Boundedness of Vector-Valued Intrinsic Square Functions in Morrey Type Spaces

Abstract

We will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the weighted Morrey spaces when , , and in the generalized Morrey spaces for , where is a growth function on satisfying the doubling condition.

1. Introduction and Main Results

The intrinsic square functions were first introduced by Wilson in [1, 2]; they are defined as follows. For , let be the family of functions defined on such that has support containing in , , and, for all , For and , we set Then we define the intrinsic square function of (of order ) by where denotes the usual cone of aperture one: Let be a sequence of locally integrable functions on . For any , Wilson [2] also defined the vector-valued intrinsic square functions of by

In [2], Wilson has established the following two theorems.

Theorem A (see [2]). Let , , and (Muckenhoupt weight class). Then there exists a constant independent of such that

Theorem (see [2]). Let and . Then, for any given weight function and , there exists a constant independent of and such that where denotes the standard Hardy-Littlewood maximal operator.

If we take , then for a.e. by the definition of weights (see Section 2). Hence, as a straightforward consequence of Theorem , we obtain the following.

Theorem B. Let , , and . Then there exists a constant independent of such that

In particular, if we take to be a constant function, then we immediately get the following.

Theorem C. Let and . Then there exists a constant independent of such that

Theorem D. Let and . Then there exists a constant independent of such that

On the other hand, the classical Morrey spaces were originally introduced by Morrey in [3] to study the local behavior of solutions to second order elliptic partial differential equations. Since then, these spaces play an important role in studying the regularity of solutions to 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 [4–6]. In [7], Mizuhara introduced the generalized Morrey spaces which was later extended and studied by many authors (see [8–12]). In [13], Komori and Shirai defined the weighted Morrey spaces which could be viewed as an extension of weighted Lebesgue spaces and then discussed the boundedness of the above classical operators in harmonic analysis on these weighted spaces. Recently, in [14–16], we have established the strong type and weak type estimates for intrinsic square functions on and .

For the boundedness of vector-valued intrinsic square functions in the weighted Morrey spaces for all and , we will prove the following.

Theorem 1. Let , , , and . Then there is a constant independent of such that

Theorem 2. Let , , , and . Then there is a constant independent of such that

For the continuity properties of in for all , we will show the following.

Theorem 3. Let and . Assume that satisfies (15) and ; then there is a constant independent of such that

Theorem 4. Let and . Assume that satisfies (15) and ; then there is a constant independent of such that

2. Notations and Definitions

2.1. Generalized Morrey Spaces

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 5 (see [7]). Let . We denote 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 we let be the smallest constant satisfying (16) and becomes a Banach space with norm .

Obviously, when with , is just the classical Morrey spaces introduced in [3]. We also denote by the generalized weak Morrey spaces of all measurable functions for which for every and all . The smallest constant satisfying (17) is also denoted by .

2.2. Weighted Morrey Spaces

A weight is a positive, locally integrable function on ; denotes the ball with the center and radius . 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 . For , a weight function is said to belong to , if there is a constant such that, for every ball , For the case , , if there is a constant such that, for every ball , A weight function if it satisfies the condition for some . It is well known that if with , then, for any ball , there exists an absolute constant such that Moreover, if , then, for all balls and all measurable subsets of , there exists a number independent of and such that

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 .

Definition 6 (see [13]). Let , , and 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

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.

3. Proofs of Theorems 1 and 2

Proof of Theorem 1. Let with and . Fix a ball and decompose , where and denotes the characteristic function of , . Then we write Using Theorem A and inequality (20), we have Let us now turn to estimate the other term . For any , , , and , we have For any , , and , then, by a direct computation, we can easily see that Thus, by using the above inequalities (29) and (30), together with Minkowski's inequality for integrals, we deduce Then, by duality and Cauchy-Schwarz inequality, we get Furthermore, it follows from Hölder’s inequality, (32), and the condition that where we denote the conjugate exponent of by . Note that for all . Hence, we apply inequality (21) to obtain where the last series is convergent since and . Summarizing the above two estimates for and and then taking the supremum over all balls , we complete the proof of Theorem 1.

Proof of Theorem 2. Let with . Fix a ball ; we set , where , . Then, for any given , one writes
Theorem B and inequality (20) imply We now turn to deal with the other term . In the proof of Theorem 1, we have already shown that, for any (see (32)), It follows directly from the condition that In addition, since , then, by inequality (21), we can see that, for all , where in the last inequality we have used the fact that . If , then the inequality holds trivially. Now if instead we suppose that then, by the pointwise inequality (39), we have which is equivalent to Therefore Summing up the above estimates for and and then taking the supremum over all balls and all , we finish the proof of Theorem 2.

4. Proofs of Theorems 3 and 4

Proof of Theorem 3. Let with . For any ball with and , we write , where , . Then we have Applying Theorem C and the doubling condition (15), we obtain We now turn to estimate the other term . We first use inequality (32) and Hölder's inequality to obtain Hence Since , then, by using the doubling condition (15) of , we know Therefore Combining the above estimates for and and then taking the supremum over all balls , we complete the proof of Theorem 3.

Proof of Theorem 4. Let . For each fixed ball , we again decompose as , where , . For any given , then we write Theorem D and the doubling condition (15) imply We turn our attention to the estimate of . Using the preceding estimate (32), we can deduce that, for all , Note that . Arguing as in the proof of (49), we can get Hence, for any , If , then the inequality holds trivially. Now we may suppose that Then, by the pointwise inequality (55), we can see which 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 4.

Conflict of Interests

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