Estimates of Intrinsic Square Functions on Generalized Weighted Morrey Spaces

Abstract

We prove the boundedness of the intrinsic functions on generalized weighted Morrey spaces [superscript] M p , [straight phi] [/superscript] ( w ) , including the strong type estimates and weak type estimates. Moreover, we define the kth-order commutators generated by BMO [superscript] R n [/superscript] functions and intrinsic functions, and obtain their strong type estimates on [superscript] M p , [straight phi] [/superscript] ( w ) . In some cases, we improve previous results.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Guilian Gao 1 and Xiaomei Wu 2

Academic Editor:Aurelian Gheondea

1, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
2, Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 24 March 2014; Accepted 8 July 2014; 22 July 2014

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.

1. Introduction

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 u ( x , t ) = ( _{P t} * f ) ( x ) is the Poisson integral of f , where _{P t} ( x ) denotes the Poisson kernel in ^{R + n + 1} = ^{R n} × ( 0 , ∞ ) . The Littlewood-Paley g -function and the Lusin area integral (square function) _{S β} are defined, respectively, by [figure omitted; refer to PDF] where _{Γ β} ( x ) = { ( y , t ) ∈ ^{R + n + 1} : | x - y | < β t } for any β > 0 . If β = 1 , set _{S β} ( f ) = S ( f ) . The corresponding ^{g λ *} -function ^{g λ *} ( f ) is given by [figure omitted; refer to PDF]

Let ψ ∈ ^{C c ∞} ( ^{R n} ) be real and radial and have support contained in { x ∈ ^{R n} : | x | ...4; 1 } , _{∫ ^{R n}} ... [varphi] ( x ) d x = 0 . The continuous square functions _{g ψ} ( f ) and _{S ψ , β} ( f ) are also defined by [figure omitted; refer to PDF] where _{ψ t} denotes the usual ^{L 1} dilation of ψ : _{ψ t} ( y ) = ^{t - n} ψ ( y / t ) .

Recently, Wilson [4] 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 _{S ψ , β} ( f ) . For 0 < α ...4; 1 , let _{C α} be the family of functions [varphi] having their support in { x ∈ ^{R n} : | x | ...4; 1 } , _{∫ ^{R n}} ... [varphi] ( x ) d x = 0 , and for all x and y , | [varphi] ( x ) - [varphi] ( y ) | ...4; | x - y ^{| α} . If f ∈ ^{L loc ... 1} ( ^{R n} ) and ( y , t ) ∈ ^{R + n + 1} , we define [figure omitted; refer to PDF] where _{[varphi] t} ( y ) = ^{t - n} [varphi] ( y / t ) . Then the intrinsic square of f (of order α ) is defined by [figure omitted; refer to PDF] Here and below, we drop the subscript β if β = 1 . Although the function _{G α , β} ( f ) is depend of kernels with uniform compact support, there is pointwise relation between _{G α , β} ( f ) with different β ...5; 1 : [figure omitted; refer to PDF] See [4] for more details. The intrinsic Littlewood-Paley g -function and the intrinsic ^{g λ *} -function are defined by [figure omitted; refer to PDF] 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 ^{L p , λ} ( ^{R n} ) were introduced by Morrey [18] to study the local behavior of solutions to second-order elliptic partial differential equations. Mizuhara [19] introduced generalized Morrey spaces. Later, Guliyev defined the generalized Morrey spaces ^{M p , [straight phi]} ( ^{R n} ) with normalized norm. Recently, Komori and Shirai [20] first defined the weighted Morrey spaces ^{L p , κ} ( w ) and studied the boundedness of some classical operators on these spaces. Guliyev et al. [21, 22] defined the generalized weighted Morrey spaces ^{M p , [straight phi]} ( w ) as follows.

Definition 1.

Let [straight phi] ( x , r ) be a positive measurable function on ^{R n} × ^{R +} and let w ( x ) be a weight function on ^{R n} . The generalized weighted Morrey space ^{M p , [straight phi]} ( w ) is the space of all functions f ∈ ^{L loc ... p , w} ( ^{R n} ) with finite norm [figure omitted; refer to PDF] where w ( B ( x , r ) ) = _{∫ B ( x , r )} ... w ( y ) d y . The weak generalized weighted Morrey space W ^{M p , [straight phi]} ( w ) consists of all functions f ∈ W ^{L loc ... p , w} ( ^{R n} ) with finite norm [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

If w ( x ) ...1; 1 and [straight phi] ( x , r ) = ^{r ( λ - n ) / p} with 0 < λ < n , then ^{M p , [straight phi]} ( w ) = ^{L p , λ} ( ^{R n} ) ; if w ( x ) ...1; 1 , then ^{M p , [straight phi]} ( 1 ) = ^{M p , [straight phi]} ( ^{R n} ) ; if [straight phi] ( x , r ) = w ( B ( x , r ) ^{) ( κ - 1 ) / p} for 0 < κ < 1 , then ^{M p , [straight phi]} ( w ) = ^{L p , κ} ( w ) . There are many papers that discussed the conditions on [straight phi] ( x , r ) to obtain the boundedness of operators on the generalized Morrey spaces ^{M p , [straight phi]} ( ^{R n} ) . For example, see [6, 23-29]. Recently, Guliyev introduced the generalized conditions: [figure omitted; refer to PDF] where C does not depend on x and r . 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 k th-order commutators on vector-valued generalized weighted Morrey spaces. In this paper, we prove the boundedness of the intrinsic functions and their k th-order commutators on generalized weighted Morrey spaces ^{M p , [straight phi]} ( w ) under the conditions (11), (12), and w ∈ _{A p} , 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, A ... B means that there is a positive constant C independent of all essential variables such that A ...4; C B . If A ... B and B ... A , then we write A [approximate] B . We denote the conjugate exponent of p > 1 by ^{p [variant prime]} = p / ( p - 1 ) and ^{p [variant prime]} = ∞ if p = 1 .

2. Definitions and Main Results

In this paper, B = B ( _{x 0} , r ) denotes the ball with center _{x 0} and radius r . Given a ball B and λ > 0 , λ B denotes the ball with the same center as B whose radius is λ times that of B , and ^{B c} = ^{R n} \ B . A weight is a locally integrable function on ^{R n} which takes values in ( 0 , ∞ ) almost everywhere. For a weight w and a measurable set E , we denote the characteristic function of E by _{χ E} , the Lebesgue measure of E by | E | and the weighted measure of E by w ( E ) , where w ( E ) = _{∫ E} w ( x ) d x . Moreover, for a locally integrable function b ( x ) , we define [figure omitted; refer to PDF]

For simplicity, we denote _{b B ( x , r ) , w} by _{b B ( x , r )} if w = 1 .

Definition 2 (see [1]).

A locally integrable function b ( x ) is in BMO ( ^{R n} ) if [figure omitted; refer to PDF]

Definition 3 (see [3]).

We say that a weight function w ∈ _{A p} , 1 < p < ∞ , if [figure omitted; refer to PDF] where the supremum is taken over all balls B in ^{R n} . A weight function w ∈ _{A ∞} if w ∈ _{A p} for some 1 < p < ∞ . A weight function w ∈ _{A 1} , if [figure omitted; refer to PDF]

Definition 4.

Let b be a locally integrable function on ^{R n} and [figure omitted; refer to PDF] The k th-order commutators [ b , _{G α} ^{] k} and [ b , _{g α} ^{] k} are defined by [figure omitted; refer to PDF] respectively. Analogously, the k th-order commutators [ b , ^{g λ , α * ] k} are defined by [figure omitted; refer to PDF]

Theorem 5.

Let 0 < α ...4; 1 , 1 ...4; p < ∞ , w ∈ _{A p} and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (11). Then, [figure omitted; refer to PDF]

Theorem 6.

Let 0 < α ...4; 1 , 1 ...4; p < ∞ , w ∈ _{A p} , and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (11). Then for λ > 3 + 2 α / n , we have [figure omitted; refer to PDF]

Theorem 6[variant prime]. Let 0 < α ...4; 1 , 1 < p < ∞ , w ∈ _{A p} and let ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (11). Then, for λ > m a x { p , 3 } , we have [figure omitted; refer to PDF]

Remark 7.

If p > 3 + 2 α / n , comparing Theorem 6 with Theorem 6[variant prime], we obtain that the conclusion of Theorem 6 is better. So in this sense, our Theorem 6 improves the result in [12]. If 1 < p ...4; 3 , the conclusion of Theorem 6[variant prime] is better than that of Theorem 6. Therefore, we improve the corresponding result in [6, 17]. Here, we point out that Theorem 6[variant prime] and Theorem 9[variant prime] can be extended to vector-valued cases; see [7-9].

Theorem 8.

Let 0 < α ...4; 1 , 1 < p < ∞ , w ∈ _{A p} , and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (12). Let b ∈ B M O ( ^{R n} ) ; then [figure omitted; refer to PDF]

Theorem 9.

Let 0 < α ...4; 1 , 1 < p < ∞ , w ∈ _{A p} , and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (12). If b ∈ B M O ( ^{R n} ) and λ > 3 + 2 α / n , then [figure omitted; refer to PDF]

Theorem 9[variant prime]. Let 0 < α ...4; 1 , 1 < p < ∞ , w ∈ _{A p} , and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (12). If b ∈ B M O ( ^{R n} ) and λ > m a x { p , 3 } , then [figure omitted; refer to PDF]

For 0 < α ...4; 1 , Wilson [4] showed that _{G α} f and _{g α} f are pointwise comparable. Therefore, by Theorems 5 and 8, we have the following.

Corollary 10.

Let 0 < α ...4; 1 , 1 ...4; p < ∞ , w ∈ _{A p} , and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (11); then [figure omitted; refer to PDF]

Corollary 11.

Let 0 < α ...4; 1 , 1 < p < ∞ , w ∈ _{A p} , and ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (12). If b ∈ B M O ( ^{R n} ) , then [figure omitted; refer to PDF]

Remark 12.

If [straight phi] ( x , r ) = w ( B ( x , r ) ^{) ( κ - 1 ) / p} for 0 < κ < 1 , then ^{M p , [straight phi]} ( w ) = ^{L p , κ} ( w ) . Let w ∈ _{A p} ( 1 ...4; p < ∞ ) ; the pair ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfies the condition (12) with k ...5; 0 . See [30] for its proof. Therefore, Theorems 5, 6[variant prime], 8, 9[variant prime] and Corollaries 10 and 11 contain the results in [12, 16].

Remark 13.

If 1 < p < ∞ , w = 1 , and k = 1 , then Theorems 5-9 and Corollaries 10 and 11 contain the results in [6, 17].

3. Lemmas

For any x ∈ ^{R n} and β ...5; 1 , the following inequality proved in [4] holds: [figure omitted; refer to PDF] So we can readily obtain the following.

Lemma 14.

Let 0 < α ...4; 1 and 1 < p < ∞ . If w is a weight, we have [figure omitted; refer to PDF]

By the similar argument as in [12], we can get the following.

Lemma 15.

Let 0 < α ...4; 1 , 1 < p < ∞ , and w ∈ _{A p} . Then, the k th-order commutators [ b , _{G α} ^{] k} and [ b , ^{g λ , α * ] k} are all bounded from ^{L w p} ( ^{R n} ) to itself whenever b ∈ B M O ( ^{R n} ) .

In the following, we will give a lemma about the Hardy type operator: [figure omitted; refer to PDF] where μ is a nonnegative Borel measure on ( 0 , ∞ ) ; we have the following.

Lemma 16 (see [30]).

The inequality [figure omitted; refer to PDF] holds for all nonnegative and nonincreasing f on ( 0 , ∞ ) if and only if [figure omitted; refer to PDF] and C [approximate] A .

Note that if k = 0 and d μ ( t ) = d t , Lemma 16 was proved in [31].

Lemma 17 (see [30]).

Let w ∈ _{A ∞} and b ∈ B M O ( ^{R n} ) . Suppose k > 0 and _{r 1} , _{r 2} > 0 . Then,

(i) for 1 ...4; p < ∞ , we have [figure omitted; refer to PDF]

: where C is independent of b , w , x , _{r 1} , and _{r 2} .

(ii) for 1 < p < ∞ , we have [figure omitted; refer to PDF]

: where C is independent of b , w , x , _{r 1} , and _{r 2} .

Lemma 18 (see [12]).

Let 0 < α ...4; 1 , 1 < p < ∞ , and w ∈ _{A p} . Then, for any j ∈ ^{Z +} , we have [figure omitted; refer to PDF]

4. Proofs of Main Theorems

Proof of Theorem 5.

We will adopt the idea used in [27]. Fix a ball B = B ( _{x 0} , r ) and decompose f = _{f 1} + _{f 2} , where _{f 1} = f _{χ 2 B} and _{f 2} = f _{χ ( 2 B ^{) c}} . Then, for 1 < p < ∞ , we have [figure omitted; refer to PDF] Since w ∈ _{A p} for 1 < p < ∞ , we can obtain the following inequality from [5] [figure omitted; refer to PDF] For _{|| f || ^{L w p} ( 2 B )} and w ∈ _{A p} with 1 ...4; p < ∞ , a calculation shows that [figure omitted; refer to PDF] See [30] for its proof. Therefore, for 1 < p < ∞ , we get [figure omitted; refer to PDF] On the other hand, for [varphi] ∈ _{C α} with 0 < α ...4; 1 , we have [figure omitted; refer to PDF] Since x ∈ B , z ∈ ( 2 B ^{) c} , and ( y , t ) ∈ Γ ( x ) , we have r ...4; | z - x | ...4; 2 t . So we obtain [figure omitted; refer to PDF] By Minkowski's inequality and | z - x | ...5; ( 1 / 2 ) | z - _{x 0} | for x ∈ B and z ∈ ( 2 B ^{) c} , we have [figure omitted; refer to PDF] It follows from Fubini's theorem, Hölder's inequality, and w ∈ _{A p} with 1 ...4; p < ∞ that [figure omitted; refer to PDF] Thus, for 1 ...4; p < ∞ , the following estimate is valid: [figure omitted; refer to PDF] Combining the above estimates (36), (39), and (44), we have [figure omitted; refer to PDF] Hence, applying the definition of ^{M p , [straight phi]} ( w ) and substitution of variables, we have [figure omitted; refer to PDF] Suppose W ( r ) = _{[straight phi] 2} ( _{x 0} , ^{r - 1 ) - 1} r and V ( x ) = _{[straight phi] 1} ( _{x 0} , ^{r - 1 ) - 1} w ( B ( _{x 0} , ^{r - 1} ) ^{) - 1 / p} . Since ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfies condition (11), we can verify that W ( r ) , V ( r ) satisfy condition (32). Obviously, _{|| f || ^{L w p} ( B ( x 0 , ^{s - 1} ) )} is decreasing on variable s . So, by Lemma 16 with k = 0 , we can conclude [figure omitted; refer to PDF]

Let p = 1 . From w ∈ _{A 1} , [5], (38) and (44), it follows that [figure omitted; refer to PDF] By the definition of W _{M 1 , [straight phi]} ( w ) and Lemma 16 with k = 0 , we get [figure omitted; refer to PDF]

Proof of Theorem 6.

By the definition of ^{g λ , α *} ( f ) , we have [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] By Theorem 5, we have [figure omitted; refer to PDF] It remains to estimate _{|| G α , ^{2 j} ( f ) || ^{M p , w 2} ( ^{R n} )} . To estimate it, we will divide _{|| G α , ^{2 j} ( f ) || ^{L w p} ( B )} into two terms. As before, [figure omitted; refer to PDF] where f = _{f 1} + _{f 2} and _{f 1} = f _{χ 2 B} . For the first term, by Lemma 14, Theorem 5, and (38), we can easily deduce [figure omitted; refer to PDF] Now, we estimate the second term. For [varphi] ∈ _{C α} with 0 < α ...4; 1 , we have [figure omitted; refer to PDF] Notice that | x - y | ...4; ^{2 j} t and | z - y | ...4; t , so | z - x | ...4; ^{2 j + 1} t . Thus, [figure omitted; refer to PDF] Because | z - x | ...5; ( 1 / 2 ) | z - _{x 0} | for x ∈ B = B ( _{x 0} , r ) and | z - _{x 0} | > 2 r , by Fubini's theorem and Hölder's inequality, [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Combining (53), (54), and (58), we have [figure omitted; refer to PDF] Thus, by changing of variables and Lemma 16 with k = 0 , we get [figure omitted; refer to PDF] Therefore, using (51), (52), and (60), it follows that [figure omitted; refer to PDF] where the series are convergent since λ > 3 + 2 α / n .

Proof of Theorem 6[variant prime].

By Lemma 18, using the arguments as the proofs of Theorem 6, we can finish the proof. The details are omitted here.

Proof of Theorem 8.

Fix a ball B = B ( _{x 0} , r ) and decompose f = _{f 1} + _{f 2} , where _{f 1} = f _{χ 2 B} . Then, [figure omitted; refer to PDF] By Lemma 15 and (38), we have that [figure omitted; refer to PDF] To estimate _{|| [ b , G α ^{] k} ( f 2 ) || ^{L w p} ( B )} , we divide [ b , _{G α} ^{] k} ( _{f 2} ) into two parts: [figure omitted; refer to PDF] First, J = | b ( x ) - _{b B , w} ^{| k} _{G α} ( _{f 2} ) ( x ) . From the proof of Theorem 5, we know [figure omitted; refer to PDF] Hence, using Lemma 17, we obtain [figure omitted; refer to PDF] For J J , note that [varphi] ∈ _{C α} with 0 < α ...4; 1 and | x - z | < 2 t ; thus, by Minkowski's inequality, [figure omitted; refer to PDF] Applying Fubini's theorem, Hölder's inequality, and Lemma 17, we have [figure omitted; refer to PDF] where we use w ∈ _{A p} in the last inequality. Combining the previous estimates for J and J J , we obtain [figure omitted; refer to PDF] Therefore, it follows from the above estimates (62), (63), and (69) that [figure omitted; refer to PDF] By substitution of variables, we obtain [figure omitted; refer to PDF] Let W ( r ) = _{[straight phi] 2} ( _{x 0} , ^{r - 1 ) - 1} r and let V ( r ) = _{[straight phi] 1} ( _{x 0} , ^{r - 1 ) - 1} w ( B ( _{x 0} , ^{r - 1} ) ^{) - 1 / p} . Since ( _{[straight phi] 1} , _{[straight phi] 2} ) satisfy condition (12), we can easily prove that W ( r ) and V ( r ) satisfy condition (32). Then, by Lemma 16, we obtain the following estimate: [figure omitted; refer to PDF]

Proof of Theorem 9.

We can prove it by using the the argument as in the proofs of Theorems 6 and 8. Here, the details are omitted.

Proof of Theorem 9[variant prime].

The method of proof is the same as those of Theorems 6[variant prime] and 9. Here, we ignore its proof.

Conflict of Interests

The authors declare that they have no conflict of interests.

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