Abstract
The authors consider the multilinear Riesz potential operator defined by 
, where 
denotes the 
-tuple 
, 
the nonnegative integers with 
, 
, 
and 
is a nonnegative 
-dimensional Borel measure. In this paper, the boundedness for the operator 
on the product of homogeneous Morrey-Herz spaces in nonhomogeneous setting is found.
Similar content being viewed by others
1. Introduction
Let 
denote a ball centered at 
with radius 
, and for any 
, 
will mean the ball with the same center as 
and with radius 
. A Borel measure 
on 
is called a doubling measure if it satisfies the so-called doubling condition; that is, there exists a constant 
such that
for every ball 
. 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 
(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) measure space, here we will consider only 
, equipped with a nonnegative 
-dimensional Borel measure 
, that is, a measure satisfying the growth condition
for any ball 
and 
is a fixed real number such that 
. 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 
defined by
was studied by García-Cuerva and Martell [1] in 2001. García-Cuerva and Martell proved that 
is bounded from 
to 
for all 
and 
and that 
is bounded from 
to 
. Here 
and 
denote the Lebesgue spaces and weak Lebesgue spaces with measure 
, 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
in the nonhomogeneous case, where, and throughout this paper, we denote by 
the 
-tuple 
, 
the nonnegative integers with 
. They obtained the following.
Proposition 1.1 (see [4]).
Let 
, 
with 
, 
, then
(a)if each 
, 
is bounded from 
to 
,
(b)if 
for some 
, 
is bounded from 
to 
.
Obviously, it is Lemma 7 in [5] if 
is the Lebesgue measure in the proposition above and it is a multilinear setting of the result of García-Cuerva and Martell [1].
In addition, in the article [6–8], we have obtained the boundedness of the operator 
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 
in the product of (weak) homogeneous Morrey-Herz spaces in the nonhomogeneous setting.
The definitions of the (weak) homogeneous Morrey-Herz spaces 
and (weak) homogeneous Herz spaces 
will be given in Section 2, here we only point out that 
and 
for 
.
We will establish the following boundedness of the multilinear Riesz potential operator 
on the homogeneous Morrey-Herz spaces.
Theorem 1.2.
Let 
, 
, 
, 
and 
for 
. Suppose that 
, 
, 
, 
then
with a constant 
independent of 
.
In the case 
for 
, we will use the weak homogeneous Morrey-Herz spaces 
and weak homogeneous Herz spaces 
to derive the following boundedness for the operator 
.
Theorem 1.3.
Let 
, 
, 
and 
for 
. Suppose that 
, 
, 
then
with a constant 
independent of 
.
Remark 1.4.
The restriction 
in Theorem 1.3 cannot be removed see [9] for an counter-example when 
and 
.
In addition, we remark that the (weak) homogeneous Morrey-Herz spaces generalize the (weak) homogeneous Herz spaces. Particularly, 
and 
for 
and 
. Moreover, we have 
, the weighted 
spaces 
for 
and 
, for details, see Section 2.
Hence, it is easy to obtain the following corollaries from the theorems above.
Corollary 1.5.
Let 
, 
, 
.
-
(i)
If
, 
, 
with 
for 
. Then
, 
, 
with 
for 
. Then
with a constant 
independent of 
.
-
(ii)
If
and 
for 
then
and 
for 
then
with a constant 
independent of 
.
Corollary 1.6.
Let 
with 
with 
for 
then
with a constant 
independent of 
.
Throughout this paper, the letter 
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 
, 
and 
for 
the characteristic function of the set 
.
Definition 2.1.
Let 
, 
and 
. The homogeneous Herz spaces 
are defined to be the following space of functions:
where
and the usual modification should be made when 
.
Definition 2.2.
Let 
, 
and 
. The weak homogeneous Herz spaces 
are defined by
where
and the usual modification should be made when 
.
Definition 2.3.
Let 
, 
, 
and 
. The homogeneous Morrey-Herz spaces 
are defined by
where
and the usual modifications should be made when 
.
Definition 2.4.
Let 
, 
, 
and 
. The weak homogeneous Morrey-Herz spaces 
are defined by
where
and the usual modifications should be made when 
.
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 
. Actually, the similar procedure works for all 
.
Indeed, we decompose 
as
To shorten the formulas below, we set
It is easy to see that the case for 
is analogous to the case for 
, the case for 
is similar to the case for 
, and the case for 
is analogous to the case for 
, respectively. Thus, by the symmetry of 
and 
in the operator 
, we will only discuss the cases for 
belong to 
, 
, 
, 
, 
and 
, respectively.
By a direct computation, we have the following fact that, for 
,
We will use estimates (3.3) in the proof of theorems below. In addition, we always let 
, 
, 
for 
, and use the notations
It is easy to see that 
when 
, and that 
when 
.
We will also use repeatedly the inequality 
for 
.
Now we are ready to the proof of Theorem 1.2. Suppose that 
, by the decomposition of 
above, we get
where
To estimate the term 
, we first note that inequality (3.3) and the growth condition (1.2) of 
imply that
Hence, by the fact 
, the Cauchy inequality and the growth condition (1.2) of 
, we can show that
where
In case 
, using the fact 
, we have
In case 
, using the Hölder inequality, we get
In case 
, recalling the definition of Morrey-Herz spaces we get
Therefore, for any 
, we have obtained that
For 
, by the growth condition (1.2) of 
, inequality (3.3) and the Cauchy inequality, we use the analogous arguments as that of 
to deduce that
We observe that 
is equal to 
, and so we have
with a constant 
independent of 
.
For 
, noting that 
, one can see easily that
with a constant 
independent of 
.
Combining inequalities (3.15) and (3.16), we obtain
For 
, by using the growth condition (1.2) of 
, estimate (3.3), and the Cauchy inequality, we obtain
Noting that 
, by (3.15), we get
with a constant 
independent of 
.
As for 
, we can also write
Now, we estimate 
and 
respectively. For 
, using similar methods as that for 
, we consider the following three cases.
If 
, the fact 
implies that
If 
, the Hölder inequality and the fact 
yield that
If 
, we get
Thus, we get
with a constant 
independent of 
.
For 
, by the fact 
, we have
Therefore, the inequality above and inequalities (3.19), (3.20), and (3.24) yield
To estimate the term 
, using Proposition 1.1, the 
-boundedness for 
, we obtain
Thus, a similar argument shows that
For 
, by condition (1.2) and inequality (3.3), we have
Finally to estimate the term 
, by the Hölder inequality, condition (1.2), and inequality (3.3), one sees that
Combining all the estimates for 
for 
, we get
This is the desired estimate of Theorem 1.2.
The proof of Theorem 1.2 is completed.
Next we turn to the proof of Theorem 1.3. Let 
, 
be functions in 
and 
respectively. Obviously, to prove the theorem, we only need to find a constant 
independent of 
such that
for all 
.
By the decomposition of 
above, we get
where
Similar to the proof of Theorem 1.2, we only need to estimate 
, 
, 
, 
, 
and 
respectively.
For 
, using the Chebychev inequality, the Hölder inequality, (1.2), and (3.3), we obtain
Therefore, the facts 
, 
and the Cauchy inequality imply that
Now, we consider the estimate of the term 
. Using a similar argument as that of 
, we can deduce that
as desired.
To estimate 
, we point out the fact
So we can show that
From the estimates of 
, we know 
, so we only need to show that 
.
For 
, we write
First, the fact 
yields that
Second, the fact 
implies
as desired.
To estimate the term 
, by Proposition 1.1, the weak 
-boundedness for 
, we obtain
Similar to the estimates of 
, we have
For 
, we obtain
Finally, we have to estimate the term 
. It can be deduced that
Here the estimate of 
and 
is similar to that of 
.
Finally, a combination for the estimates of 
(
) finishes the proof of Theorem 1.3.
References
García-Cuerva J, Martell JM: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana University Mathematics Journal 2001, 50(3):1241–1280.
Sawano Y, Tanaka H: Morrey spaces for non-doubling measures. Acta Mathematica Sinica 2005, 21(6):1535–1544. 10.1007/s10114-005-0660-z
Xu J: Boundedness of multilinear singular integrals for non-doubling measures. Journal of Mathematical Analysis and Applications 2007, 327(1):471–480. 10.1016/j.jmaa.2006.04.049
Lian J, Wu H: A class of commutators for multilinear fractional integrals in nonhomogeneous spaces. Journal of Inequalities and Applications 2008, 2008:-17.
Kenig CE, Stein EM: Multilinear estimates and fractional integration. Mathematical Research Letters 1999, 6(1):1–15.
Tao XX, Shi YL, Zhang SY: Boundedness of multilinear Riesz potential on the product of Morrey and Herz-Morrey spaces. Acta Mathematica Sinica. Chinese Series 2009, 52(3):535–548.
Shi Y, Tao X: Boundedness for multilinear fractional integral operators on Herz type spaces. Applied Mathematics, Journal of Chinese Universities B 2008, 23(4):437–446. 10.1007/s11766-008-1995-x
Shi Y, Tao X: Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces. Hokkaido Mathematical Journal 2009, 38(4):635–662.
Komori Y: Weak type estimates for Calderón-Zygmund operators on Herz spaces at critical indexes. Mathematische Nachrichten 2003, 259: 42–50. 10.1002/mana.200310093
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Shi, Y., Tao, X. & Zheng, T. Multilinear Riesz Potential on Morrey-Herz Spaces with Non-Doubling Measures. J Inequal Appl 2010, 731016 (2010). https://doi.org/10.1155/2010/731016
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/731016