WEIGHTED BOUNDEDNESS OF MULTILINEAR LITTLEWOOD{PALEY OPERATORS FOR THE EXTREME CASES OF p

Nova S´erie

WEIGHTED BOUNDEDNESS OF MULTILINEAR LITTLEWOOD–PALEY OPERATORS

FOR THE EXTREME CASES OF p Liu Lanzhe*

Recommended by A. Ferreira dos Santos

Abstract: In this paper, we prove the boundedness of multilinear Littlewood–Paley operators for the extreme cases ofp.

1 – Preliminaries and results

Throughout this paper, Q will denote a cube of Rⁿ with sides parallel to the axes. For a cube Q and a locally integral function f on Rⁿ, denote that f(Q) =^R_Qf(x)dx,fQ=|Q|^−1RQf(x)dxandf^#(x) = sup

x∈Q|Q|^−1RQ|f(y)−fQ|dy.

For a weight functions w∈A₁(see[10]), f is said to belong to BM O(w) iff^# ∈ L^∞(w) and define that kfkBM O(w) = kf^#kL^∞(w); If w = 1, we denote that BM O(Rⁿ) = BM O(w). Also, we give the concepts of atom and weighted H¹ space. A function a is called a H¹(w) atom if there exists a cube Q such that a is supported on Q, kakL^∞(w) ≤ w(Q)⁻¹ and ^R a(x)dx = 0. It is well known that, forw∈A₁, the weighted Hardy spaceH¹(w) has the atomic decomposition characterization (see[2]).

In this paper, we will consider a class of multilinear operators related to Littlewood–Paley operators, whose definition are the following.

Received: September 17, 2003; Revised: December 24, 2003.

AMS Subject Classification: 42B20, 42B25.

Keywords: Littlewood–Paley operator; multilinear operator; weighted Hardy space; BMO space.

* Supported by the NNSF (Grant: 10271071).

Let ψbe a fixed function which satisfies the following properties:

(1) ^R ψ(x)dx= 0,

(2) |ψ(x)| ≤C(1 +|x|)⁻⁽ⁿ⁺¹⁾,

(3) |ψ(x+y)−ψ(x)| ≤C|y|(1 +|x|)⁻⁽ⁿ⁺²⁾ when 2|y|<|x|.

Let m be a positive integer and A be a function on Rⁿ. We denote that Γ(x) ={(y, t)∈Rⁿ⁺¹₊ :|x−y|< t}and the characteristic function of Γ(x) by χ_Γ(x). The multilinear Littlewood–Paley operator is defined by

S_ψ^A(f)(x) =

"

Z

Γ(x)|F_t^A(f)(x, y)|²dy dt tⁿ⁺¹

#1/2

, where

F_t^A(f)(x, y) = Z

Rⁿ

R_m+1(A;x, z)

|x−z|^m f(z)ψ_t(y−z)dz , Rm+1(A;x, y) = A(x)− ^X

|α|≤m

1

α!D^αA(y) (x−y)^α ,

the derivatives D^αA are understood in the distributional sense and ψ_t(x) = t⁻ⁿψ(x/t) for t >0. We denote thatF_t(f) =f∗ψ_t. We also define

S_ψ(f)(x) = ÃZ

Γ(x)|Ft(f)(y)|² dy dt tⁿ⁺¹

!1/2

, which is the Littlewood–Paley operator (see [18]).

LetH be the Hilbert space H=ⁿh:khk=^{³R R}_Rn+1

+ |h(t)|²dy dt/t^n+1´1/2<∞^o. Then for each fixed x ∈ Rⁿ, F_t^A(f)(x, y) may be viewed as a mapping from (0,+∞) to H, and it is clear that

S^A_ψ(f)(x) =^°°_°χ_Γ(x)F_t^A(f)(x, y)^°°_° and S_ψ(f)(x) =^°°_°χ_Γ(x)F_t(f)(y)^°°_° . We also consider the variant ofS_ψ^A, which is defined by

S˜_ψ^A(f)(x) =

"

Z

Γ(x)|F˜_t^A(f)(x, y)|²dy dt tⁿ⁺¹

#1/2

, where

F˜_t^A(f)(x, y) = Z

Rⁿ

Q_m+1(A;x, z)

|x−z|^m ψ_t(y−z)f(z) dz and

Q_m+1(A;x, z) = R_m(A;x, z)− ^X

|α|=m

D^αA(x) (x−z)^α .

Note that whenm= 0,S_ψ^Ais just the commutator of Littlewood–Paley opera- tor (see [1],[15],[16]). It is well known that multilinear operators, as an extension of commutators, are of great interest in harmonic analysis and have been widely studied by many authors (see [4–9],[12–14]). In [11],[17], the endpoint bound- edness properties of commutators generated by the Calderon–Zygmund operator or fractional integral operator with BMO functions are obtained. The main pur- pose of this paper is to study the boundedness of the multilinear Littlewood–Paley operators for the extreme cases of p. We shall prove the following theorems in Section 3.

Theorem 1. Let D^αA ∈ BM O(Rⁿ) for |α| = m and w ∈ A₁. Then S_ψ^A mapsL^∞(w) continuously intoBM O(w).

Theorem 2. Let D^αA ∈ BM O(Rⁿ) for |α| = m and w ∈ A₁. Then S˜_ψ^A mapsH¹(w) continuously intoL¹(w).

Theorem 3. Let D^αA ∈ BM O(Rⁿ) for |α| = m and w ∈ A₁. Then S_ψ^A mapsH¹(w) continuously into weakL¹(w).

Remark. In general,S_ψ^A is not (H¹(w), L¹(w)) bounded.

2 – Some Lemmas

We begin with some preliminary lemmas.

Lemma 1 (see [7]). Let A be a function on Rⁿ and D^αA ∈ L^q(Rⁿ) for

|α|=m and someq > n. Then

|R_m(A;x, y)| ≤ C|x−y|^{m X}

|α|=m

à 1

|Q(x, y)˜ | Z

Q(x,y)˜ |D^αA(z)|^qdz

!1/q

,

whereQ(x, y)˜ is the cube centered atx and having side length 5√

n|x−y|. Lemma 2 (see [3]). LetT_b be the commutator defined by

T_bf(x) =

Z b(x)−b(y)

|x−y|ⁿ f(y)dy .

If w∈A₁,1< p <∞ and b∈BM O(Rⁿ). Then T_b is bounded on L^p(w).

Lemma 3 (see [8],[9]). LetT_A be the multilinear operator defined by T_Af(x) =

Z

Rⁿ

R_m+1(A;x, y)

|x−y|^m+n f(y)dy .

If w∈A1,1< p <∞,1< r ≤ ∞, 1/q = 1/p+ 1/r and D^αA ∈BM O(Rⁿ) for

|α|=m. Then T_A is bounded from L^p(w) toL^q(w), that is kTA(f)kL^q(w)≤CkfkL^p(w) .

Lemma 4. Let w ∈ A₁, 1 < p < ∞, 1 < r ≤ ∞, 1/q = 1/p+ 1/r and D^αA∈BM O(Rⁿ) for|α|=m. Then S_ψ^A is bounded fromL^p(w) toL^q(w), that is

kS_ψ^A(f)kL^q(w) ≤ C ^X

|α|=m

kD^αAkBM OkfkL^p(w) .

Proof: By Minkowski inequality and the condition ofψ, we have S_ψ^A(f)(x)≤

Z

Rⁿ

|f(z)| |Rm+1(A;x, z)|

|x−z|^m

ÃZ

Γ(x)|ψt(y−z)|² dydt t¹⁺ⁿ

!1/2

dz

≤C Z

Rⁿ

|f(z)| |Rm+1(A;x, z)|

|x−z|^m

ÃZ ∞ 0

Z

|x−y|≤t

t⁻²ⁿ (1 +|y−z|/t)²ⁿ⁺²

dydt t¹⁺ⁿ

!1/2

dz

≤C Z

Rⁿ

|f(z)| |Rm+1(A;x, z)|

|x−z|^m

ÃZ ∞ 0

Z

|x−y|≤t

2²ⁿ⁺²·t¹⁻ⁿ

(2t+|y−z|)²ⁿ⁺²dydt

!1/2

dz , noting that 2t+|y−z| ≥2t+|x−z| − |x−y| ≥t+|x−z|when|x−y| ≤tand

Z _∞

0

t dt

(t+|x−z|)²ⁿ⁺² = C|x−z|⁻²ⁿ , we obtain

S_ψ^A(f)(x) ≤ C Z

Rⁿ

|f(z)| |R_m+1(A;x, z)|

|x−z|^m

µZ _∞

0

t dt (t+|x−z|)²ⁿ⁺²

¶1/2

dz

= C Z

Rⁿ

|f(z)| |Rm+1(A;x, z)|

|x−z|^m+n dz , thus, the lemma follows from Lemma 3.

3 – Proofs of Theorems

Proof of Theorem 1: We have only to prove that there exists a constant C_Q such that

1 w(Q)

Z

Q|S_ψ^A(f)(x)−C_Q|w(x) dx ≤ CkfkL^∞(w)

holds for any cube Q. Fix a cube Q = Q(x₀, l). Let ˜Q = 5√

nQ and ˜A(x) = A(x)−^P

|α|=m 1

α!(D^αA)Q˜x^α, thenR_m(A;x, y) =R_m( ˜A;x, y) andD^αA˜=D^αA−(D^αA)Q˜

for|α|=m. We write, for f₁=f χ_Q˜ and f₂ =f χ_Rn\Q˜, F_t^A(f)(x) = F_t^A(f₁)(x) +F_t^A(f₂)(x), then

1 w(Q)

Z

Q|S_ψ^A(f)(x)−S_ψ^A(f₂)(x₀)|w(x) dx =

= 1

w(Q) Z

Q

¯

¯

¯kχ_Γ(x)F_t^A(f)(x, y)k − kχ_Γ(x)F_t^A(f₂)(x₀, y)k^¯¯¯w(x) dx

≤ 1

w(Q) Z

Q

S_ψ^A(f₁)(x)w(x) dx

+ 1

w(Q) Z

Q

°

°

°χ_Γ(x)F_t^A(f2)(x, y)−χ_Γ(x)F_t^A(f2)(x0, y)^°°_°w(x)dx := I+II .

Now, let us estimate I and II . First, by the L^∞ boundedness of S_ψ^A (see Lemma 4), we get

I ≤ kS_ψ^A(f₁)kL^∞(w) ≤ CkfkL^∞(w) . To estimateII, we write

χ_Γ(x)F_t^A(f₂)(x, y)−χ_Γ(x0)F_t^A(f₂)(x₀, y) =

=

Z · 1

|x−z|^m − 1

|x₀−z|^m

¸

χ_Γ(x)ψt(y−z)Rm( ˜A;x, z)f2(z) dz +

Z χ_Γ(x)ψ_t(y−z)f₂(z)

|x₀−z|^m

hRm( ˜A;x, z)−Rm( ˜A;x0, z)ⁱdz +

Z

(χ_Γ(x)−χ_Γ(x0))ψ_t(y−z)R_m( ˜A;x₀, z)f₂(z)

|x₀−z|^m dz

− ^X

|α|=m

1 α!

Z "

χ_Γ(x)(x−z)^α

|x−z|^m − χ_Γ(x0)(x₀−z)^α

|x₀−z|^m

#

ψt(y−z)D^αA(z)˜ f2(z) dz := II₁^t(x) +II₂^t(x) +II₃^t(x) +II₄^t(x).

Note that|x−z| ∼ |x₀−z|forx ∈Qand z∈Rⁿ\Q, similarly to the proof of Lemma 4 and by Lemma 1, we have

1 w(Q)

Z

QkII₁^t(x)kw(x) dx ≤

≤ C

w(Q) Z

Q

ÃZ

Rⁿ\Q˜

|x−x₀| |f(z)|

|x−z|^n+m+1 |R_m( ˜A;x, z)|dz

!

w(x) dx

≤ C

w(Q) Z

Q

Ã_∞ X

k=0

Z

2^k+1Q\2˜ ^kQ˜

|x−x₀| |f(z)|

|x−z|^n+m+1 |R_m( ˜A;x, z)|dz

!

w(x) dx

≤ C

∞

X

k=0

k l(2^kl)^m (2^kl)^n+m+1

X

|α|=m

kD^αAkBM O

µZ

2^kQ˜|f(z)|dz

¶

≤ C ^X

|α|=m

kD^αAkBM OkfkL^∞(w)

∞

X

k=0

k2^−k

≤ C ^X

|α|=m

kD^αAkBM OkfkL^∞(w) ; ForII₂^t(x), by the formula (see [7]):

R_m( ˜A;x, z)−R_m( ˜A;x₀, z) =R_m( ˜A;x, x₀) + ^X

0<|β|<m

1

β!R_m−|β|(D^βA;˜ x₀, z)(x−x₀)^β and Lemma 1, we get

|R_m( ˜A;x, z)−R_m( ˜A;x₀, z)| ≤

≤ C ^X

|α|=m

kD^αAkBM O

µ

|x−x₀|^m+ ^X

0<|β|<m

|x₀−z|^m−|β||x−x₀|^|β|´, thus, forx∈Q,

kII₂^t(x)k ≤ C Z

Rⁿ

|f₂(z)|

|x−z|^m+n|R_m( ˜A;x, z)−R_m( ˜A;x₀, z)|dz

≤ C ^X

|α|=m

kD^αAkBM O

Z

Rⁿ

|x−x₀|^m+ ^X

0<|β|<m

|x₀−z|^m−|β||x−x₀|^|β|

|x₀−z|^m+n |f₂(z)|dz

≤ C ^X

|α|=m

kD^αAk^{BM O}kfkL^∞(w)

∞

X

k=0

kl^m (2^kl)^m+n

Z

2^kQ˜|f(z)|dz

≤ C ^X

|α|=m

kD^αAkBM OkfkL^∞(w)

∞

X

k=1

k2^−km

≤ C ^X

|α|=m

kD^αAkBM OkfkL^∞(w) ;

For II₃^t(x), note that |x+y−z| ∼ |x₀ +y−z| for x ∈ Q and z ∈ Rⁿ\Q, we obtain, similarly to the estimate ofII₁^t(x),

kII₃^t(x)k ≤

≤C Z

Rⁿ



 ZZ

Rⁿ⁺¹₊

"

|ψ_t(y−z)||f₂(z)||R_m( ˜A;x₀, z)|

|x₀−z|^m |χ_Γ(x)(y, t)−χ_Γ(x0)(y, t)|

#2

dydt tⁿ⁺¹





1/2

dz

≤C Z

Rⁿ

|f₂(z)||R_m( ˜A;x₀, z)|

|x₀−z|^m

¯

¯

¯

¯

¯ ZZ

Γ(x)

t¹⁻ⁿdydt (t+|y−z|)²ⁿ⁺² −

ZZ

Γ(x0)

t¹⁻ⁿdydt (t+|y−z|)²ⁿ⁺²

¯

¯

¯

¯

¯

1/2

dz

≤C Z

Rⁿ

|f₂(z)||R_m( ˜A;x₀, z)|

|x₀−z|^m

· ÃZZ

|y|≤t

¯

¯

¯

¯

1

(t+|x+y−z|)²ⁿ⁺² − 1

(t+|x₀+y−z|)²ⁿ⁺²

¯

¯

¯

¯ dydt tⁿ⁻¹

!1/2

dz

≤C Z

Rⁿ

|f₂(z)| |R_m( ˜A;x₀, z)|

|x₀−z|^m

ÃZ Z

|y|≤t

|x−x₀|t¹⁻ⁿdydt (t+|x+y−z|)²ⁿ⁺³

!1/2

dz

≤C Z

Rⁿ

|f₂(z)| |x−x₀|^1/2|R_m( ˜A;x₀, z)|

|x0−z|^m+n+1/2 dz

≤C

∞

X

k=0

kl^1/2(2^kl)^m (2^kl)^n+m+1/2

X

|α|=m

kD^αAkBM O

µZ

2^kQ˜|f(z)|dz

¶

≤C ^X

|α|=m

kD^αAkBM OkfkL^∞(w)

∞

X

k=0

k2^−k/2

≤C ^X

|α|=m

kD^αAkBM OkfkL^∞(w) ;

ForII₄^t(x), similarly to the estimates of II₁^t(x) and II₃^t(x), we get

kII₄^t(x)k ≤ C Z

Rⁿ\Q˜

à |x−x₀|

|x−z|ⁿ⁺¹ + |x−x₀|^1/2

|x−z|^n+1/2

! X

|α|=m

|D^αA(z)˜ | |f(z)|dz

≤ C ^X

|α|=m

kD^αAkBM OkfkL^∞(w)

∞

X

k=0

k(2^−k+ 2^−k/2)

≤ C ^X

|α|=m

kD^αAk^{BM O}kfkL^∞(w) .

Combining these estimates, we complete the proof of Theorem 1.

Proof of Theorem 2: It suffices to show that there exists a constantC >0 such that for everyH¹(w)-atoma, the following holds:

kS˜_ψ^A(a)kL¹(w) ≤C . We write

Z

Rⁿ

S˜_ψ^A(a)(x)w(x) dx =

"

Z

2Q

+ Z

(2Q)^c

#

S˜_ψ^A(a)(x)w(x)dx := J+JJ . ForJ, by the following equality

Q_m+1(A;x, y) = R_m+1(A;x, y)−^X

|α|=m

1

α!(x−y)^α³D^αA(x)−D^αA(y)^´ , we get, similarly to the proof of Lemma 4,

S˜_ψ^A(a)(x) ≤ S_ψ^A(a)(x) + C ^X

|α|=m

Z |D^αA(x)−D^αA(y)|

|x−y|ⁿ |a(y)|dy , thus, ˜S_ψ^A isL^∞-bounded by Lemma 2 and Lemma 4. We see that

J ≤ CkS˜_ψ^A(a)kL^∞(w)w(2Q) ≤ CkakL^∞(w)w(Q) ≤ C . To obtain the estimate ofJJ, we denote that ˜A(x) =A(x)−^P|α|=m 1

α!(D^αA)_2Qx^α. Then Q_m(A;x, y) = Q_m( ˜A;x, y). We write, by the vanishing moment of a and Q_m+1(A;x, y) =R_m(A;x, y)−^P|α|=m 1

α!(x−y)^αD^αA(x), for x∈(2Q)^c, F˜_t^A(a)(x, y) =

Z ψ_t(y−z)R_m( ˜A;x, z)

|x−z|^m a(z) dz

− ^X

|α|=m

1 α!

Z ψt(y−z)D^αA(z) (x˜ −z)^α

|x−z|^m a(z)dz

= Z "

ψt(y−z)Rm( ˜A;x, z)

|x−z|^m −ψt(y−x0)Rm( ˜A;x, x0)

|x−x₀|^m

#

a(z) dz

−^X

|α|=m

1 α!

Z ·ψ_t(y−z)(x−z)^α

|x−z|^m −ψ_t(y−x₀)(x−x₀)^α

|x−x0|^m

¸

D^αA(x)˜ a(z)dz , thus, similarly to the proof ofII in Theorem 1, we obtain

kF˜_t^A(a)(x, y)k ≤C|Q|^1+1/n w(Q)



 X

|α|=m

kD^αAkBM O|x−x₀|⁻ⁿ⁻¹+|x−x₀|⁻ⁿ⁻¹|D^αA(x)˜ |



;

Note that if w ∈ A1, then ^w(Q_|Q²⁾

2|

|Q1|

w(Q1) ≤C for all cubes Q1, Q2 with Q1 ⊂ Q2. Thus, by Holder’inequality and the reverse of Holder’ inequality for w ∈ A1, choosep >1 and 1/p+ 1/p⁰ = 1, we obtain

JJ ≤ C ^X

|α|=m

kD^αAkBM O

∞

X

k=1

2^−k à |Q|

w(Q)

w(2^k+1Q)

|2^k+1Q|

!

+C ^X

|α|=m

∞

X

k=1

2^−k |Q| w(Q)

µ 1

|2^k+1Q| Z

2^k+1Q|D^αA(x)˜ |^pdx

¶1/p

·

µ 1

|2^k+1Q| Z

2^k+1Q

w(x)^p0dx

¶1/p⁰

≤ C ^X

|α|=m

kD^αAkBM O

∞

X

k=1

k2^−k

Ãw(2^k+1Q)

|2^k+1Q|

|Q| w(Q)

!

≤ C ,

which together with the estimate ofJ yields the desired result. This finishes the proof of Theorem 2.

Proof of Theorem 3: By the equality R_m+1(A;x, y) = Q_m+1(A;x, y) + ^X

|α|=m

1

α!(x−y)^α³D^αA(x)−D^αA(y)^´ and similarly to the proof of Lemma 4, we get

S_ψ^A(f)(x) ≤ S˜_ψ^A(f)(x) + C ^X

|α|=m

Z |D^αA(x)−D^αA(y)|

|x−y|ⁿ |f(y)|dy , by Theorem 1,2 and Lemma 2, we obtain

w µn

x∈Rⁿ: S_ψ^A(f)(x)> λ^o

¶

≤

≤ w µn

x∈Rⁿ: ˜S_ψ^A(f)(x)> λ/2^o

¶

+w











x∈Rⁿ: ^X

|α|=m

Z |D^αA(x)−D^αA(y)|

|x−y|ⁿ |f(y)|dy > Cλ











≤ CkfkH¹(w)/λ .

This completes the proof of Theorem 3.

ACKNOWLEDGEMENT– The author would like to express his deep gratitude to the referee for his very valuable comments and suggestions.

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Liu Lanzhe,

College of Mathematics and Computer, Changsha University of Science and Technology, Changsha 410077 – P.R. OF CHINA

E-mail: [email protected]