b_δ and the H¨older’s inequality, we have

II ≤ C





∞

X

k=−∞

2^kαp





∞

X

j=k−2

|λ_j|||S^~b_δ(a_j)χ_j||_L^q2





p



1/p

≤ C





∞

X

−∞

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|^pPj+2

k=−∞2^(k−j)p/2i1/p

, 1< p <∞

≤ C





∞

X

j=−∞

|λ_j|^p





1/p

≤ C||f||_HK˙^α,p

q1,~b

.

ForI, when m=1, let b¹_j =|B_j|⁻¹R

Bjb1(x)dx. We have S_δ^b1(a_j)(x) =

"

Z Z

Γ(x)

| Z

Bj

(b₁(x)−b₁(z))ψ_t(y−z)a_j(z)dz|²dydt tⁿ⁺¹

#1/2

≤



 Z Z

Γ(x)

Z

Bj

|ψ_t(y−z)−ψ_t(y)||b₁(y)−b₁(z)||a_j(z)|dz

!2

dydt tⁿ⁺¹





1/2

≤ C

Z Z

Γ(x)

t¹⁻ⁿ

(t+|y|)^{2(n+1+ε−δ)}dydt

!1/2

Z

Bj

|z|^ε|a_j(z)||b₁(x)−b1(z)|dz

≤ C

Z Z

Γ(x)

t¹⁻ⁿ

(t+|x|)^{2(n+1+ε−δ)}dydt

!1/2

Z

Bj

|z|^ε|a_j(z)||b₁(x)−b1(z)|dz

≤ C

Z ∞ 0

tdt

(t+|x|)^{2(n+1+ε−δ)}

1/2 Z

Bj

|z|^ε|a_j(z)||b₁(x)−b1(z)|dz

!

≤ C|x|^{−(n+ε−δ)} Z

Bj

|z|^ε|a_j(z)|b₁(x)−b1(z)|dz

≤ C|x|^{−(n+ε−δ)} Z

Bj

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

Z

Bj

|z|^ε|a_j(z)||b₁(z)−b¹_j|dz

≤ 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} .

So

||S_δ^b1(aj)χk||_L^q2

≤ C2j(ε+n(1−1/q₁)−α)

[ Z

Bk

|b₁(x)−b¹_j||x|^{−q2(n+ε−δ)}dx 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}2[j(ε+n(1−1/q₁)−α)−k(n+ε−δ)+kn/q₂];

thus

I = C





∞

X

j=−∞

2^kαp





k−3

X

j=−∞

|λ_j|||S_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|2[j(ε+n(1−1/q₁)−α)−k(n+ε−δ)+kn/q₂]





p



1/p

≤ C||b₁||_{BM O}

×













 hP∞

k=−∞2^kαpPk−3

j=−∞|λ_j|^p2[j(ε+n(1−1/q₁)−α)−k(n+ε−δ)+kn/q₂]pi1/p

, 0< p≤1 hP∞

k=−∞2^kαp Pk−3

j=−∞|λ_j|^p2p[j(ε+n(1−1/q1)−α)−k(n+ε−δ)+kn/q2]/2

× Pk−3

j=−∞2^p0[j(ε+n(1−1/q1)−α)−k(n+ε−δ)+kn/q2]/2p/p⁰1/p

, 1< p <∞

≤ C||b₁||_{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/2i1/p

, 1< p <∞

≤ C||b₁||_{BM O}





∞

X

j=−∞

|λ_j|^p





1/p

≤ C||f||_HK˙α,p q1,~b

.

When m >1, Let bⁱ_j =|B_j|⁻¹R

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

"

Z Z

Γ(x)

| Z

Bj

m

Y

i=1

(b_i(x)−b_i(z))ψ_t(y−z)a_j(z)dz|²dydt tⁿ⁺¹

#1/2

≤



 Z Z

Γ(x)

Z

Bj

m

Y

i=1

|b_i(x)−b_i(z)||ψ_t(y−z)−ψ_t(y)||a_j(z)|dz

!2

dydt tⁿ⁺¹





1/2

≤ C|x|^{−(n+ε−δ)} Z

Bj

|z|^ε|a_j(z)|

m

Y

i=1

|b_i(x)−bi(z)|dz

≤ C|x|^{−(n+ε−δ)}

m

X

i=0

X

σ∈C_i^m

|(~b(x)−~b)_σ| Z

Bj

|z|^ε|a_j(z)||(~b(y)−~b)_σ^c|dz

≤ C|x|^{−(n+ε−δ)}

m

X

i=0

X

σ∈C_i^m

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

≤ C|x|^{−(n+ε−δ)}·2j(ε+n(1−1/q₁)−α) m

X

i=0

X

σ∈C_i^m

|(~b(x)−~b)_σ|||~b_σ^c||_{BM O};

So

||S^~b_δ(a_j)χ_k||_L^q

≤ C2j(ε+n(1−1/q1)−α)||~bσ^c||_{BM O}



 Z

Bk



|x|^−(n+ε)

m

X

i=0

X

σ∈C_i^m

|(~b(x)−~b)σ|





q

dx





1/q

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

then

I = C





∞

X

j=−∞

2^kαp





k−3

X

j=−∞

|λ_j|||S^~b_δ(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(n+ε−δ)+kn/q₂]





p



1/p

≤ C||~b||_{BM O}

×













 hP∞

k=−∞2^kαpPk−3

j=−∞|λ_j|^p2[j(ε+n(1−1/q₁)−α)−k(n+ε−δ)+kn/q₂]pi1/p

, 0< p≤1 hP∞

k=−∞2^kαp Pk−3

j=−∞|λ_j|^p2p[j(ε+n(1−1/q1)−α)−k(n+ε−δ)+kn/q2]/2

× Pk−3

j=−∞2^p0[j(ε+n(1−1/q₁)−α)−k(n+ε−δ)+kn/q₂]p/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/2i1/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.

References

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[12] C. P´erez, Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 128(1995), 163-185.

[13] C. P´erez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65(2002), 672-692.

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Jiasheng Zeng

Department of Mathematics Hunan Business College

Changsha, 410205, P. R. of China E-mail: [email protected]