Paper-23-Acta24.2010. 151KB Jun 04 2011 12:03:13 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 24/2010
pp. 255-270

ENDPOINT ESTIMATES FOR MULTILINEAR COMMUTATOR OF
MARCINKIEWICZ OPERATOR

Gao Xinshan and Liu Lanzhe

Abstract. In this paper, we prove the endpoint estimates for the multilinear
commutator of Marcinkiewicz operator.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction
As the development of singular integral operators, their commutators have been
well studied. Let b ∈ BM O(Rn ) and T be the Calderón-Zygmund operator, the
commutator [b, T ] generated by b and T is defined by
[b, T ](f )(x) = b(x)T (f )(x) − T (bf )(x).
A classical result of Coifman, Rochberb and Weiss (see [3]) proved that the commutator [b, T ] is bounded on Lp (Rn ), (1 < p < ∞). In [2][5], the boundedness properties
of the commutators for the extreme values of p are obtained. And note that [b, T ]

is not bounded for the end point boundedness (that is p = 1 and p = ∞). In this
paper, we will introduce the multilinear commutator of Marcinkiewicz operator and
prove the boundedness properties of the operator for the extreme cases.
First let us introduce some notations (see [1][4][7][8]). In this paper, Q = Q(x, r)
will denote a cube of Rn with sides parallel to the axes and center atR x and edge is
r. For a cube Q and a locally integrable function f , let fQ = |Q|−1 Q f (x)dx and
R
f (Q) = Q f (x)dx, the sharp function of f is defined by
1
f (x) = sup
Q∋x |Q|
#

Z

Q

255

|f (y) − fQ |dy.

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...
f is said to belong to BM O(Rn ) if f # ∈ L∞ (Rn ) and define ||f ||BM O = ||f # ||L∞ .
We have |f2Q − fQ | ≤ C||f ||BM O and ||f − f2k Q ||BM O ≤ Ck||f ||BM O for k ≥ 1 (see
[4][8]). We also define the central BM O space by CM O(Rn ), which is the space of
those functions f ∈ Lloc (Rn ) such that
Z
−1
||f ||CM O = sup |Q(0, r)|
|f (y) − fQ |dy < ∞.
r>1

Q

It is well-known that
−1

||f ||CM O ≈ sup inf |Q(0, r)|
r>1 c∈C

Z

|f (x) − c|dx.

Q

Let M be the Hardy-Littlewood maximal operator, that is
Z
1
M (f )(x) = sup
|f (y)|dy.
Q∋x |Q| Q
Definition 1. A function a is called a H 1 (Rn )−atom, if there exists a cube Q,
such that
1) supp a ⊂ Q = Q(x0 , r),
2) R||a||L∞ ≤ |Q|−1 ,
3) Rn a(x)dx = 0.
It is well known that the Hardy space H 1 (Rn ) has the atomic decomposition
characterization (see [4][8]).
The Ap weight is defined by (see [4])

(
)

p−1

Z
Z
1
1
w(x)dx
w(x)−1/(p−1) dx
Ap = w : sup
< ∞ , 1 < p < ∞,
|Q| Q
|Q| Q
Q
and
A1 = {w : M (w)(x) ≤ Cw(x), a.e.}.
Definition 2. Let 0 < δ < n and 1 < p < n/δ. We shall call Bpδ (Rn ) the space
of those functions f on Rn such that

||f ||Bpδ = sup r−n(1/p−δ/n) ||f χQ(0,r) ||Lp < ∞.
r>1

n+1
We denote Γ(x) = {(y, t) ∈ R+
: |x − y| < t} and the characteristic function
of Γ(x) by χΓ(x) .
Definition 3. Let bj (j = 1, · · · , m) be the fixed locally integrable functions on
Rn , 0 < δ < n and 0 < γ ≤ 1. Suppose that S n−1 is the unit sphere of Rn (n ≥ 2)

256

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...
equipped with normalized Lebesgue measure dσ = dσ(x′ ). Let Ω be homogeneous of
degree zero and satisfy the following two conditions:
(i) Ω(x) is continuous on S n−1 and satisfies the Lipγ condition on S n−1 , i.e.
|Ω(x′ ) − Ω(y ′ )| ≤ M |x′ − y ′ |γ ,

x′ , y ′ ∈ S n−1 ;

R
(ii) S n−1 Ω(x′ )dx′ = 0;
The Marcinkiewicz multilinear commutator is defined by
~
µbs,δ (f )(x)

=

"Z Z

Γ(x)

dydt
~
|Ftb (f )(x, y)|2 n+3
t

#1/2

,

where
~

Ftb (f )(x, y) =

Z

|y−z|≤t

Set



m
Ω(y − z)  Y
(bj (x) − bj (z)) f (z)dz.
|y − z|n−1−δ
j=1

Ft (f )(y) =

Z

|y−z|≤t

Ω(y − z)
f (z)dz.
|y − z|n−1−δ

We also define that
dydt
|Ft (f )(y)|2 n+3
t
Γ(x)

Z Z

µs,δ (f )(x) =

!1/2

,

which is the Marcinkiewicz operator (see [5],[6] and [10]).
Remark. Fixed λ > max(1, 2n/(n+2−2δ)). Another Marcinkiewicz multilinear
operators is defined by
~
µbλ,δ (f )(x)

=

"Z Z

n+1
R+



t

t + |x − y|

nλ

dydt
~
|Ftb (f )(x, y)|2 n+3
t

#1/2

,

where
~

Ftb (f )(x, y) =
Set

Z

|y−z|≤t



Ω(y − z) 
|y − z|n−1−δ

Ft (f )(x) =

m
Y

j=1

Z

|x−y|≤t

(bj (x) − bj (z)) f (z)dz.

Ω(x − y)
f (y)dy.
|x − y|n−1−δ

257



G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

We also define
µλ (f )(x) =

Z Z

n+1
R+



t
t + |x − y|

nλ

dydt
|Ft (f )(y)|2 n+3
t

!1/2

,

which is another Marcinkiewicz operators.


1/2
R R
2
n+3
1, set ~bQ = ((b1 )Q , · · · , (bm )Q ) ∈ Rn , where (bj )Q = |Q|−1 Q bj (y)dy, 1 ≤

261

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

j ≤ m, we have
~

Ftb (f )(x, y) = (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)
+(−1)m Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(y)
Z
m−1
X X
Ω(y − z)
m−j
f (z)dz
(b(z) − b(x))σc
+
(−1)
(b(x) − (b)2Q )σ
|y − z|n−1−δ
|y−z|≤t
m
j=1 σ∈Cj

= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)
+(−1)m Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(y)
+

m−1
X

X

~

cm,j (b(x) − (b)2Q )σ Ftbσ (f )(x, y),
c

j=1 σ∈Cjm

thus,
~

|µbs,δ (f )(x) − µs,δ ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q ))f2 )(x0 )|
~

≤ ||χΓ(x) Ftb (f )(x, y) − χΓ(x0 ) Ft (((b1 )2Q − b1 ) · · · ((bm )2Q − bm )f2 )(y)||
≤ ||χΓ(x) (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)||
m−1
X

+

X

~

||χΓ(x) (b̃(x) − (bm )2Q )σ Ftbσ (f )(x, y)||
c

j=1 σ∈Cjm

+ ||χΓ(x) Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f1 )(y)||
m
m
Y
Y
+ ||χΓ(x) Ft ( (bj − (bj )2Q )f2 )(y) − χΓ(x0 ) Ft ( (bj − (bj )2Q )f2 )(y)||
j=1

j=1

= S1 (x) + S2 (x) + S3 (x) + S4 (x).

For S1 (x), taking 1 < p < n/δ, and 1/q = 1/p − δ/n, by the Hölder’s inequality and
Lemma 1,2,3, we have
1
|Q|

Z

Q



S1 (x)dx ≤ 

≤ C||~b||BM O |Q|−1/q

1
|Q|

Z

Q

≤ C||~b||BM O ||f ||Ln/δ .

Z

|

m
Y

Q j=1

′

1/q′

(bj (x) − (bj )Q )|q dx

1/p
p
|f (x)| dx
|Q|(1−(δp/n))/p

262



1
|Q|

Z

Q

|µs,δ (f )(x)|q dx

1/q

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

For S2 (x), taking 1 < p < n/δ and 1/q = 1/p − δ/n, then
Z
1
S2 (x)dx
|Q| Q
1/q′ 
1/q
Z
m−1
X X  1 Z
1
q′
q
~
~
~
~
≤ C
|(b(x) − bQ )σ | dx
|µs,δ ((b − bQ )σc )f )(x)| dx
|Q| Q
|Q| Q
m
j=1 σ∈Cj

≤ C

m−1
X

||~bσ ||BM O |Q|

1/q

j=1

≤ C

~ − ~bQ )σc )f (x)|p χQ (x)dx
|(b(x)



Z

Rn

m−1
X

X

||~bσ ||BM O

m−1
X

X

||~bσ ||BM O ||~bσc ||BM O ||f ||Ln/δ

j=1 σ∈Cjm

≤ C

Z

j=1

1
|Q|

|(~b(x) − ~bQ )σc |q dx

Q

1/q

1/p

||f ||Ln/δ

σ∈Cjm

≤ C||~b||BM O ||f ||Ln/δ .
For S3 (x), taking 1 < p < n/δ and 1/q = 1/p − δ/n, we get
Z
1
S3 (x)dx
|Q| Q

1/q
Z
1
q
≤
|µs,δ ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f1 )(x)| dx
|Q| Q

≤ C|Q|−1/q ||((b1 (x) − (b1 )Q ) · · · (bm (x) − (bm )Q )f1 (x)||Lp
1/q

Z
1
q
|(b1 − (b1 )Q ) · · · (bm − (bm )Q )| dx
||f ||Ln/δ
≤ C
|2Q| 2Q
≤ C||~b||BM O ||f || n/δ .
L

263

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

For S4 (x), similar to the proof of C(x) in Case m = 1, we obtain
S4 (x) ≤ C

∞ Z
X
k=1

1/2

2k+1 Q\2k Q

|x − x0 |

−(n+1/2−δ)

|x0 − z|

m
Y

|bj (z) − (bj )2Q ||f (z)|dz

j=1




m
Z
Y


1
−k/2
 (bj (z) − (bj )2Q ) |f (z)|dz
≤ C
2


1−δ/n
k+1
|2 Q|
2k+1 Q j=1

k=1


n/(n−δ) (n−δ)/n


Z
∞
m
X
Y

1

−k/2 


||f ||Ln/δ
dz 
≤ C
2
(b
(z)
−
(b
)
)
 k+1
j
j
2Q

|2 Q| 2k+1 Q 

∞
X

j=1

k=1

≤ C||~b||BM O ||f ||Ln/δ .

This completes the total proof of Theorem 1.
Theorem 2. Let 0 < δ < n, 1 < p < n/δ and ~b = (b1 , · · · , bm ) with bj ∈
~
BM O(Rn ) for 1 ≤ j ≤ m. Then µbs,δ is bounded from Bpδ (Rn ) to CM O(Rn ).
Proof. It suffices to prove that there exist constant CQ , such that
Z
1
~
|µbs,δ (f )(x) − CQ |dx ≤ C||f ||Bpδ
|Q| Q
holds for any cube Q = Q(0, d) with d > 1. Fix a cube Q = Q(0, d) with d >
1. SetR f1 = f χ2Q , f2 = f χRn \2Q and ~bQ = ((b1 )Q , · · · , (bm )Q ), where (bj )Q =
|Q|−1 Q |bj (y)|dy, 1 ≤ j ≤ m, we have
~

|µbs,δ (f )(x) − µs,δ ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q ))f2 )(x0 )|
~

≤ ||χΓ(x) Ftb (f )(x, y) − χΓ(x0 ) Ft (((b1 )2Q − b1 ) · · · ((bm )2Q − bm )f2 )(y)||
≤ ||χΓ(x) (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)||
+

m−1
X

X

~

||χΓ(x) (b̃(x) − (bm )2Q )σ Ftbσ (f )(x, y)||
c

j=1 σ∈Cjm

+ ||χΓ(x) Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f1 )(y)||
m
m
Y
Y
+ ||χΓ(x) Ft ( (bj − (bj )2Q )f2 )(y) − χΓ(x0 ) Ft ( (bj − (bj )2Q )f2 )(y)||
j=1

j=1

= H1 (x) + H2 (x) + H3 (x) + H4 (x).

Taking 1 < p < n/δ, 1/s = 1/r − δ/n, by the Hölder’s inequality and Lemma 1,2,3,
264

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

we have
Z
1
H1 (x)dx
|Q| Q

1/q′
1/q

Z Y
Z
m
1
1
′
≤ 
|
|µs,δ (f )(x)|q dx
(bj (x) − (bj )Q )|q dx
|Q| Q
|Q| Q
j=1

≤ C||~b||BM O |Q|

−1/q

Z

p

|f (x)| dx

Q
−n(1/p−δ/n)

≤ C||~b||BM O d
≤ C||~b||BM O ||f ||

1/p

||f χQ ||Lp

Bpδ .

For H2 (x), taking 1 < p < n/δ, 1/s = 1/r − δ/n, and 1/s′ + 1/s = 1, then
Z
1
H2 (x)dx
|Q| Q
1/s′ 
1/s
Z
m−1
X X  1 Z
1
s′
s
~
~
~
~
≤ C
|(b(x) − bQ )σ | dx
|µs,δ ((b − bQ )σc )f )(x)| dx
|Q| Q
|Q| Q
m
j=1 σ∈Cj

≤ C

m−1
X

||~bσ ||BM O |Q|−1/s

j=1

≤ C

Rn

~ − ~bQ )σc )f (x)|r χQ (x)dx
|(b(x)

m−1
X

X

||~bσ ||BM O

m−1
X

X

||~bσ ||BM O ||~bσc ||BM O d−n(1/p−δ/n) ||f χQ ||Lp

j=1 σ∈Cjm

≤ C

Z



1
|Q|

Z

|(~b(x) − ~bQ )σc |pr/(p−r) dx

Q

j=1 σ∈Cjm

≤ C||~b||BM O ||f ||Bpδ .

265

1/r

(p−r)/pr

|Q|(δ/n−1/p) ||f χQ ||Lp

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...
For H3 (x), taking 1 < p < n/δ , 1/s = 1/r − δ/n and 1/s′ + 1/s = 1, we get
Z
1
H3 (x)dx
|Q| Q

1/s
Z
1
s
≤
|µs,δ ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f1 )(x)| dx
|Q| Q

≤ C|Q|−1/s ||((b1 (x) − (b1 )Q ) · · · (bm (x) − (bm )Q )f χ2Q ||Lr
(p−r)/pr

Z
1
pr/(p−r)
|(b1 − (b1 )Q ) · · · (bm − (bm )Q )|
dx
≤ C
d−n(1/p−δ/n) ||f χ2Q ||Lp
|2Q| 2Q
≤ C||~b||BM O ||f || δ .
Bp

For H4 (x), we have
∞ Z
X
S4 (x) ≤ C
k=1

1/2

2k+1 Q\2k Q

|x − x0 |

1
≤ C
2−k/2 k+1 1−δ/n
|2 Q|
k=1
≤ C

×

Z

2−k/2

k=1

1
|2k+1 Q|1−δ/n
p

|f (z)| dz

2k+1 Q

|x0 − z|

m
Y

|bj (z) − (bj )2Q ||f (z)|dz

j=1

∞
X

∞
X

−(n+1/2−δ)

1/p



m


Y
 (bj (z) − (bj )2Q ) |f (z)|dz


2k+1 Q j=1


p/(p−1) (p−1)/p


m
Z

Y




dz 
(b
(z)
−
(b
)
)

j
j
2Q


k+1
2
Q j=1

Z

p/(p−1) (p−1)/p


Y
m


1

−k/2 
 (bj (z) − (bj )2Q )
dz 
≤ C
2
 k+1


|2 Q| 2k+1 Q 

j=1
k=1
∞
X



Z

×|2k+1 Q|−(1/p−δ/n) ||f χ2k+1 Q ||Lp
≤ C||~b||BM O ||f ||Bpδ .

This completes the total proof of Theorem 2.
Theorem 3. Let 0 < δ < n and ~b = (b1 , · · · , bm ) with bj ∈ BM O(Rn ) for
1 ≤ j ≤ m. If for any H 1 (Rn )−atom a supported on certain cube Q and u ∈ Q,
there is
Z
n/(n−δ)

m X Z
X


Ω(y
−
u)

|(b(x) − bQ )σc |  (~b(z) − ~bQ )σ a(z)dz
dx ≤ C,

n−1−δ
|y − u|
Q
m (2Q)c
j=1 σ∈Cj

266

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...
~

then µbs,δ is bounded from H 1 (Rn ) to Ln/(n−δ) (Rn ).
Proof. Let a be an atom supported in some cube Q. We write
Z
Z
Z
~
~
~
|µbs,δ (a)(x)|n/(n−δ) dx = I+II.
|µbs,δ (a)(x)|n/(n−δ) dx =
|µbs,δ (a)(x)|n/(n−δ) dx+
Rn

(2Q)c

2Q

For I, taking 1 < p < n/δ and 1/q = 1/p − δ/n, we have
~

n/(n−δ)

I ≤ ||µbs,δ (a)||Lq

n/(n−δ)

|2Q|1−n/((n−δ)q) ≤ C||a||Lp

|Q|1−n/((n−δ)q) ≤ C.

~

For II, we first calculate Ftb (a)(x), we have



Y
Z
m


Ω(y
−
z)
~b
a(z)dz 
|Ft (a)(x)| ≤  (bj (x) − (bj )Q )
n−1−δ
|y−z|≤t |y − z|

j=1




Z
m X 

X
Ω(y
−
u)
Ω(y
−
z)

~
~
~
~
c
+
(
b(z)
−
b
)
a(z)dz
−
(
b(x)
−
b
)


Q σ
Q σ
n−1−δ
n−1−δ


|y
−
z|
|y
−
u|
|y−z|≤t
j=1 σ∈Cjm


Z
m X 

X
Ω(y
−
u)
~
~b(z) − ~bQ )σ a(z)dz 
(
+
(b(x) − ~bQ )σc
n−1−δ


|y−z|≤t |y − u|
m
j=1 σ∈Cj

= ν1 + ν2 + ν3 ,

~
µbδ (a)(x)

=

~
||Ftb (a)(x)||

+

Z Z

Γ(x)

≤

Z Z

Γ(x)

|ν3 |2

dydt
tn+3

dydt
|ν1 |2 n+3
t

!1/2

!1/2

+

Z Z

Γ(x)

= A(x) + B(x) + C(x).

267

dydt
|ν2 |2 n+3
t

!1/2

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

For A(x), we have
!1/2 m

2
Y
 dydt
 |χΓ(x) − χΓ(y) |


|bj (x) − (bj )Q |
A(x) ≤ C
|a(z)|dz
 |y − z|n−1−δ
 tn+3
Rn
j=1
1/2
Z Z
Z

1
dt
dt
1


≤ C
|a(z)|dz
−

2n−2−2δ t3
2n−2−2δ t3 

|x
−
y|
|x
−
u|
n
R
|u−z|≤t
|x−z|≤t
Z Z

m
Y

×

|bj (x) − (bj )Q |

j=1

Z

≤ C

Rn

m
Y

×

 !1/2

 dt

1
1


|a(z)|dz
 |x + y − u|2n−2−2δ − |x|2n−2−2δ  t3
|x|≤t,|x+y−u|≤t

Z

|bj (x) − (bj )Q |

j=1

Z

≤ C

Rn

Z

≤ C

Rn

Z

|x|≤t,|x+y−u|≤t

!1/2

|a(z)|dz

m
Y
|y − u|1/2
|a(z)|dz
|bj (x) − (bj )Q |
|x + y − u|n+1/2−δ
j=1

1/2n

≤ C|Q|

|y − u|
dt
2n−1−2δ
t3
|x + y − u|

−(n+1/2−δ)

|x − u|

m
Y

m
Y

|bj (x) − (bj )Q |

j=1

|bj (x) − (bj )Q |,

j=1

thus
Z

!(n−δ)/n

(A(x))n/(n−δ) dx

(2Q)c

≤ C

∞
X
k=1




2−k/2 

≤ C||~b||BM O .

1
|2k+1 Q|

Z

2k+1 Q




m
Y

j=1

268

n/(n−δ)

|bj (x) − (bj )Q |

(n−δ)/n


dx

G. Xinshan, L. Lanzhe - Endpoint estimates for multilinear commutator of...

For B(x), we have
m X
X

(~b(x) − ~bQ )σc

Γ(x)

σ∈Cjm

j=1

≤

Z Z

m X
X

(~b(x) − ~bQ )σc

j=1 σ∈Cjm

× a(z)(~b(z) − ~bQ )σ dz

Z

Rn

Z

!1/2


 Ω(y − z)
 dydt
~
~


 |y − z|n−1−δ a(z)(b(z) − bQ )σ dz  tn+3

Rn

χΓ(y)



|Ω(x − y)|
|Ω(x − u)|
−
n−1−δ
|x − y|
|x − u|n−1−δ



2 dt 1/2
,
t3

similarly, we get
|B(x)| ≤ C

m X
X

|(~b(x) − ~bQ )σc |

j=1 σ∈Cjm

|Q|1/2n
||~bσ ||BM O ,
|x − u|n+1/2−δ

thus
Z

n/(n−δ)

(B(x))

!(n−δ)/n

dx

(2Q)c

≤ C

∞
m X X
X
j=1

σ∈Cjm

k=1

−k/2

2



1
|2k+1 Q|

Z

2k+1 Q

≤ C||~b||BM O .


n/(n−δ) (n−δ)/n
~

dx
||~bσ ||BM O
(b(x) − ~bQ )σc 

So, if
m X Z
X
j=1 σ∈Cjm

then

(2Q)c



 Z
 

Ω(y − u)  n/(n−δ)
~
~


|(b(x) − bQ )σc |  (b(z) − bQ )σ a(z)dz
dx ≤ C,
|y − u|n−1−δ 
Q

Z

Rn

~

|µbs,δ (a)(x)|n/(n−δ) dx ≤ C.

This completes the proof of the Theorem 3.
Remark. Theorem 1, 2 and 3 also hold for µb̃λ , we omit the details.
References
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[8] E. M. Stein, Harmonic analysis: real variable methods, orthogonality and
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Math., 60/61(1990), 235–243.
Gao Xinshan and Liu Lanzhe
Department of Mathematics
Changsha University of Science and Technology
Changsha, 410077, P. R. of China
E-mail: lanzheliu@163.com

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