Volume 2008, Article ID 373050, 17 pagesdoi:10.1155/2008/373050 Research Article A Class of Commutators for Multilinear Fractional Integrals in Nonhomogeneous Spaces Jiali Lian and Huoxi
Trang 1Volume 2008, Article ID 373050, 17 pages
doi:10.1155/2008/373050
Research Article
A Class of Commutators for Multilinear Fractional Integrals in Nonhomogeneous Spaces
Jiali Lian and Huoxiong Wu
School of Mathematical Sciences, Xiamen University, Xiamen Fujian, 361005, China
Correspondence should be addressed to Huoxiong Wu,huoxwu@xmu.edu.cn
Received 3 March 2008; Accepted 16 July 2008
Recommended by Nikolaos Papageorgiou
Let μ be a nondoubling measure onRd A class of commutators associated with multilinear fractional integrals and RBMOμ functions are introduced and shown to be bounded on product
of Lebesgue spaces with μ.
Copyrightq 2008 J Lian and H Wu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
In recent years, the study of multilinear operators and their commutator has been attracting many researchers Many results which parallel to the linear theory of classical integral
rapidly Many results of singular integrals and the related operators on Euclidean spaces with Lebesgue measure have been generalized to the Lebesgue spaces with nondoubling measures
of multilinear fractional integrals and RBMO functions with nondoubling measure, which
Before stating our results, we recall some definitions and notations Let μ be a Radon
n ∈ 0, d, such that
side length of Q For r > 0, rQ will denote the cube with the same center as Q and with
lrQ rlQ.
Trang 2Let 0≤ β < n, given two cubes Q ⊂ R in R d, we set
K Q,R β 1
NQ,R
k1
μ
2k Q
l
2k Qn
1−β/n
where N Q,R is the first integer k such that l2 k Q ≥ lR If β 0, then K0Q,R K Q,R The later
1
μηQ
Q
by − m
Q b dμy ≤ C1,
m Q b − m R b ≤ C1K Q,R , for any two doubling cubes Q ⊂ R,
1.3
different parameters η > 1 and βd > 2 n
We consider the following multilinear fractional integral operator
I α,m
f1, , f m
x
Rdm
f1
x − y1
f2
x − y2
· · · f m
x − y m
y1, y2, , y m mn−α dμ
y1
· · · dμy m
σ {σ1, , σj} of {1, 2, , m} of j different elements For any σ ∈ C j m , we denote σ
{1, 2, , m} \ σ {σj 1, , σm} Moreover, for b j ∈ RBMOμ, j 1, 2, , m, let
b b1, b2, , b m and denote by b σ b σ1 , , b σj and by b σ x b σ1 x · · · b σj x Also, we denote f f1, , f m , f σ f σ1 , , f σj , b σf σ b σj1 f σj1 , , b σm f σm
b, I α,m
f
j0
σ∈C m j
−1m−j
b σ xI α,m
f σ , b σf σ
In particular, for m 2, we define
b1, b2, I α,2
f1, f2
x b1xb2xI α,2
f1, f2
x − b1xI α,2
f1, b2f2
x
− b2xI α,2
b1f1, f2
x I α,2
b1f1, b2f2
Trang 3
Theorem 1.1 Let μ be defined as above and μ ∞, b j ∈ RBMORd , j 1, 2, 0 < α < 2n Then
b1, b2, I α,2 is a bounded operator from L q1× L q2 to L q with 1/q 1/q1 1/q2− α/2n > 0 and
1 < q1, q2 < ∞.
Remark 1.2 ByLemma 2.2inSection 2,Theorem 1.1for the caseμ < ∞ also holds provided
I α,2,b1, b2, I α,2 , b1, I α,2 , and b2, I α,2 satisfy certain T1 type conditions For instance, if I α,2
satisfies the T1 condition, that is, I α,2∗1 0, then we can easily obtain I α,2 f1, f2xdμx 0
see 3 for the notation I∗1
α,2
More generally, we have the following theorem
Theorem 1.3 Let m ∈ N, μ be defined as above, and μ ∞, b j ∈ RBMORd , j 1, 2, , m,
0 < α < mn Then
b, I α,m
f L q μ ≤ Cm
j1
where 1/q 1/q1 1/q2 · · · 1/q m − α/mn > 0 and 1 < q j < ∞, j 1, 2, , m.
this paper, we always use the letter C to denote a positive constant that may vary at each
occurrence but is independent of the essential variable
2 Proofs of theorems
Before proving our results, we need to recall some notation and establish some lemmas which play important roles in the proofs
M β p,η fx sup
1
μηQ 1−βp/n
Q
fy p
dμy
1/p
and the sharp maximal function
M #,β fx sup 1
μ
Q
fy − m Qf dμy sup
Q,R doubling
m Q f − m R f
where the supremum is taken over all cubes Q with sides parallel to the coordinate axes,
M #,0 f by M#f.
Nfx sup
Q doubling
1
μQ
Q
fy dμy. 2.3
Trang 4Lemma 2.1 see 11 Let 1 ≤ p < ∞ and 1 < ρ < ∞ Then b ∈ RBMOμ, if and only if for any
cube Q ⊂ R d ,
1
μρQ
Q
bx − m Qb p
dμx ≤ Cb p∗, 2.4
and for any doubling cubes Q ⊂ R,
Lemma 2.2 see 5 Let f ∈ L1
locμ with fdμ 0 if μ < ∞ For 1 < p < ∞, if inf1, Nf ∈
L p μ, then for 0 ≤ β < n we have
Nf L p μ ≤ C M #,β f L p μ 2.6
Lemma 2.3 see 5 Let p < r < n/α and 1/q 1/r − α/n Then
M α p,η f L q μ ≤ Cf L r μ , 2.7
where η > 1 and 0 ≤ α < n/p.
Lemma 2.4 Suppose μ is a Radon measure satisfying 1.1 Let m ∈ N and 1/s 1/r1· · ·1/r m−
α/n > 0 with 0 < α < mn, 1 ≤ r j ≤ ∞ Then,
a if each r j > 1,
I α,m
f1, , f m L s μ ≤ Cm
j1
b if r j 1 for some j,
I α,m
f1, , f m L s,∞ μ ≤ Cm
j1
Proof The proof follows the idea that, for the classical setting, can be found in4 For the sake of completeness, we will show it again
1/r1 · · · 1/r l ≤ l, so that mn − α > m − ln, integration in y l1 , , y mreduces matters to
that if 0 < c i , i 1, , m, and 0 < α < m
i1 c i , we can find 0 < α i < c i such that α m
i1 α i
i1 1/s i 1/s, 0 < α i /n ≤ 1,
1 < s i < ∞, and
y1 n−α1 y2 n−α2· · · y m n−α m ≤ y1, , y m nm−α
i1 α i It follows that
I α,m
f1, , f m
i1
I α i
f i
Trang 5
Lemma 2.5 Let b1, b2, I α,2 be as in 1.6, 0 < α < 2n, τ > 1, b1, b2∈ RBMOμ Then there exists
a constant C > 0 such that for all f1∈ L q1μ, f2∈ L q2μ, and x ∈ R d ,
M #,α
b1, b2, I α,2
f1, f2
I α,2
f1, f2
x
b1 ∗M τ,3/2
b2, I α,2
f1, f2
x
b2 ∗M τ,3/2
b1, I α,2
f1, f2
x
b1 ∗ b2 ∗M α p
1,9/8 f1xM α p
2,9/8 f2x ,
2.12
M #,α
b1, I α,2
f1, f2
M τ,3/2
I α,2
f1, f2
x
M p α
1,9/8
f1
xM α p
2,9/8
f2
M #,α
b2, I α,2
f1, f2
M τ,3/2
I α,2
f1, f2
x
M p α1,9/8f1
xM α p2,9/8f2
where
b1, I α,2
f1, f2
x b1xI α,2
f1, f2
x − I α,2
b1f1, f2
x,
b2, I α,2
f1, f2
x b2xI α,2
f1, f2
x − I α,2
f1, b2f2
Proof By the definition, to obtain2.12, it suffices to prove that for any x ∈ R d and a cube
1
μ
Q
b1, b2, I α,2
f1, f2
z − h Q dμz ≤ C b1
∗ b2 ∗M τ,3/2
I α,2
f1, f2
x
b1 ∗M τ,3/2
b2, I α,2f1, f2
x
b2 ∗M τ,3/2
b1, I α,2f1, f2
x
b1 ∗ b2 ∗M α p1,9/8 f1xM α p2,9/8 f2x ,
2.16
and for any cubes Q ⊂ R, where Q is an arbitrary cube and R is doubling,
h Q − h R ≤ CK2
Q,R K α Q,R b1 ∗ b2 ∗M τ,3/2
I α,2
f1, f2
x
b1 ∗M τ,3/2
b2, I α,2
f1, f2
x
b2 ∗M τ,3/2
b1, I α,2
f1, f2
x
b1 ∗ b2 ∗M α p1,9/8 f1xM α p
2,9/8 f2x ,
2.17
where
h Q m Q
I α,2
m Q
b1
− b1
f1χRd \4/3Q ,
m Q
b2
− b2
f2χRd \4/3Q
,
h R m R
I α,2
m R
b1
− b1
f1χRd \4/3R ,
m R
b2
− b2
f2χRd \4/3R
Trang 6First of all, it is easy to see that
b1, b2, I α,2
f1, f2
z − h Q ≤ b1z − m Q
b1
b2z − m Q
b2
I α,2
f1, f2
z
b1z − m Q
b1
I α,2
f1,
b2z − b2
f2
z
b2z − m
Q
b2
I α,2
b1z − b2
f1, f2
z
I α,2
b1− m Qb1f1,
b1− m Qb1f2
z − h Q : Iz IIz IIIz IVz
2.19
Consequently,
1
μ
Q
b1, b2, I α,2
f1, f2
In what follows, we estimate I–IV, respectively For I, by H ¨older’s inequality and
Lemma 2.1, we have
μ
Q
Izdμz
≤ C
1
μ
Q
b1z − m
Q
b1 τ1
dμz
1/τ1
×
1
μ
Q
b2z − m Q
b2 τ2
dμz
1/τ2
×
1
μ
Q
I α,2
f1, f2 τ
dμz
1/τ
≤ C b1 ∗ b2 ∗M τ,3/2
I α,2
f1, f2
x,
2.21
where τ1> 1, τ2> 1 and 1/τ 1/τ1 1/τ2 1.
For II, we have
μ
Q
IIzdμz
≤ C
1
μ
Q
b1z − m Q
b1 s
dμz
1/s
μ
Q
b2, I α,2
f1, f2
z τ
dμz
1/τ
≤ C b1 ∗M τ,3/2
b2, I α,2
f1, f2
x,
2.22
where s > 1 and 1/s 1/τ 1.
Trang 7Similarly, we have
III≤ C b2 ∗M τ,3/2
b2, I α,2
f1, f2
It remains to estimate IV For convenience, we set f j0 f j χ 4/3Q , f j f0
j f∞
j , j 1, 2.
Then,
IVz ≤ Iα,2
b1− m Qb1
f10,
b2− m Qb2
f20
z
I α,2
b1− m Q
b1
f10,
b2− m Q
b2
f2∞
z
I α,2
b1− m Qb1
f1∞,
b2− m Qb2
f20
z
I α,2
b1− m Qb1
f1∞,
b2− m Qb2
f2∞
z − h Q
IV1z IV2z IV3z IV4z,
2.24
and so we have
1
μ
Q
IVz dμz ≤4
j1
1
μ
Q
j1
To estimate IV1, set s1 √p1, s2 √p2, and 1/v 1/s1 1/s2− α/n It follows from H¨older’s
IV1≤ μQ1−1/v
μ
b1− m Q
b1
f10,
b2− m Q
b2
f20 L v μ
≤ Cμ3/2Q−1/v b1− m Qb1
f0
1 L s1 μ b2− m Qb2
f0
2 L s2 μ
μ
4/3Q
f1y1 p1
dμy1
1/p1
×
4/3Q
b1y1 − m Q
b1 p1/√p
1 −1
dμy1
√p1−1/p1
×
4/3Q
f2y2 p2
dμy2
1/p2
4/3Q
b2y1 − m Q
b i p2/√p
2 −1
dμy i
√p2−1/p2
i1
1
μ
3/2Q1−αpi /2n
4/3Q
f i y i p i
dμy i
1/p i
×
1
μ
4/3Q
b i y i − m Q
b i p i /√p
i−1
dμy i
√p i −1/p i
≤ C b1 ∗ b2 ∗M p α1,9/8 f1xM p α2,9/8 f2x.
2.26
Trang 8For term IV2, byLemma 2.1, we have
μ
Q
μ
Q
Rd \4/3Q
4/3Q
× b1
y1
− m Qb1
f0 1
y1 b2
y2
− m Qb2
f2∞
y2 z − y1, z − y2 2n−α dμ
y1
dμ
y2
dμz
μ
Q
4/3Q
b1y − m Q
b1
f10
y1 dμ
y1
×
Rd \4/3Q
b2
y2
− m Qb2
f2∞
y2
z − y2 2n−α dμ
y2
dμz
≤ C
1
μ
3/2Q1−αp1/2n
4/3Q
f1
y1 p1
dμ
y1
1/p1
×
1
μ
4/3Q
b1− m Q
b1 p1
dμ
y1
1/p1
× μ
3
−α/2n
μQ
∞
k1
2k 4/3Q\2 k−1 4/3Q
b2
y2
− m Q
b2
f2
y2
2k2n−α lQ 2n−α dμ
y2
≤ C b1 ∗M p α
1,9/8 f1x∞
k1
2−kn−α/2 l
2k3
−nα/2
×
2k 4/3Q
b2
y2
− m Q
b2
f2
y2 dμ
y2
≤ C b1 ∗M p α1,9/8 f1x∞
k1
2−kn−α/2 l
2k3
−nα/2
×
2k 4/3Q
b2
y2
− m2k4/3Qb2
f2
y2 dμ
y2
2k 4/3Q
b2
− m Qb2
2k 3/2Q
f2
y2 dμ
y2
≤ C b1 ∗M p α1,9/8 f1x
×
∞
k1
2−kn−α/2
1
l
2k 3/2Qn
2k 4/3Q
b2
y2
− m2k4/3Qb2 p2
dμ
y2
1/p2
×
1
l
2k 3/2Qn−αp2/2
2k 4/3Q
f2
y2 p2
dμ
y2
1/p2
k1
k2 −kn−α/2 b2 ∗
1
l
2k 3/2Qn−α/2
2k 4/3Q
f2
y2 dμ
y2
≤ C b1 ∗ b2 ∗M α p
1,9/8 f1xM α p
2,9/8 f2x,
2.27
Trang 9where the last inequality follows from the following two facts:
1
l
2k 3/2Qn−α/2
2k 4/3Q
f2
y2 dμ
y2
2k 4/3Q1−1/p2
l
2k 3/2Qn−α/2
2k 4/3Q
f2
y2 p2
dμ
y2
1/p2
2k1 4/3Q1−1/p21/p2−α/2n
l
2k1 4/3Qn−α/2
1
μ
2k1 4/3Q1−αp2/2n
2k1 4/3Q
f2
y2 p2
dμ
y2
1/p2
≤ CM p α2,9/8 f2x,
2.28 andsee11
2k 4/3Q
b j
− m Qb j j ∗K Q, 2k 4/3Q ≤ C b j ∗K Q,2 k 4/3Q ≤ Ck b j ∗, j 1, 2.
2.29 Similarly,
IV3 ≤ C b1 ∗ b2 ∗M α p1,9/8 f1xM α p
2,9/8 f2x. 2.30
I α,2
b1− m Q
b1
f1∞,
b2− m Q
b2
f2∞
z − I α,2
b1− m Q
b1
f1∞,
b2− m Q
b2
f2∞
y
≤
Rd \4/3Q
Rd \4/3Q
z − y1, z − y2 2n−α− y − y1, y − y1 2 2n−α
×
2
i1
b i
y i
− m Q
b i
f i∞
y i dμ
y1
dμ
y2
≤
Rd \4/3Q
Rd \4/3Q
|z − y|
y − y1, y − y2 2n−α1
×
2
i1
b i
y i
− m Qb i
f i∞
y i dμ
y1
dμ
y2
i1
Rd \4/3Q
|z − y| 1/2
y − y i n−α/21/2 b i
y i
− m Qb i
f i∞
y i dμ
y i
i1
∞
k1
2k 4/3Q\2 k−1 4/3Q2−k/2 1
l
2k Qn−α/2 b i
y i
− m Qb i f∞
i
y i dμ
y i
i1
∞
k1
2−k/2
1
l
2k 3/2Qn
2k 4/3Q
b i
y i
− m Q
b i pi
dμ
y i
1/p i
×
1
l
2k 3/2Qn−αp i /2
2k 4/3Q
f i
y i p i
dμ
y i
1/p i
Trang 10≤ C2
i1
∞
k1
2−k/2 M p α
i ,9/8 f i x
×
1
l
2k 3/2Qn
2k 4/3Q
b i
y i
−m2k4/3Qb i
m2k4/3Qb i
−m Q
b i pi
dμ
y i
1/pi
i1
∞
k1
2−k/2 k b i ∗M α p i ,9/8 f i x
≤ C b1 ∗ b2 ∗M α p
1,9/8 f1xM α p
2,9/8 f2x.
2.31
Taking the mean over y ∈ Q, we obtain
I α,2
b1−m Qb1
f1∞,
b1−m Qb1
f2∞
z−h Q 1 ∗ b2 ∗M α p1,9/8 f1xM α p
2,9/8 f2x.
2.32 Thus,
μ
Q
IV4zdμz ≤ C b1 ∗ b2 ∗M α p1,9/8 f1xM p α2,9/8 f2x. 2.33 Combing2.20–2.33, we obtain 2.16
h Q − h R m Q
I α,2
b1− m Qb1
f1∞,
b2− m Qb2
f2∞
I α,2
b1− m R b1
f1∞,
b2− m R b2
f2∞
≤ m R
I α,2
b1− m Qb1
f1χRd\2N Q ,
b2− m Qb2
f2χRd\2N Q
I α,2
b1− m Qb1
f1χRd\2N Q ,
b2− m Qb2
f2χRd\2N Q
m R
I α,2
b1− m R b1
f1χRd\2N Q ,
b2− m R b2
f2χRd\2N Q
I α,2
b1− m Qb1
f1χRd\2N Q ,
b2− m Qb2
f2χRd\2N Q
m Q
I α,2
b1− m Qb1
f1χ2N Q\4/3Q ,
b2− m Qb2
f2χRd \4/3Q
m Q
I α,2
b1− m Qb1
f1χRd\2N Q ,
b2− m Qb2
f2χ2N Q\4/3Q
m R
I α,2
b1− m R b1
f1χRd \4/3R ,
b2− m R b2
f2χ2N Q\4/3R
m R
I α,2
b1− m R b1
f1χ2N Q\4/3R ,
b2− m R b2
f2χRd\2N Q
i1
A i
2.34
A1 ≤ C K Q,R
2
b1 ∗ b2 ∗M α p ,9/8 f1xM α p ,9/8 f2x. 2.35
Trang 11To estimate A2, we write
I α,2
b1− m R
b1
f1χRd\2N Q ,
b2− m R
b2
f2χRd\2N Q
z
− I α,2
b1− m Qb1
f1χRd\2N Q ,
b2− m Qb2
f2χRd\2N Q
z
m R
b2
− m Q
b2
I α,2
b1− m R
b1
f1χRd\2N Q , f2χRd\2N Q
z
m R
b1
− m Q
b1
I α,2
f1χRd\2N Q ,
b2− m R
b2
f2χRd\2N Q
z
m R
b1
− m Q
b1
m R
b2
− m Q
b2
I α,2
f1χRd\2N Q , f2χRd\2N Q
z.
2.36
Then,
A2≤ m R
b2
− m Q
b2 μR1
R
I α,2
b1− m R
b1
f1χRd\2N Q , f2χRd\2N Q
zdμz
m R
b1
− m Q
b1 μR1
R
I α,2
f1χRd\2N Q ,
b2− m R
b2
f2χRd\2N Q
zdμz
m R
b1
− m Qb1 m R
b2
− m Qb2
μR1
R
I α,2
f1χRd\2N Q , f2χRd\2N Q
zdμz
A21 A22 A23.
2.37
It is obvious that
A23≤ CK2
Q,R b1 ∗ b2 ∗M τ,3/2
I α,2
f1, f2
I α,2
b1− m R b1
f1χRd\2N Q , f2χRd\2N Q
z
I α,2
b1− m R b1
f1, f2
z − I α,2
b1− m R b1
f1χ2N Q χ 4/3R , f2χ 4/3R
z
− I α,2
b1− m R b1
f1χ 4/3R , f2χ2N Q χ 4/3R
z
I α,2
b1− m R b1
f1χ2N Q χ 4/3R , f2χ2N Q χ 4/3R
z
− I α,2
b1− m R b1
f1χRd \4/3R , f2χ2N Q
z
− I α,2
b1− m R b1
f1χ2N Q , f2χRd \4/3R
z
I α,2
b1− m R b1
f1χ2N Q\4/3R , f2χ2N Q\4/3R
z
j1
B j z.
2.39
I α,2
b1− m R b1
f1, f2
z ≤ I α,2
b1− b1zf1, f2
z I α,2
b1z − m R b1
f1, f2
z .
2.40
Trang 12By H ¨older’s inequality and the fact that R is doubling, we have
1
μR
R
I α,2
b1− m R b1
f1, f2
I α,2
f1, f2
x,
1
μR
R
I α,2
b1− b1zf1, f2
b1, I α,2
f1, f2
x,
2.41
which imply
∗M τ,3/2
I α,2
f1, f2
x M τ,3/2
b1, I α,2
f1, f2
Lemma 2.4, we have
1
μR
R
μR I α,2
b1− m R
b1
f1χ2N Q χ 4/3R , f2χ 4/3R L v μ
≤ Cμ
3
−1/v
b1− m R
b1
f1χ2N Q χ 4/3R L s1 μ f2χ 4/3R L s2 μ
≤ Cμ
3
4/3R
f2y p2
dμy
1/p2
4/3R
f1x p1
dμy
1/p1
×
4/3R
b1y − m R
b1 p1/√p
1 −1
dμy
√p1−1/p1
≤ C
1
μ
3/2R1−αp1/2n
4/3R
f1y p1
dμy
1/p1
×
1
μ
4/3R
b1y − m R
b1 p1/√p
1 −1
dμy
√p1−1/p1
×
1
μ
3/2R1−αp2/2n
4/3Q
f2y p2
dμy
1/p2
≤ C b1 ∗M α p
1,9/8 f1xM α p
2,9/8 f2x,
2.43 which implies
p1,9/8 f1xM p α2,9/8 f2x. 2.44 Similarly,
∗M α p
1,9/8 f1xM α p
2,9/8 f2x,
∗M α p ,9/8 f1xM α p ,9/8 f2x. 2.45
... b2f2Trang 3
Theorem 1.1 Let μ be defined as above and... f2x.
2.26
Trang 8For term IV2, byLemma 2.1, we have
μ... and 1/s 1/τ 1.
Trang 7Similarly, we have
III≤ C b2 ∗M τ,3/2