Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 731016, 21 pages
doi:10.1155/2010/731016
Research Article
Multilinear Riesz Potential on Morrey-Herz Spaces
with Non-Doubling Measures
Yanlong Shi,1 Xiangxing Tao,2 and Taotao Zheng3
1
Department of Fundamental Courses, Zhejiang Pharmaceutical College, Ningbo, Zhejiang 315100, China
Department of Mathematics, Zhejiang University of Science & Technology, Hangzhou,
Zhejiang 310023, China
3
Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, China
2
Correspondence should be addressed to Xiangxing Tao, xxtao@hotmail.com
Received 19 November 2009; Accepted 10 March 2010
Academic Editor: Nikolaos Papageorgiou
Copyright q 2010 Yanlong Shi et al. 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.
→
−
The authors consider the multilinear Riesz potential operator defined by Iα,m f x →
−
f y f y · · · fm ym /|x − y1 , . . . , x − ym |mn−α dμy1 · · · dμym , where f denotes the mRd m 1 1 2 2
tuple f1 , f2 , . . . , fm , m, n the nonnegative integers with n ≥ 2, m ≥ 1, 0 < α < mn, and μ is a
nonnegative n-dimensional Borel measure. In this paper, the boundedness for the operator Iα,m on
the product of homogeneous Morrey-Herz spaces in nonhomogeneous setting is found.
1. Introduction
Let Bx, r denote a ball centered at x ∈ Rd with radius r > 0, and for any > 0, Bx, r will
mean the ball with the same center as Bx, r and with radius r. A Borel measure μ on Rd is
called a doubling measure if it satisfies the so-called doubling condition; that is, there exists
a constant C > 0 such that
μBx, 2r ≤ CμBx, r
1.1
for every ball Bx, r ⊂ Rd . The doubling condition is a key feature for a homogeneous
metric measure space. Many classical theories in Fourier analysis have been generalized
to the homogeneous setting without too much difficulties. In the last decade, however, some
researchers found that many results are still true without the assumption of the doubling
condition on μ see, e.g., 1–4. This fact has encouraged other researchers to study various
theories in the nonhomogeneous setting. By a nonhomogeneous space we mean a metric
2
Journal of Inequalities and Applications
measure space, here we will consider only Rd , equipped with a nonnegative n-dimensional
Borel measure μ, that is, a measure satisfying the growth condition
1.2
μBx, r ≤ Cr n
for any ball Bx, r ⊂ Rd and n is a fixed real number such that 0 < n ≤ d. Unless otherwise
stated, throughout this paper we will always work in the nonhomogeneous setting.
As one of the most important operators in harmonic analysis and its applications, the
Riesz potential operator Iα defined by
Iα fx :
f x−y
n−α dμ y ,
Rd y 0 < α < n,
1.3
was studied by Garcı́a-Cuerva and Martell 1 in 2001. Garcı́a-Cuerva and Martell proved
that Iα is bounded from Lp μ to Lq μ for all p > 1 and 1/q 1/p − α/n > 0 and that Iα
is bounded from L1 μ to WLn/n−α μ. Here Lp μ and WLp μ denote the Lebesgue spaces
and weak Lebesgue spaces with measure μ, respectively.
Simultaneously many classical multilinear operators on Euclidean spaces with Lebesgue measure have been generalized for nondoubling measures, that is, the case
nonhomogeneous setting see 3, 4. For example, based on the work of Kenig and Stein 5,
Lian and Wu 4 studied multilinear Riesz potential operator
→
−
Iα,m f x f1 y1 f2 y2 · · · fm ym
mn−α dμ y1 · · · dμ ym
Rd m x − y1 , . . . , x − ym 1.4
→
−
in the nonhomogeneous case, where, and throughout this paper, we denote by f the m-tuple
f1 , f2 , . . . , fm , m, n the nonnegative integers with n ≥ 2, m ≥ 1. They obtained the following.
Proposition 1.1 see 4. Let m ∈ N, 1/s 1/r1 1/r2 · · · 1/rm − α/n > 0 with 0 < α < mn,
1 ≤ ri ≤ ∞, then
a if each ri > 1, Iα,m is bounded from Lr1 μ × · · · × Lrm μ to Ls μ,
b if ri 1 for some i, Iα,m is bounded from Lr1 μ × · · · × Lrm μ to WLs μ.
Obviously, it is Lemma 7 in 5 if μ is the Lebesgue measure in the proposition above
and it is a multilinear setting of the result of Garcı́a-Cuerva and Martell 1.
In addition, in the article 6–8, we have obtained the boundedness of the operator
Iα,m on the product of Morrey type spaces, weak homogeneous Morrey-Herz spaces, and
Herz type Hardy spaces in the classical case and extended the result of Kenig and Stein. As
a continuation of previous work in 4, 6–8, in this paper, we will study the operator Iα,m in
the product of weak homogeneous Morrey-Herz spaces in the nonhomogeneous setting.
Journal of Inequalities and Applications
3
σ,λ
μ
The definitions of the weak homogeneous Morrey-Herz spaces WMK̇p,q
σ,p
σ,p
and weak homogeneous Herz spaces W K̇q μK̇q μ will be given in
0,0
0,0
μ WLp μ and MK̇p,p
μ Lp μ for
Section 2, here we only point out that WMK̇p,p
1 ≤ p < ∞.
We will establish the following boundedness of the multilinear Riesz potential
operator Iα,m on the homogeneous Morrey-Herz spaces.
σ,λ
μ
MK̇p,q
Theorem 1.2. Let 0 < α < mn, 0 ≤ λi < n − α/m, 0 < pi ≤ ∞, 1 < qi < ∞, and λi α/m −
n/qi < σi < n1 − 1/qi for i 1, 2, . . . , m. Suppose that λ λ1 · · · λm , σ σ1 · · · σm ,
1/p 1/p1 · · · 1/pm − α/n > 0, 1/q 1/q1 · · · 1/qm − α/n > 0, then
→
− Iα,m f σ,λ
MK̇p,q
μ
≤C
m fi σi ,λi
MK̇
μ
i1
pi ,qi
1.5
→
−
with a constant C > 0 independent of f .
In the case σi n1 − 1/qi for i 1, 2, . . . , m, we will use the weak homogeneous
σ,p
σ,λ
Morrey-Herz spaces WMK̇p,q
μ and weak homogeneous Herz spaces W K̇q μ to derive
the following boundedness for the operator Iα,m .
Theorem 1.3. Let 0 < α < mn, 0 ≤ λi < n − α/m, 0 < pi ≤ 1 and 1 ≤ qi < ∞ for i 1, 2, . . . , m.
Suppose that λ λ1 · · ·λm , 1/p 1/p1 · · ·1/pm −α/n > 0, 1/q 1/q1 · · ·1/qm −α/n > 0,
then
→
− Iα,m f nm−1/q−α,λ
WMK̇p,q
μ
≤C
m fi n1−1/qi ,λi
MK̇
μ
i1
pi ,qi
1.6
→
−
with a constant C > 0 independent of f .
Remark 1.4. The restriction 0 < pi ≤ 1 in Theorem 1.3 cannot be removed see 9 for an
counter-example when m 1 and n d.
In addition, we remark that the weak homogeneous Morrey-Herz spaces generalize
σ,p
σ,0
σ,0
μ K̇q μ and WMK̇p,q
μ the weak homogeneous Herz spaces. Particularly, MK̇p,q
σ,p
σ,p
p
W K̇q μ for 0 < p, q < ∞ and σ ∈ R. Moreover, we have K̇p μ L|x|σ μ, the weighted Lp
p
spaces Lw μ {f : fw ∈ Lp μ} for 1 ≤ p < ∞ and σ ∈ R, for details, see Section 2.
Hence, it is easy to obtain the following corollaries from the theorems above.
Corollary 1.5. Let 0 < α < mn, 1/p 1/p1 · · · 1/pm − α/n > 0, 1/q 1/q1 · · · 1/qm − α/
n > 0.
i If 0 < pi ≤ ∞, 1 < qi < ∞, σ σ1 · · · σm with α/m − n/qi < σi < n1 − 1/qi for
4
Journal of Inequalities and Applications
i 1, 2, . . . , m. Then
→
− Iα,m f ≤C
σ,p
K̇q μ
m fi σi ,pi
K̇q μ
i1
1.7
i
→
−
with a constant C > 0 independent of f .
(ii) If 0 < pi ≤ 1 and 1 ≤ qi < ∞ for i 1, 2, . . . , m, then
→
− Iα,m f ≤C
nm−1/q−α,p
W K̇q
μ
m fi n1−1/qi ,pi
K̇
μ
i1
qi
1.8
→
−
with a constant C > 0 independent of f .
Corollary 1.6. Let 0 < α < mn, 1/p 1/p1 · · ·1/pm −α/n > 0 with 1 < pi < ∞, σ σ1 · · ·σm
with α/m − n/pi < σi < n1 − 1/pi for i 1, 2, . . . , m, then
→
− Iα,m f ≤C
p
L|x|σ μ
m fi pi
L σ
i1
|x| i
1.9
μ
→
−
with a constant C > 0 independent of f .
Throughout this paper, the letter C always remains to denote a positive constant that
may vary at each occurrence but is independent of all essential variables.
2. The Definitions of Some Function Spaces
We start with some notations and definitions. Here and in what follows, denote by Bk B0, 2k {x ∈ Rd : |x| ≤ 2k }, Ek Bk \ Bk−1 , and χk χEk for k ∈ Z the characteristic function
of the set Ek .
σ,p
Definition 2.1. Let σ ∈ R, 0 < p ≤ ∞, and 0 < q < ∞. The homogeneous Herz spaces K̇q μ
are defined to be the following space of functions:
σ,p q
K̇q μ f ∈ Lloc Rd \ {0} : f K̇qσ,p μ < ∞ ,
2.1
where
f σ,p K̇q μ
∞
2
k−∞
p
fχk Lq μ
kσp 1/p
,
2.2
and the usual modification should be made when p ∞.
Definition 2.2. Let σ ∈ R, 0 < p ≤ ∞ and 0 < q < ∞. The weak homogeneous Herz spaces
σ,p
W K̇q μ are defined by
σ,p W K̇q μ f is a measurable function on Rd and f W K̇qσ,p μ < ∞ ,
2.3
Journal of Inequalities and Applications
5
where
f σ,p sup γ
W K̇q μ
γ>0
∞
kσp
2
p/q
μ x ∈ Ek : fx > γ
1/p
2.4
,
k−∞
and the usual modification should be made when p ∞.
Definition 2.3. Let σ ∈ R, 0 < p ≤ ∞, 0 < q < ∞, and 0 ≤ λ < ∞. The homogeneous Morreyσ,λ
Herz spaces MK̇p,q
μ are defined by
q
σ,λ <∞ ,
MK̇p,q
μ f ∈ Lloc Rn \ {0} : f MK̇p,q
σ,λ
μ
2.5
where
f σ,λ sup2−k0 λ
MK̇p,q μ
k0 ∈Z
k0
p
fχk Lq μ
kσp 2
k−∞
1/p
2.6
,
and the usual modifications should be made when p ∞.
Definition 2.4. Let σ ∈ R, 0 < p ≤ ∞, 0 < q < ∞ and 0 ≤ λ < ∞. The weak homogeneous
σ,λ
Morrey-Herz spaces WMK̇p,q
μ are defined by
σ,λ <∞ ,
WMK̇p,q
μ f is a measurable function on Rn and f WMK̇p,q
σ,λ
μ
2.7
where
f σ,λ
WMK̇p,q
μ
−k0 λ
sup γ sup 2
γ>0
k0 ∈Z
k0
kσp
2
p/q
μ x ∈ Ek : fx > γ
1/p
,
2.8
k−∞
and the usual modifications should be made when p ∞.
3. Proof of Theorems 1.2 and 1.3
Without loss of generality, in order to simplify the proof, we only consider the situation when
m 2. Actually, the similar procedure works for all m ∈ N.
6
Journal of Inequalities and Applications
Indeed, we decompose fi as
fi x ∞
fi xχli x :
li −∞
∞
fli x,
i 1, 2, li ∈ Z.
li −∞
3.1
To shorten the formulas below, we set
Λ1 {l1 , l2 : l1 , l2 ≤ k − 2},
Λ2 {l1 , l2 : l1 ≤ k − 2, k − 1 ≤ l2 ≤ k 1},
Λ3 {l1 , l2 : l1 ≤ k − 2, l2 ≥ k 2},
Λ4 {l1 , l2 : k − 1 ≤ l1 ≤ k 1, l2 ≤ k − 2},
3.2
Λ5 {l1 , l2 : k − 1 ≤ l1 , l2 ≤ k 1},
Λ6 {l1 , l2 : k − 1 ≤ l1 ≤ k 1, l2 ≥ k 2},
Λ7 {l1 , l2 : l1 ≥ k 2, l2 ≤ k − 2},
Λ8 {l1 , l2 : l1 ≥ k 2, k − 1 ≤ l2 ≤ k 1},
Λ9 {l1 , l2 : l1 , l2 ≥ k 2}.
It is easy to see that the case for l1 , l2 ∈ Λ2 is analogous to the case for l1 , l2 ∈ Λ4 ,
the case for l1 , l2 ∈ Λ3 is similar to the case for l1 , l2 ∈ Λ7 , and the case for l1 , l2 ∈ Λ6 is
analogous to the case for l1 , l2 ∈ Λ8 , respectively. Thus, by the symmetry of f1 and f2 in the
operator Iα,2 , we will only discuss the cases for l1 , l2 belong to Λ1 , Λ2 , Λ3 , Λ5 , Λ6 and Λ9 ,
respectively.
By a direct computation, we have the following fact that, for x ∈ Ek ,
⎧ −k2n−α fl 1 fl 1
C2
⎪
1 L μ
2 L μ
⎪
⎪
⎨ kl α/2−n fl 1 fl 1
Iα,2 fl , fl x ≤ C2 2
1 L μ
2 L μ
1
2
⎪
⎪
⎪
⎩ l1 l2 α/2−n fl 1 fl 1
C2
1 L μ
2 L μ
if l1 , l2 ∈ Λ1 , Λ2 ,
if l1 , l2 ∈ Λ3 , Λ6 ,
3.3
if l1 , l2 ∈ Λ9 .
We will use estimates 3.3 in the proof of theorems below. In addition, we always let 1/g 1/q1 1/q2 , 1/l 1/p1 1/p2 , 1/hi 1/qi − α/2n for i 1, 2, and use the notations
Gi s 2sn1/qi −1σi ,
Hi s 2sn/qi −α/2σi ,
i 1, 2.
3.4
Journal of Inequalities and Applications
7
∞
It is easy to see that ∞
s2 Gi s < ∞ when n1/qi − 1 σi < 0, and that
s2 H2 −s sλ2
2
H
−s
<
∞
when
α/2
−
n/q
≤
λ
α/2
−
n/q
<
σ
.
2
2
2
2
2
s2
We will also use repeatedly the inequality k |aκ |γ ≤ k |aκ |γ for 0 < γ ≤ 1.
Now we are ready to the proof of Theorem 1.2. Suppose that f1 × f2 ∈ MK̇pσ11,q,λ11 μ ×
MK̇pσ22,q,λ22 μ, by the decomposition of fi above, we get
∞
→
− Iα,2 f σ,λ
MK̇p,q
μ
k0
−k0 λ
sup 2
k0 ∈Z
2
L
k−∞
−k0 λ
≤ sup 2
k0 ∈Z
≤C
→
− p
Iα,2 f χk q
kσp ⎧
k0
⎨
1/p
μ
⎫1/p
⎬
p
∞ I f , f χk l ,l −∞ α,2 l1 l2
kσp 2
⎩k−∞
Lq μ
1 2
⎭
3.5
9
V ,
1
where
⎧
⎪
k0
⎨
p
Iα,2 fl , fl χk V sup 2−k0 λ
2kσp 1
2
⎪
l1 ,l2 ∈Λ
q
⎩k−∞
k0 ∈Z
L μ
⎫1/p
⎪
⎬
⎪
⎭
.
3.6
To estimate the term V1 , we first note that inequality 3.3 and the growth condition
1.2 of μ imply that
Iα,2 fl , fl χk 1
2
l1 ,l2 ∈Λ1
2
≤ Cχk Lq μ
i1
Lq μ
k−2
fli L1 μ
−kn−α/2 2
li −∞
2
k−2
1/q k
−kn−α/2 fli L1 μ
Cμ B 0, 2
2
i1
≤C
2
i1
k−2
fli L1 μ .
−kn−α/2kn/hi 2
li −∞
li −∞
3.7
8
Journal of Inequalities and Applications
Hence, by the fact p < l, the Cauchy inequality and the growth condition 1.2 of μ, we
can show that
−k0 λ
V1 ≤ C sup 2
k0 ∈Z
kσp
2
≤ C sup 2
k0 ∈Z
k0
kσp
2
≤ C sup 2
k0 ∈Z
k0 2
≤ C sup 2
k0 ∈Z
2
−k0 λi
2
2
2
k0 ∈Z i1
≤ Csup
2
2
k0 ∈Z i1
k0
k0
2
k0
2
k−2
fli L1 μ
2
k−2
fli L1 μ
2
l ⎫1/l
⎬
⎭
3.8
fli L1 μ
pi 1/pi
kn1/qi −1kσi 2
kn1/qi −1kσi
2
li −∞
k−2
p 1/p
kn1/qi −1 li −∞
p 1/p
kn1/qi −1 li −∞
k−∞
k−2
kσi
2
k−∞
−k0 λi
fli L1 μ
li −∞
k−∞
−k0 λi
k−2
kσi
⎩k−∞ i1
k0 ∈Z i1
≤ C sup
⎧
k0 2
⎨
L1 μ
i
kn1/qi −1 li −∞
k−∞ i1
−k0 λ
2
2
k−2
p 1/p
fl −kn−α/2kn/hi li −∞
i1
k−∞
−k0 λ
2
k−2
i1
k−∞
−k0 λ
≤ C sup
k0
1−1/qi fl q
μ B 0, 2li
i L i μ
fli Lqi μ
pi 1/pi
pi 1/pi
kn1/qi −1kσi li n1−1/qi 2
2
li −∞
: CsupV11 k0 × V12 k0 ,
k0 ∈Z
where
−k0 λi
V1i k0 2
k0
k−∞
k−2
Gi k − li 2
li −∞
fli Lqi μ
li σi pi 1/pi
.
3.9
Journal of Inequalities and Applications
9
In case 0 < pi ≤ 1, using the fact n1/qi − 1 σi < 0, we have
k0
k−2
−k0 λi
V1i k0 ≤ 2
p
Gi pi k − li 2li σi pi fli Liqi μ
1/pi
k−∞ li −∞
−k0 λi
≤2
k −2 k
0
0
p
Gi pi k − li 2li σi pi fli Liqi μ
1/pi
li −∞ kli 2
−k0 λi
≤2
k −2
0
pi
fli Lqi μ
2
li −∞
−k0 λi
≤2
k −2
0
≤ C2
k0
Gi pi k − li 1/pi
3.10
kli 2
1/pi
∞
pi
fli Lqi μ
Gi spi
li σi pi 2
li −∞
−k0 λi
li σi pi k −2
0
s2
li σi pi 2
li −∞
p
fi χli Liqi μ
1/pi
≤ Cfi MK̇σi ,λi μ .
pi ,qi
In case 1 < pi < ∞, using the Hölder inequality, we get
−k0 λi
V1i k0 2
k0
≤ 2−k0 λi
Gi k − li 2
⎧
k0
k−2
⎨
⎩k−∞
≤ C2
≤ C2
p
Gi k − li 2li σi pi fli Liqi μ
k0
k−2
≤ C2
k −2
0
≤ Cfi MK̇σi ,λi μ .
pi ,qi
pi
fli Lqi μ
Gi k − li 2
1/pi
1/pi
1/pi
∞
pi
fli Lqi μ
Gi s
li σi pi 2
li −∞
pi
fli Lqi μ
Gi k − li 2
s2
Gi k − li Gi s
s2
li σi pi li −∞ kli 2
−k0 λi
∞
li σi pi li −∞
k −2 k
0
0
k−2
pi −1 ⎫1/pi
⎬
li −∞
li −∞
k−∞
−k0 λi
li −∞
⎩k−∞
pi 1/pi
p
Gi k − li 2li σi pi fli Liqi μ
⎧
k0
k−2
⎨
−k0 λi
fli Lqi μ
li σi li −∞
k−∞
≤ 2−k0 λi
k−2
⎭
pi −1 ⎫1/pi
⎬
⎭
3.11
10
Journal of Inequalities and Applications
In case pi ∞, recalling the definition of Morrey-Herz spaces we get
−k0 λi
V1i k0 2
fli Lqi μ
li σi sup sup 2
li ≤k−2
k≤k0
k−2
Gi k − li li −∞
∞
fli Lqi μ
Gi s
li σi −k0 λi
≤ C2
sup sup 2
k≤k0
li ≤k−2
≤ C2−k0 λi sup sup 2li σi fli Lqi μ
s2
3.12
li ≤k−2
k≤k0
≤ C2−k0 λi sup 2k−2λi fi MK̇σi ,λi μ
pi ,qi
k≤k0
≤ Cfi MK̇σi ,λi μ .
pi ,qi
Therefore, for any 0 < pi ≤ ∞, we have obtained that
V1 ≤ C sup V11 k0 × V12 k0 ≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
p2 ,q2
p1 ,q1
k0 ∈Z
3.13
For V2 , by the growth condition 1.2 of μ, inequality 3.3 and the Cauchy inequality,
we use the analogous arguments as that of V1 to deduce that
k0
−k0 λ1
V2 ≤ C sup 2
k0 ∈Z
k−∞
−k0 λ2
×2
k0
k−∞
k1
k−2
G1 k − l1 2
l1 −∞
fl2 Lq2 μ
p1 1/p1
p2 1/p2
l2 σ2 G2 k − l2 2
l2 k−1
fl1 Lq1 μ
l1 σ1 3.14
: C sup V21 k0 × V22 k0 .
k0 ∈Z
We observe that V21 k0 is equal to V11 k0 , and so we have
V21 k0 ≤ Cf1 MK̇σ1 ,λ1 μ
p1 ,q1
with a constant C independent of k0 .
3.15
Journal of Inequalities and Applications
11
For V22 k0 , noting that 1 < q2 , one can see easily that
−k0 λ2
V22 k0 2
k0
2
k0
2
≤ C2
2
k0
≤ C2
p2 1/p2
fl2 Lq2 μ
p2 1/p2
k−l2 n1/q2 −1 2
l2 k−1
p
f2 χ{2k−2 <|·|≤2k1 } ·L2q2 μ
1/p2
3.16
kσ2 p2 2
k−∞
−k0 λ2
k1
kσ2 p2
k−∞
−k0 λ2
fl2 Lq2 μ
k−l2 n1/q2 −1kσ2 l2 k−1
k−∞
−k0 λ2
k1
k 1
0
p
f2 χk L2q2 μ
1/p2
kσ2 p2 2
k−∞
≤ Cf2 MK̇σ2 ,λ2 μ
p2 ,q2
with a constant C independent of k0 .
Combining inequalities 3.15 and 3.16, we obtain
V2 ≤ C sup V21 k0 × V22 k0 ≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
p2 ,q2
p1 ,q1
k0 ∈Z
3.17
For V3 , by using the growth condition 1.2 of μ, estimate 3.3, and the Cauchy
inequality, we obtain
−k0 λ
V3 ≤ C sup 2
k0 ∈Z
k0
kσp
2
×
−k0 λ
≤ C sup 2
k0 ∈Z
∞
2
fl2 L1 μ
2
l2 k2
kσl
∞
p
p 1/p
−l2 n−α/2kn/h2 2
⎩k−∞
×
fl1 L1 μ
−kn−α/2kn/h1 l1 −∞
k−∞
⎧
k0
⎨
k−2
k−2
2
l1 −∞
−kn−α/2kn/h1 fl1 L1 μ
2−l2 n−α/2kn/h2 fl2 L1 μ
l2 k2
l
l ⎫1/l
⎬
⎭
12
Journal of Inequalities and Applications
k0
−k0 λ1
≤ C sup 2
k0 ∈Z
2
×2
k0
kσ2 p2
2
k0
−k0 λ1
≤ C sup 2
×2
k0
∞
k−2
∞
fl2 L1 μ
2
fl1 Lq1 μ
p2 1/p2
p1 1/p1
l1 σ1 G1 k − l1 2
fl2 Lq2 μ
l2 σ2 H2 k − l2 2
l2 k2
k−∞
p1 1/p1
−l2 n−α/2kn/h2 l1 −∞
k−∞
−k0 λ2
2
l2 k2
k−∞
k0 ∈Z
fl1 L1 μ
kn1/q1 −1 l1 −∞
k−∞
−k0 λ2
k−2
kσ1 p1
p2 1/p2
: C sup V31 k0 × V32 k0 .
3.18
k0 ∈Z
Noting that V31 k0 V21 k0 V11 k0 , by 3.15, we get
V31 k0 ≤ Cf1 MK̇σ1 ,λ1 μ
3.19
p1 ,q1
with a constant C independent of k0 .
As for V32 k0 , we can also write
−k0 λ2
V32 k0 ≤ 2
k0
l2 k2
k0
k−∞
fl2 Lq2 μ
∞
p2 1/p2
l2 σ2 H2 k − l2 2
k−∞
−k0 λ2
2
k0
H2 k − l2 2
l2 k0 1
l2 σ2 fl2 Lq2 μ
p2 1/p2
3.20
1
2
: V32
k0 V32
k0 .
1
2
1
Now, we estimate V32
k0 and V32
k0 , respectively. For V32
k0 , using similar methods
as that for V1i k0 , we consider the following three cases.
Journal of Inequalities and Applications
13
If 0 < p2 ≤ 1, the fact α/2 − n/q2 ≤ λ2 α/2 − n/q2 < σ2 implies that
1
V32
k0 k0
k0
−k0 λ2
≤2
p
H2 p2 k − l2 2l2 σ2 p2 fl2 L2q2
k−∞ l2 k2
k0
l
2 −2
−k0 λ2
≤2
1/p2
μ
p
H2 p2 k − l2 2l2 σ2 p2 fl2 L2q2 μ
1/p2
l2 −∞ k−∞
k0
−k0 λ2
≤2
1/p2
∞
p2
fl2 Lq2 μ
H2 −sp2 l2 σ2 p2 2
l2 −∞
−k0 λ2
≤ C2
3.21
s2
k0
p2
fl2 Lq2 μ
1/p2
l2 σ2 p2 2
l2 −∞
≤ Cf2 MK̇σ2 ,λ2 μ .
p2 ,q2
If 1 < p2 < ∞, the Hölder inequality and the fact α/2 − n/q2 < σ2 yield that
1
V32
k0 −k0 λ2
≤2
k0
≤ C2
k0
l
2 −2
fl2 Lq2 μ
H2 k − l2 2
p2
fl2 Lq2 μ
≤ C2
k0
H2 k − l2 2
l2 −∞
−k0 λ2
≤ C2
k0
s2
p2
fl2 Lq2 μ
l2 σ2 p2 2
l2 −∞
≤ Cf2 MK̇σ2 ,λ2 μ .
p2 ,q2
1/p2
∞
p2
fl2 Lq2 μ
H2 −s
l2 σ2 p2 2
1/p2
l2 σ2 p2 l2 −∞ k−∞
−k0 λ2
p2 1/p2
l2 σ2 l2 k2
k−∞
−k0 λ2
k0
1/p2
3.22
14
Journal of Inequalities and Applications
If p2 ∞, we get
1
V32
k0 −k0 λ2
≤2
sup
k0
H2 k − l2 2
l2 k2
k≤k0
−k0 λ2
≤2
fl2 Lq2 μ
fl2 Lq2 μ
k0
l2 σ2 sup sup2
l2 ≤k0
k≤k0
l2 σ2 H2 k − l2 3.23
l2 k2
∞
−k0 λ2
k0 λ 2 f2 MK̇σ2 ,λ2 μ
≤2
sup 2
H2 −z
p2 ,q2
k≤k0
z2
≤ Cf2 MK̇σ2 ,λ2 μ .
p2 ,q2
Thus, we get
1
V32
k0 ≤ Cf2 MK̇σ2 ,λ2 μ
3.24
p2 ,q2
with a constant C independent of k0 .
2
k0 , by the fact α/2 − n/q2 ≤ λ2 α/2 − n/q2 < σ2 , we have
For V32
2
V32
k0 −k0 λ2
≤2
k0
k−∞
∞
fl2 Lq2 μ
l2 σ2 H2 k − l2 2
l2 k0 1
p2 1/p2
⎧
⎡
⎞1/p2 ⎤p2 ⎫1/p2
⎛
⎪
⎪
k0
l
∞
⎨
⎬
2
p
⎢ ⎥
≤ 2−k0 λ2
H2 k − l2 ⎝
2jσ2 p2 f2 χj L2q2 μ ⎠
⎣
⎦
⎪
⎪
⎩k−∞ l2 k0 1
⎭
j−∞
−k0 λ2
≤ C2
k0
(
∞
H2 k − l2 2
k−∞ l2 k0 1
≤ Cf2 MK̇σ2 ,λ2 μ 2−k0 λ2
p2 ,q2
f2 MK̇σ2 ,λ2 μ
l2 λ2 k0
3.25
p2 ,q2
1/p2 H2 kp2 )p2 1/p2
∞
l2 λ2
2
H2 −l2 l2 k0 1
k−∞
≤ Cf2 MK̇σ2 ,λ2 μ .
p2 ,q2
Therefore, the inequality above and inequalities 3.19, 3.20, and 3.24 yield
V3 ≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
p1 ,q1
p2 ,q2
3.26
To estimate the term V5 , using Proposition 1.1, the Lq -boundedness for Iα,2 , we obtain
Iα,2 fl fl χk q ≤ Cfl q fl q .
1
2
1 L 1 μ
2 L 2 μ
L μ
3.27
Journal of Inequalities and Applications
15
Thus, a similar argument shows that
⎧
k0
⎨
⎞p ⎫1/p
⎬
Iα,2 fl , fl χk q ⎠
V5 ≤ sup 2−k0 λ
2kσp ⎝
1
2
L
μ
⎩k−∞
⎭
k0 ∈Z
l ,l ∈Λ
⎛
1 2
≤ C sup 2−k0 λ
k0 ∈Z
−k0 λ
≤ C sup 2
k0 ∈Z
⎧
k0
⎨
2kσl
⎩k−∞
2
i1
5
k1 fl q
1 L 1 μ
l1 k−1
k0
l kσ1 pi
2
l ⎫1/l
⎬
l2 k−1
⎭
3.28
pi 1/pi
k1 fl Lqi μ
i
li k−1
k−∞
k1 fl q
2 L 2 μ
≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
p2 ,q2
p1 ,q1
For V6 , by condition 1.2 and inequality 3.3, we have
−k0 λ1
V6 ≤ sup 2
k0 ∈Z
k0
×2
k0
fl1 Lq1 μ
∞
G1 k − l1 2
fl2 Lq2 μ
l2 σ2 H2 k − l2 2
l2 k2
k−∞
p1 1/p1
l1 σ1 l1 k−1
k−∞
−k0 λ2
k1
p2 1/p2
3.29
: sup V61 k0 × V62 k0 ≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
p2 ,q2
p1 ,q1
k0 ∈Z
Finally to estimate the term V9 , by the Hölder inequality, condition 1.2, and inequality 3.3, one sees that
−k0 λ1
V9 ≤ sup 2
k0 ∈Z
×2
k0
k−∞
∞
l1 σ1 H1 k − l1 2
l1 k2
k−∞
−k0 λ2
k0
∞
fl1 Lq1 μ
fl2 Lq2 μ
l2 σ2 H2 k − l2 2
l2 k2
p1 1/p1
p2 1/p2
3.30
: sup V91 k0 × V92 k0 ≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
p2 ,q2
p1 ,q1
k0 ∈Z
Combining all the estimates for Vi for i 1, 2, . . . , 9, we get
Iα,2 fl fl σ,λ ≤ Cf1 σ1 ,λ1 f2 σ2 ,λ2 .
1
2
MK̇p,q μ
MK̇
μ
MK̇
μ
p1 ,q1
This is the desired estimate of Theorem 1.2.
The proof of Theorem 1.2 is completed.
p2 ,q2
3.31
16
Journal of Inequalities and Applications
n1−1/q1 ,λ1
Next we turn to the proof of Theorem 1.3. Let f1 , f2 be functions in MK̇p1 ,q1
and
n1−1/q ,λ
MK̇p2 ,q2 2 2 μ,
μ
respectively. Obviously, to prove the theorem, we only need to find a
→
−
constant C > 0 independent of f such that
−k0 λ
γ sup 2
k0 ∈Z
k0
kn2−1/gp
2
→
p/q
− μ x ∈ Ek : Iα,2 f x > 9γ
1/p
3.32
k−∞
≤ Cf1 MK̇n1−1/p1 ,λ1 μ f2 MK̇n1−1/p2 ,λ2 μ
p2 ,q2
p1 ,q1
for all γ > 0.
By the decomposition of fi above, we get
−k0 λ
γ sup 2
k0 ∈Z
k0
kn2−1/gp
2
→
p/q
− μ x ∈ Ek : Iα,2 f x > 9γ
1/p
k−∞
≤ Cγ sup 2−k0 λ
k0 ∈Z
≤C
9
⎧
k0
⎨
2kn2−1/gp μ
⎩k−∞
−k0 λ
γ sup 2
1
k0 ∈Z
G k0 : C
∞ Iα,2 fl , fl x > 9γ
x ∈ Ek :
1
2
p/q ⎫1/p
⎬
li −∞
⎭
9
H ,
1
3.33
where
⎧
⎪
k0
⎨
⎫⎞p/q ⎫1/p
⎛⎧
⎪
⎨
⎬
⎬
kn2−1/gp ⎝
⎠
x ∈ Ek :
Iα,2 fl1 , fl2 x > γ
G k0 2
μ
.
⎪
⎪
⎩
⎭
⎩k−∞
⎭
l ,l ∈Λ
1 2
3.34
Similar to the proof of Theorem 1.2, we only need to estimate H1 , H2 , H3 , H5 , H6 , and
H9 , respectively.
For H1 , using the Chebychev inequality, the Hölder inequality, 1.2, and 3.3, we
obtain
⎫⎞1/q
⎛⎧
⎨
⎬
μ⎝ x ∈ Ek :
|Iα,2 fl1 , fl2 x| > γ ⎠
⎩
⎭
l ,l ∈Λ
1 2
≤ Cγ −1
1
2
k−2
i1
2nli −k1−1/qi fli Lqi μ .
li −∞
3.35
Journal of Inequalities and Applications
17
Therefore, the facts l < p, 0 < pi ≤ 1 and the Cauchy inequality imply that
H1 ≤ C
2
−k0 λi
sup 2
i1 k0 ∈Z
k0
k−2
pi
fli Lqi μ
2
k−∞ li −∞
2 ≤ C fi MK̇n1−1/qi ,λi μ sup2−k0 λi
pi ,qi
i1
≤ Cf1 k0 ∈Z
n1−1/q1 ,λ1
MK̇p1 ,q1
μ
1/pi
li n1−1/qi pi f2 k0
1/pi
3.36
k−2λi pi
2
k−∞
n1−1/q2 ,λ2
MK̇p2 ,q2
μ
.
Now, we consider the estimate of the term H2 . Using a similar argument as that of H1 ,
we can deduce that
k0
−k0 λ1
H2 ≤ C sup 2
k0 ∈Z
2
×2
k0
kn1−1/q2 p2
2
k0
k−2
≤ C sup 2
k0 ∈Z
k1
fl2 Lq2 μ
×2
k 1
0
2
l1 n1−1/q1 p1 2
p
f2 χk L2q2
kn1−1/q2 p2 2
k−∞
p1 1/p1
p2 1/p2
nl2 −k1−1/q2 k−∞ l1 −∞
−k0 λ2
2
l2 k−1
k−∞
−k0 λ1
fl1 Lq1 μ
nl1 −k1−1/q1 l1 −∞
k−∞
−k0 λ2
k−2
kn1−1/q1 p1
p
fl1 L1q1 μ
1/p1
3.37
1/p2
μ
≤ Cf1 MK̇n1−1/q1 ,λ1 μ f2 MK̇n1−1/q2 ,λ2 μ ,
p1 ,q1
p2 ,q2
as desired.
To estimate H3 , we point out the fact
μ
∞ k−2 Iα,2 fl , fl x > γ
x ∈ Ek :
1
2
1/q
l1 −∞ l2 k2
≤ Cγ −1
k−2
2nl1 −k1−1/q1 fl1 Lq1 μ
l1 −∞
∞
2l2 −kα/2−n/q2 fl2 Lq2 μ .
l2 k2
3.38
18
Journal of Inequalities and Applications
So we can show that
−k0 λ1
H3 ≤ C sup 2
k0 ∈Z
k0
k−2
p1
fl1 Lq1 μ
2
k−∞ l1 −∞
k0
∞
−k0 λ2
×2
1/p1
l1 n1−1/q1 p1 p2
fl2 Lq2 μ
1/p2
l2 −kα/2−np2 l2 n1−1/q2 p2 2
k−∞ l2 k2
3.39
: H31 k0 × H32 k0 .
From the estimates of H1 , we know H31 k0 ≤ Cf1 MK̇n1−1/q1 ,λ1 μ , so we only need to
p1 ,q1
show that H32 k0 ≤ Cf2 MK̇n1−1/q2 ,λ2 μ .
p2 ,q2
For H32 k0 , we write
−k0 λ2
H32 k0 ≤ 2
k0
k0
p2
fl2 Lq2 μ
2
k−∞ l2 k2
−k0 λ2
2
k0
1/p2
l2 −kα/2−np2 l2 n1−1/q2 p2 ∞
p2
fl2 Lq2 μ
1/p2
l2 −kα/2−np2 l2 n1−1/q2 p2 2
k−∞ l2 k0 1
3.40
: H132 k0 H232 k0 .
First, the fact 0 < α < 2n yields that
H132 k0 −k0 λ2
≤2
k0
l
2 −2
p2
fl2 Lq2 μ
2
l2 −∞ k−∞
−k0 λ2
≤2
k0
l2 n1−1/q2 p2 2
l2 −∞
≤ Cf2 MK̇n1−1/q2 ,λ2 μ .
p2 ,q2
1/p2
l2 −kα/2−np2 l2 n1−1/q2 p2 1/p2
∞
p2
sα/2−np2
fl2 Lq2 μ
2
s2
3.41
Journal of Inequalities and Applications
19
Second, the fact λ2 < n − α/2 implies
H232 k0 −k0 λ2
≤2
k0
∞
p2
fl2 Lq2 μ
1/p2
l2 −kα/2−np2 l2 n1−1/q2 p2 2
2
k−∞ l2 k2
⎞⎫1/p2
⎬
p
≤ 2−k0 λ2
2l2 −kα/2−np2 ⎝
2jn1−1/q2 p2 f2 χj L2q2 μ ⎠
⎭
⎩k−∞ l k2
j−∞
⎧
k0
∞
⎨
⎛
l2
2
≤ Cf2 n1−1/q2 ,λ2
MK̇p2 ,q2
μ
−k0 λ2
2
k0
∞
3.42
1/p2
l2 −kα/2−np2 l2 λ2
2
2
k−∞ l2 k2
≤ Cf2 MK̇n1−1/q2 ,λ2 μ ,
p2 ,q2
as desired.
To estimate the term H5 , by Proposition 1.1, the weak Lq -boundedness for Iα,2 , we
obtain
Iα,2 fl fl χk q ≤ Cfl q fl q .
1
2
1 L 1 μ
2 L 2 μ
WL μ
3.43
Similar to the estimates of V5 , we have
⎧
k0
⎨
⎞p ⎫1/p
⎬
fl q fl q ⎠
H5 ≤ C sup 2−k0 λ
2kn2−1/gp ⎝
1 L 1 μ
2 L 2 μ
⎩k−∞
⎭
k0 ∈Z
l ,l ∈Λ
⎛
1 2
2
≤ C sup
2−k0 λi
k0 ∈Z i1
k0
5
kn1−1/qi pi
2
k1 fl q
i L i μ
li k−1
k−∞
≤ Cf1 MK̇n1−1/q1 ,λ1 μ f2 MK̇n1−1/q2 ,λ2 μ .
p1 ,q1
p2 ,q2
pi 1/pi
3.44
20
Journal of Inequalities and Applications
For H6 , we obtain
H6 ≤ C sup 2
k0 ∈Z
k0
−k0 λ1
2
k0
×2
≤ Csup2
2
fl2 Lq2 μ
l2 −kα/2−n/q2 2
p
f1 χk L1q1 μ
2
k0
∞
×2
p1 1/p1
p2 1/p2
1/p1
3.45
kn1−1/q1 p1 k−∞
−k0 λ2
2
l2 k2
k 1
0
k0 ∈Z
∞
kn1−1/q2 p2
k−∞
−k0 λ1
fl1 Lq1 μ
nl1 −k1−1/q1 li k−1
k−∞
−k0 λ2
k1
kn1−1/q1 p1
p2
fl2 Lq2 μ
1/p2
l2 −kα/2−np2 l2 n1−1/q2 p2 2
k−∞l2 k2
≤ Cf1 MK̇n1−1/q1 ,λ1 μ f2 MK̇n1−1/q2 ,λ2 μ .
p2 ,q2
p1 ,q1
Finally, we have to estimate the term H9 . It can be deduced that
−k0 λ1
H9 ≤ sup 2
k0 ∈Z
k0
kn1−1/q1 p1
2
k0
×2
kn1−1/q2 p2
2
k−∞
l1 −kα/2−n/q1 2
l1 k2
k−∞
−k0 λ2
∞
∞
fl1 Lq1 μ
fl2 Lq2 μ
l2 −kα/2−n/q2 2
l2 k2
p1 1/p1
p2 1/p2
3.46
: sup H91 k0 × H92 k0 ≤ Cf1 MK̇σ1 ,λ1 μ f2 MK̇σ2 ,λ2 μ .
k0 ∈Z
p1 ,q1
p2 ,q2
Here the estimate of H91 k0 and H92 k0 is similar to that of H32 k0 .
Finally, a combination for the estimates of Hi i 1, 2, . . . , 9 finishes the proof of
Theorem 1.3.
Acknowledgment
This work was supported partly by the National Natural Science Foundation of China under
Grant no. 10771110 and sponsored by the Natural Science Foundation of Ningbo City under
Grant no. 2009A610084.
Journal of Inequalities and Applications
21
References
1 J. Garcı́a-Cuerva and J. M. Martell, “Two-weight norm inequalities for maximal operators and
fractional integrals on non-homogeneous spaces,” Indiana University Mathematics Journal, vol. 50, no. 3,
pp. 1241–1280, 2001.
2 Y. Sawano and H. Tanaka, “Morrey spaces for non-doubling measures,” Acta Mathematica Sinica, vol.
21, no. 6, pp. 1535–1544, 2005.
3 J. Xu, “Boundedness of multilinear singular integrals for non-doubling measures,” Journal of
Mathematical Analysis and Applications, vol. 327, no. 1, pp. 471–480, 2007.
4 J. Lian and H. Wu, “A class of commutators for multilinear fractional integrals in nonhomogeneous
spaces,” Journal of Inequalities and Applications, vol. 2008, Article ID 373050, 17 pages, 2008.
5 C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,” Mathematical Research
Letters, vol. 6, no. 1, pp. 1–15, 1999.
6 X. X. Tao, Y. L. Shi, and S. Y. Zhang, “Boundedness of multilinear Riesz potential on the product of
Morrey and Herz-Morrey spaces,” Acta Mathematica Sinica. Chinese Series, vol. 52, no. 3, pp. 535–548,
2009.
7 Y. Shi and X. Tao, “Boundedness for multilinear fractional integral operators on Herz type spaces,”
Applied Mathematics, Journal of Chinese Universities B, vol. 23, no. 4, pp. 437–446, 2008.
8 Y. Shi and X. Tao, “Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey
spaces,” Hokkaido Mathematical Journal, vol. 38, no. 4, pp. 635–662, 2009.
9 Y. Komori, “Weak type estimates for Calderón-Zygmund operators on Herz spaces at critical indexes,”
Mathematische Nachrichten, vol. 259, pp. 42–50, 2003.