Multilinear Riesz Potential on Morrey-Herz Spaces with Non-Doubling Measures

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.

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

with a constant independent of .

• (ii) If and for then

(1.8)

with a constant independent of .

Corollary 1.6.

Let with with for then

(1.9)

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:

(2.1)

where

(2.2)

and the usual modification should be made when .

Definition 2.2.

Let , and . The weak homogeneous Herz spaces are defined by

(2.3)

where

(2.4)

and the usual modification should be made when .

Definition 2.3.

Let , , and . The homogeneous Morrey-Herz spaces are defined by

(2.5)

where

(2.6)

and the usual modifications should be made when .

Definition 2.4.

Let , , and . The weak homogeneous Morrey-Herz spaces are defined by

(2.7)

where

(2.8)

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

(31)

To shorten the formulas below, we set

(32)

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 ,

(33)

We will use estimates (3.3) in the proof of theorems below. In addition, we always let , , for , and use the notations

(34)

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

(35)

where

(36)

To estimate the term , we first note that inequality (3.3) and the growth condition (1.2) of imply that

(37)

Hence, by the fact , the Cauchy inequality and the growth condition (1.2) of , we can show that

(38)

where

(39)

In case , using the fact , we have

(310)

In case , using the Hölder inequality, we get

(311)

In case , recalling the definition of Morrey-Herz spaces we get

(312)

Therefore, for any , we have obtained that

(313)

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

(314)

We observe that is equal to , and so we have

(315)

with a constant independent of .

For , noting that , one can see easily that

(316)

with a constant independent of .

Combining inequalities (3.15) and (3.16), we obtain

(317)

For , by using the growth condition (1.2) of , estimate (3.3), and the Cauchy inequality, we obtain

(318)

Noting that , by (3.15), we get

(319)

with a constant independent of .

As for , we can also write

(320)

Now, we estimate and respectively. For , using similar methods as that for , we consider the following three cases.

If , the fact implies that

(321)

If , the Hölder inequality and the fact yield that

(322)

If , we get

(323)

Thus, we get

(324)

with a constant independent of .

For , by the fact , we have

(325)

Therefore, the inequality above and inequalities (3.19), (3.20), and (3.24) yield

(326)

To estimate the term , using Proposition 1.1, the -boundedness for , we obtain

(327)

Thus, a similar argument shows that

(328)

For , by condition (1.2) and inequality (3.3), we have

(329)

Finally to estimate the term , by the Hölder inequality, condition (1.2), and inequality (3.3), one sees that

(330)

Combining all the estimates for for , we get

(331)

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

(332)

for all .

By the decomposition of above, we get

(333)

where

(334)

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

(335)

Therefore, the facts , and the Cauchy inequality imply that

(336)

Now, we consider the estimate of the term . Using a similar argument as that of , we can deduce that

(337)

as desired.

To estimate , we point out the fact

(338)

So we can show that

(339)

From the estimates of , we know , so we only need to show that .

For , we write

(340)

First, the fact yields that

(341)

Second, the fact implies

(342)

as desired.

To estimate the term , by Proposition 1.1, the weak -boundedness for , we obtain

(343)

Similar to the estimates of , we have

(344)

For , we obtain

(345)

Finally, we have to estimate the term . It can be deduced that

(346)

Here the estimate of and is similar to that of .

Finally, a combination for the estimates of () finishes the proof of Theorem 1.3.