Research Article | Open Access
Estimates of Intrinsic Square Functions on Generalized Weighted Morrey Spaces
We prove the boundedness of the intrinsic functions on generalized weighted Morrey spaces , including the strong type estimates and weak type estimates. Moreover, we define the kth-order commutators generated by functions and intrinsic functions, and obtain their strong type estimates on . In some cases, we improve previous results.
It is well known that the theory of Littlewood-Paley square functions plays an important role in harmonic analysis, such as in the study of Fourier multiplier and singular integral operators. About their detailed properties and applications, we refer the readers to [1–3].
For convenience, let us recall some definitions. Suppose is the Poisson integral of , where denotes the Poisson kernel in . The Littlewood-Paley -function and the Lusin area integral (square function) are defined, respectively, by where for any . If , set . The corresponding -function is given by
Let be real and radial and have support contained in , . The continuous square functions and are also defined by where denotes the usual dilation of : .
Recently, Wilson  introduced a natural substitute for the above square functions, which he called the intrinsic square function. This function dominates pointwise all the above square functions and is independent of any particular kernel. At the same time, it is not essentially larger than any particular . For , let be the family of functions having their support in , , and for all and , . If and , we define where . Then the intrinsic square of (of order ) is defined by Here and below, we drop the subscript if . Although the function is depend of kernels with uniform compact support, there is pointwise relation between with different : See  for more details. The intrinsic Littlewood-Paley -function and the intrinsic -function are defined by respectively. In [4, 5], Wilson established the boundedness of intrinsic square functions on weighted Lesbesgue spaces. Their boundedness on various function spaces and their sharp bounds have received great attentions; see [6–17].
The classical Morrey spaces were introduced by Morrey  to study the local behavior of solutions to second-order elliptic partial differential equations. Mizuhara  introduced generalized Morrey spaces. Later, Guliyev defined the generalized Morrey spaces with normalized norm. Recently, Komori and Shirai  first defined the weighted Morrey spaces and studied the boundedness of some classical operators on these spaces. Guliyev et al. [21, 22] defined the generalized weighted Morrey spaces as follows.
Definition 1. Let be a positive measurable function on and let be a weight function on . The generalized weighted Morrey space is the space of all functions with finite norm
where . The weak generalized weighted Morrey space consists of all functions with finite norm
If and with , then ; if , then ; if for , then . There are many papers that discussed the conditions on to obtain the boundedness of operators on the generalized Morrey spaces . For example, see [6, 23–29]. Recently, Guliyev introduced the generalized conditions: where does not depend on and . Under these conditions, the boundedness of some classical operators and commutators in generalized weighted Morrey spaces were obtained, respectively. See [8, 9, 21, 22, 30].
Recently, Wang [12, 16] studied the boundedness of the intrinsic functions and their commutators on weighted Morrey spaces. In [6, 17], the authors obtained the boundedness of the intrinsic functions and their commutators on generalized Morrey spaces. Guliyev et al. [7–9] obtained the estimates for vector-valued intrinsic square functions and their th-order commutators on vector-valued generalized weighted Morrey spaces. In this paper, we prove the boundedness of the intrinsic functions and their th-order commutators on generalized weighted Morrey spaces under the conditions (11), (12), and , respectively. Our partial results coincide with some results [7–9], but we improve known results in some case. See Remark 7.
The rest of the paper is organized as follows: some definitions and our main results are stated in Section 2. In Section 3, we give some lemmas. Finally, in Section 4, we prove our main theorem. Throughout this paper, means that there is a positive constant independent of all essential variables such that . If and , then we write . We denote the conjugate exponent of by and if .
2. Definitions and Main Results
In this paper, denotes the ball with center and radius . Given a ball and , denotes the ball with the same center as whose radius is times that of , and . A weight is a locally integrable function on which takes values in almost everywhere. For a weight and a measurable set , we denote the characteristic function of by , the Lebesgue measure of by and the weighted measure of by , where . Moreover, for a locally integrable function , we define
For simplicity, we denote by if .
Definition 2 (see ). A locally integrable function is in if
Definition 3 (see ). We say that a weight function , , if where the supremum is taken over all balls in . A weight function if for some . A weight function , if
Definition 4. Let be a locally integrable function on and The th-order commutators and are defined by respectively. Analogously, the th-order commutators are defined by
Theorem 5. Let , , and satisfy condition (11). Then,
Theorem 6. Let , , , and satisfy condition (11). Then for , we have
Theorem 6′. Let , , and let satisfy condition (11). Then, for , we have
Remark 7. If , comparing Theorem 6 with Theorem 6′, we obtain that the conclusion of Theorem 6 is better. So in this sense, our Theorem 6 improves the result in . If , the conclusion of Theorem 6′ is better than that of Theorem 6. Therefore, we improve the corresponding result in [6, 17]. Here, we point out that Theorem 6′ and Theorem 9′ can be extended to vector-valued cases; see [7–9].
Theorem 8. Let , , , and satisfy condition (12). Let ; then
Theorem 9. Let , , , and satisfy condition (12). If and , then
Theorem 9′. Let , , , and satisfy condition (12). If and , then
Corollary 10. Let , , , and satisfy condition (11); then
Corollary 11. Let , , , and satisfy condition (12). If , then
For any and , the following inequality proved in  holds: So we can readily obtain the following.
Lemma 14. Let and . If is a weight, we have
By the similar argument as in , we can get the following.
Lemma 15. Let , , and . Then, the th-order commutators and are all bounded from to itself whenever .
In the following, we will give a lemma about the Hardy type operator: where is a nonnegative Borel measure on ; we have the following.
Lemma 17 (see ). Let and . Suppose and . Then,(i) for , we have where is independent of , , , , and .(ii) for , we have where is independent of , , , , and .
Lemma 18 (see ). Let , , and . Then, for any , we have
4. Proofs of Main Theorems
Proof of Theorem 5. We will adopt the idea used in . Fix a ball and decompose , where and . Then, for , we have
Since for , we can obtain the following inequality from 
For and with , a calculation shows that
See  for its proof. Therefore, for , we get
On the other hand, for with , we have
Since , , and , we have . So we obtainBy Minkowski’s inequality and for and , we have
It follows from Fubini’s theorem, Hölder’s inequality, and with that
Thus, for , the following estimate is valid:
Combining the above estimates (36), (39), and (44), we have
Hence, applying the definition of and substitution of variables, we have
Suppose and . Since satisfies condition (11), we can verify that , satisfy condition (32). Obviously, is decreasing on variable . So, by Lemma 16 with , we can conclude
Let . From , , (38) and (44), it follows that By the definition of and Lemma 16 with , we get
Proof of Theorem 6. By the definition of , we have Thus, By Theorem 5, we have It remains to estimate . To estimate it, we will divide into two terms. As before, where and . For the first term, by Lemma 14, Theorem 5, and (38), we can easily deduce Now, we estimate the second term. For with , we have Notice that and , so . Thus, Because for and , by Fubini’s theorem and Hölder’s inequality, So Combining (53), (54), and (58), we have Thus, by changing of variables and Lemma 16 with , we get Therefore, using (51), (52), and (60), it follows that where the series are convergent since .
Proof of Theorem 8. Fix a ball and decompose , where . Then, By Lemma 15 and (38), we have that To estimate , we divide into two parts: First, . From the proof of Theorem 5, we know Hence, using Lemma 17, we obtain For , note that with and ; thus, by Minkowski’s inequality, Applying Fubini’s theorem, Hölder’s inequality, and Lemma 17, we have