International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 738125, 15 pages

http://dx.doi.org/10.1155/2014/738125

Research Article

Vector-Valued Inequalities in the Morrey Type Spaces

Hua Wang

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 8 February 2014; Accepted 3 May 2014; Published 15 May 2014

Academic Editor: Ingo Witt

Copyright © 2014 Hua Wang. 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.

Abstract.

We will obtain the strong type and weak type estimates for vector-valued analogues of classical Hardy-Littlewood maximal function, weighted maximal function, and singular integral operators in the weighted Morrey spaces when and , and in the generalized Morrey spaces for , where is a growth function on satisfying the doubling condition.

1. Introduction

The classical Hardy-Littlewood maximal function is defined for a locally integrable function on by

where the supremum is taken over all balls containing . It is well known that the maximal operator maps into for all and into . Let be a sequence of locally integrable functions on . For any , we define and

This nonlinear operator was introduced by Fefferman and Stein in [1], and since then it has played an important role in the development of modern harmonic analysis. In this remarkable paper [1], Fefferman and Stein extended the classical maximal theorem to the case of vector-valued functions.

Theorem 1 (see [1]). Let . Then, for every , there exists a constant independent of such that

Theorem 2 (see [1]). Let . Then, for every , there exists a constant independent of such that

A weight is a nonnegative, locally integrable function on ; denotes the ball with the center and radius . For , a weight function is said to belong to , if there is a constant such that, for every ball (see [2, 3]),

For the case ,  , if there is a constant such that, for every ball ,

A weight function if it satisfies the condition for some . We say that , if for any ball there exists an absolute constant such that

It is well known that if with , then . Moreover, if , then, for all balls and all measurable subsets of , there exists such that

Given a ball and , denotes the ball with the same center as whose radius is times that of . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and the weighted measure of by , where .

Given a weight function on , for , the weighted Lebesgue space is defined as the set of all functions such that

We also denote by the weighted weak space consisting of all measurable functions such that

In particular, when equals to a constant function, we will denote and simply by and .

In [4], Andersen and John considered the weighted version of Fefferman-Stein maximal inequality and showed the following.

Theorem 3 (see [4]). Let and . Then, for every , there exists a constant independent of such that

Theorem 4 (see [4]). Let and . Then, for every , there exists a constant independent of such that

Given a weight , the weighted maximal function is defined as

where the supremum is taken over all balls containing . One says that with , if and there is a constant such that, for every ball ,

If is a sequence of locally integrable functions on , then, for any , we can also define and

In [5], Wu proved the following theorem.

Theorem 5 (see [5]). Let and . Then, for every , there exists a constant independent of such that

Obviously, by the definition, we have with , if . Thus, as a direct consequence of Theorem 5, we get the following theorem.

Theorem  5′. Let and . Then, for every , there exists a constant independent of such that

Let us now consider the vector-valued singular integral operators. Suppose that is the unit sphere in () equipped with the normalized Lebesgue measure . Let with be homogeneous of degree zero and satisfy the cancellation condition

where for any . Then, the homogeneous singular integral operator is defined by

For , we say that satisfies the -Dini condition and write , if and

where denotes the integral modulus of continuity of order of defined by

is a rotation in , and . When , is defined by the following expression:

Let be a sequence of locally integrable functions on . For any , as above, the vector-valued singular integral operators with kernels can be defined as and

In [4], Andersen and John derived the weighted strong and weak type estimates for vector-valued singular integral operators with kernels (see also [6, 7]).

Theorem 6 (see [4]). Let be a singular integral operator with kernel . If and , then, for every , there exists a constant independent of such that

Theorem 7 (see [4]). Let be a singular integral operator with kernel . If and , then, for every , there exists a constant independent of such that

Theorem 6 was later extended by Rubio de Francia et al. in [7].

Theorem 8 (see [7]). Let and let be a singular integral operator with kernel . If and , then, for every , there exists a constant independent of such that

In particular, if we take to be a constant function, then we immediately get the following unweighted results (see also [8]).

Theorem 9. Let be a singular integral operator with kernel and . Then, for every , there exists a constant independent of such that

Theorem 10. Let be a singular integral operator with kernel and . Then, for every , there exists a constant independent of such that

Theorem 11. Let and let be a singular integral operator with kernel . If , then, for every , there exists a constant independent of such that

2. Our Main Results

The classical Morrey spaces were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [10–12]. In [13], Mizuhara introduced the generalized Morrey spaces which were later extended and studied by many authors (see [14–18]). In [19], Komori and Shirai defined the weighted Morrey spaces which could be viewed as an extension of weighted Lebesgue spaces and then studied the boundedness of the above classical operators in harmonic analysis on these weighted spaces.

Definition A (see [19]). Let , let  , and let be a weight function on . Then, the weighted Morrey space is defined by

where

and the supremum is taken over all balls in .

For and , we also denote by the weighted weak Morrey spaces of all measurable functions satisfying

Let , , be a growth function, that is, a positive increasing function in , and satisfy the following doubling condition:

where is a doubling constant independent of .

Definition B (see [13]). Let . One denotes by the space of all locally integrable functions defined on , such that for every and all

where is the ball centered at and with radius . Then, one lets be the smallest constant satisfying (34) and becomes a Banach space with norm .

Obviously, when with , is just the classical Morrey spaces introduced in [9]. We also denote by the generalized weak Morrey spaces of all measurable functions for which

for every and all . The smallest constant satisfying (35) is also denoted by .

Let be a sequence of locally integrable functions on . Now, let us formulate our main results as follows. For the continuity properties of vector-valued Hardy-Littlewood maximal function, weighted maximal function, and singular integral operators in the weighted Morrey spaces for all and , we will prove the following.

Theorem 12. Let , , and . Then, for all , there exists a constant independent of such that

Theorem 13. Let , , and . Then, for all , there exists a constant independent of such that

Theorem 14. Let , , and . Then, for all , there exists a constant independent of such that

Theorem 15. Let be a singular integral operator with kernel . If , , and , then, for all , there exists a constant independent of such that

Theorem 16. Let be a singular integral operator with kernel . If , , and , then, for all , there exists a constant independent of such that

Theorem 17. Let and let be a singular integral operator with kernel . If , , and , then, for all , there exists a constant independent of such that

For the vector-valued inequalities in the generalized Morrey spaces for all , we will show the following.

Theorem 18. Assume that satisfies (33) and . Then, for all , there exists a constant independent of such that

Theorem 19. Assume that satisfies (33) and . Then, for and all , there exists a constant independent of such that

Theorem 20. Let be a singular integral operator with kernel . Assume that satisfies (33) and ; then, for all , there exists a constant independent of such that

Theorem 21. Let be a singular integral operator with kernel . Assume that satisfies (33) and ; then, for and all , there exists a constant independent of such that

Theorem 22. Let and let be a singular integral operator with kernel . Assume that satisfies (33) and ; then, for and all , there exists a constant independent of such that

Throughout this paper, the letter always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. By , we mean that there exists a constant such that . Moreover, we will denote the conjugate exponent of by .

3. Boundedness in the Weighted Morrey Spaces

3.1. Proofs of Theorems 12, 13, and 14

Proof of Theorem 12. Let with , and . Fix a ball and decompose , where and denotes the characteristic function of , . Then, we write

Using Theorem 3 and inequality (7), we get

Let us now turn to estimate the other term . For any and any fixed ball , by definition (1), one can write

A simple geometric observation shows that when and , then we have . This fact together with the condition and Hölder’s inequality implies that

Therefore, for any ,

So we have

Combining the above estimates for and and then taking the supremum over all balls , we complete the proof of Theorem 12.

Proof of Theorem 13. Let with and . Fix a ball ; then, we set , where , . Then, for any given , one writes

Theorem 4 and inequality (7) yield

Next, let us turn to deal with the term . We apply Chebyshev’s inequality to obtain

For any and any fixed ball , we still have if and . It then follows from (49) and the condition that

Hence, for any ,

Therefore,

Summing up the above estimates for and and then taking the supremum over all balls and all , we finish the proof of Theorem 13.

Proof of Theorem 14. Let with , and . For any ball and let , where , . Then, we have

By using Theorem 5′ and inequality (7), we thus obtain

To estimate the other term , it follows from definition (13) that, for any and any fixed ball ,

Moreover, as before, we know that if and . Applying the condition and Hölder’s inequality, we can deduce that

Therefore, for any ,

Consequently,

Collecting all the above estimates for and and then taking the supremum over all balls , we complete the proof of Theorem 14.

3.2. Proofs of Theorems 15, 16, and 17

Proof of Theorem 15. Let with , and . Fix a ball and decompose , where , . Then, we write

It follows from Theorem 6 and inequality (7) that

To estimate the term , we first observe that when and , then . This fact together with the condition yields

Then, by duality and Hölder’s inequality with exponent , we get

Furthermore, it follows from Hölder’s inequality, (68), and the condition that

Since , thus, by inequality (8), we find that

where the last series is convergent since and . Combining the above estimates for and and then taking the supremum over all balls , we complete the proof of Theorem 15.

Proof of Theorem 16. Let with and . For any ball and decompose , where ,  . Then, for any given , we can write

Theorem 7 and inequality (7) imply that

In the proof of Theorem 15, we have already showed that, for any ,

On the other hand, it follows directly from the condition that

In addition, since , then by inequality (8), we can see that for all

where in the last inequality we have used the fact that . If

then the inequality

holds trivially. Now, we may suppose that

Then, by the pointwise inequality (75), we can see that

which in turn is equivalent to

Therefore,

Summing up the above estimates for and and then taking the supremum over all balls and all , we conclude the proof of Theorem 16.

Proof of Theorem 17. Let with ,  , and . For each fixed ball , we set , where , . Then, we have

Applying Theorem 8 and inequality (7), we can get

We now turn to deal with the term . An application of Hölder’s inequality gives us that

When and , then we can easily see that . Since with , then

Consequently,

We will consider two cases. When , then it follows directly from the condition that

For the case of , we set . Then, by using Hölder’s inequality with exponent and the fact that , we deduce that

Hence, for every , by the pointwise estimates (87) and (88) together with inequality (8), we finally obtain

Combining the above estimates for and and then taking the supremum over all balls , we are done.

4. Boundedness in the Generalized Morrey Spaces

4.1. Proofs of Theorems 18 and 19

In order to prove the main theorems of this section, we need to establish the following technical lemma.

Lemma 23. Let with and the doubling constant satisfy . Then, for any and any ball , there exists a constant depending only on and such that

Proof. Let with and . Then,

By a standard estimate of maximal functions , we get

where the last inequality is due to . We are done.

Proof of Theorem 18. Let with . We first recall the following vector-valued maximal inequality established by Fefferman and Stein (see [1]); there exists a constant depending on ,  , and such that

holds for any positive functions and any weight function . Moreover, it can be easily shown using vector-valued interpolation that, for all (see [1, 20]),

It is worth pointing out that the above estimate (94) is false in general in the range (see [21]). For any ball with and , we take as the characteristic function of the ball ; then, by Lemma 23 and (94), we get

Therefore, taking the supremum over all balls , we complete the proof of Theorem 18.

Proof of Theorem 19. Let with . In [21], Pérez proved that, for each weight and any positive functions , there exists a constant depending only on and such that

holds for all . For each fixed ball , we again take to be the characteristic function of the ball ; then, by using Lemma 23 and (96), we obtain

Taking the supremum over all balls and all , we finish the proof of Theorem 19.

4.2. Proofs of Theorems 20, 21, and 22

Proof of Theorems 20 and 22. Let with ,  . For any ball with and , we write , where , . Then, we have

Applying Theorems 9 and 11 and the doubling condition (33), we obtain

Let us analyze the second term . To this end, we will consider the following two cases. (i) When , we use the estimate (68) along with Hölder’s inequality to obtain

(ii) The second case is when with . In the proof of Theorem 17, we have already proved that, for any (see (86)),

For the case of , we have

We now proceed to the case where ; in the present situation, we also set . Then, by using Hölder’s inequality with exponent , we deduce that

Summarizing the estimates (100)–(103) derived above, we conclude that

Since , then by using the doubling condition (33) of , we know that the above series is bounded by an absolute constant:

Therefore,

Combining the above estimates for and and then taking the supremum over all balls , we complete the proof of Theorems 20 and 22.

Proof of Theorem 21. Let with . For each fixed ball , we again decompose as , where , . For any given , we write

Let us start with the term . Theorem 10 and the doubling condition (33) imply that

We turn our attention to the estimate of . It follows directly from the estimate (68) that, for any ,

Notice that . Arguing as in the proof of (105), we can get

Hence, for any ,

If , then the inequality

holds trivially. Now if instead we suppose that

then, by the pointwise inequality (111), we can see that

which in turn gives us that

Therefore,

Summing up the above estimates for and and then taking the supremum over all balls and all , we conclude the proof of Theorem 21.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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