Abstract

We first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases. Then, we discuss the strong-type and weak-type estimates for a class of Calderón–Zygmund type operators in these new Morrey type spaces. Furthermore, the strong-type estimate and endpoint estimate of commutators formed by and are established. Also, we study related problems about two-weight, weak-type inequalities for and in the Morrey type spaces and give partial results.

1. Introduction

Calderón–Zygmund singular integral operators and their generalizations on the Euclidean space have been extensively studied (see [1–5], for instance). In particular, Yabuta [5] introduced certain -type Calderón–Zygmund operators to facilitate his study of certain classes of pseudodifferential operators. Following the terminology of Yabuta [5], we introduce the so-called -type Calderón–Zygmund operators.

Definition 1. Let be a nonnegative, nondecreasing function on withA measurable function on is said to be a -type kernel if it satisfies

Definition 2. Let be a linear operator from into its dual . One can say that is a -type Calderón–Zygmund operator if(1) can be extended to be a bounded linear operator on ;(2)there is a -type kernel such thatfor all and for all , where is the space consisting of all infinitely differentiable functions on with compact supports.

Note that the classical Calderón–Zygmund operator with standard kernel (see [1, 2]) is a special case of -type operator when with .

Definition 3. Given a locally integrable function defined on and given a -type Calderón–Zygmund operator , the linear commutator is defined for smooth, compactly supported functions as

Throughout the paper, let us suppose that is a nonnegative, nondecreasing function on satisfying condition (1). Let us give the following weighted result of obtained by Quek and Yang in [6].

Theorem 4 (see [6]). Let and . Then, the -type Calderón–Zygmund operator is bounded on for and bounded from into for .

Since linear commutator has a greater degree of singularity than the corresponding -type Calderón–Zygmund operator, we need a slightly stronger version of condition (8) given below. The following weighted endpoint estimate for commutator of the -type Calderón–Zygmund operator was established in [7] under a stronger version of condition (8) assumed on , if (for the unweighted case, see [8]).

Let us now recall the definition of the space of (see [9]). is the Banach function space modulo constants with the norm defined bywhere the supremum is taken over all balls in and stands for the mean value of over ; that is,

Theorem 5 (see [7]). Let be a nonnegative, nondecreasing function on withand let and . Then, for all , there is a constant independent of and such thatwhere and .

On the other hand, the classical Morrey space was originally introduced by Morrey in [10] to study the local behavior of solutions to second-order elliptic partial differential equations. Since then, this space played an important role in studying the regularity of solutions to partial differential equations. In [11], Mizuhara introduced the generalized Morrey space which was later extended and studied by many authors. In [12], Komori and Shirai defined a version of the weighted Morrey space which is a natural generalization of the weighted Lebesgue space. Let be the -type Calderón–Zygmund operator, and let be its linear commutator. The main purpose of this paper is twofold. We first define a new kind of Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases, and then we will establish the weighted strong type and endpoint estimates for and in these Morrey type spaces for all and . In addition, we will discuss two-weight, weak-type norm inequalities for and in and give some partial results.

Throughout this paper, will denote a positive constant whose value may change at each appearance. We also use to denote the equivalence of and ; that is, there exist two positive constants and independent of and such that .

2. Statements of the Main Results

2.1. Notation and Preliminaries

Let be the -dimensional Euclidean space of points with norm . For and , let denote the open ball centered at of radius , denote its complement, and be the Lebesgue measure of the ball . A weight is a nonnegative locally integrable function on that takes values in almost everywhere. A weight is said to belong to Muckenhoupt’s class for , if there exists a constant such thatfor every ball , where is the dual of such that . The class is defined replacing the inequality above byfor every ball . We also define . Given a ball and , will denote the ball with the same center as whose radius is times that of . Given a Lebesgue measurable set and a weight function , we denote the characteristic function of by , the Lebesgue measure of by , and the weighted measure of by , where . It is well known that if with (or ), then satisfies the doubling condition; that is, for any ball , there exists an absolute constant such that (see [2])Moreover, if , then for any ball and any measurable subset of a ball , there exists a number independent of and such that (see [2])

Given a weight function on , as usual, the weighted Lebesgue space for is defined as the set of all functions such thatWe also denote by () the weighted weak Lebesgue space consisting of all measurable functions such that

We next recall some basic definitions and facts about Orlicz spaces needed for the proof of the main results. For further information on the subject, one can see [13]. A function is called a Young function if it is continuous, nonnegative, convex, and strictly increasing on with and as . An important example of Young function is with some . Given a Young function , we define the -average of a function over a ball by means of the following Luxemburg norm:When , , it is easy to see thatthat is, the Luxemburg norm coincides with the normalized norm. Given a Young function , we use to denote the complementary Young function associated with . Then, the following generalized Hölder’s inequality holds for any given ball :In particular, when , we know that its complementary Young function is . In this situation, we denoteSo we have

2.2. Morrey Type Spaces

Let us begin with the definitions of the weighted Morrey space and generalized Morrey space.

Definition 6 (see [12]). Let , , and be a weight function on . Then, the weighted Morrey space is defined bywhere the supremum is taken over all balls in . We also denote by the weighted weak Morrey space of all measurable functions such that

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 7 (see [11]). Let and be a growth function in . Then, the generalized Morrey space is defined bywhere the supremum is taken over all balls in with . One can also denote by the generalized weak Morrey space of all measurable functions for which

In order to unify these two definitions, we now introduce Morrey type spaces associated with as follows. Let . Assume that is a positive increasing function defined in and satisfies the following condition:where is a constant independent of and .

Definition 8. Let , , satisfy the condition (26), and be a weight function on . We denote by the generalized weighted Morrey space, the space of all locally integrable functions with finite norm:Then, we know that becomes a Banach function space with respect to the norm . Furthermore, we denote by the generalized weighted weak Morrey space of all measurable functions for which

Definition 9. In the unweighted case (when equals a constant function), one can denote the generalized unweighted Morrey space by and weak Morrey space by . That is, let and satisfy the condition (26) with ; one can define

Note the following:(i)If , then and . Thus, our (weak) Morrey type space is an extension of the weighted (weak) Lebesgue space.(ii)If with , then is just the weighted Morrey space , and is just the weighted weak Morrey space .(iii)If , below we will show that reduces to the generalized Morrey space , and reduces to the generalized weak Morrey space .

Our main results on the boundedness of in the Morrey type spaces can be formulated as follows.

Theorem 10. Let and . Assume that satisfies the condition (26) with ; then, the -type Calderón–Zygmund operator is bounded on .

Theorem 11. Let and . Assume that satisfies the condition (26) with ; then, the -type Calderón–Zygmund operator is bounded from into .

Let be a nonnegative, nondecreasing function on satisfying condition (8), and let be the commutator formed by and BMO function . For the strong-type estimate of the linear commutator in with , we will prove the following.

Theorem 12. Let , , and . Assume that satisfies (8) and satisfies the condition (26) with ; then, the commutator operator is bounded on .

To obtain endpoint estimate for the linear commutator in , we first need to define the weighted -average of a function over a ball by means of the weighted Luxemburg norm; that is, given a Young function and , we define (see [13, 14])When , this norm is denoted by ; when , this norm is also denoted by . The complementary Young function of is with mean Luxemburg norm denoted by . For and for every ball in , we can also show the weighted version of (20). Namely, the following generalized Hölder’s inequality in the weighted settingis valid (see [14], for instance). Now we introduce new Morrey type spaces of type associated with as follows.

Definition 13. Let , , satisfy the condition (26), and be a weight function on . One can denote by the generalized weighted Morrey space of type, the space of all locally integrable functions defined on with finite norm .where

Note that for all ; then, for any ball and , we have by definition; that is, the inequalityholds for any ball . From this, we can further see that when satisfies the condition (26) with ,Hence, we have by definition.

Definition 14. In the unweighted case (when equals a constant function), one can denote by the generalized unweighted Morrey space of type. That is, let and satisfy the condition (26) with ; one can definewhere

We also consider the special case when is taken to be with and denote the corresponding space by .

Definition 15. Let , , and be a weight function on . One can denote by the weighted Morrey space of type, the space of all locally integrable functions defined on with finite norm .whereIn this situation, we have .

For the endpoint case, we will also prove the following weak-type estimate of the linear commutator in the Morrey type space associated with .

Theorem 16. Let , , and . Assume that satisfies (8) and satisfies the condition (26) with ; then, for any given and any ball , there exists a constant independent of , , and such thatwhere . From the definitions, we can roughly say that the commutator operator is bounded from into .

In particular, if we take with , then we immediately get the following strong-type estimate and endpoint estimate of and in the weighted Morrey spaces for all and .

Corollary 17. Let , , and . Then, the -type Calderón–Zygmund operator is bounded on .

Corollary 18. Let , , and . Then, the -type Calderón–Zygmund operator is bounded from into .

Corollary 19. Let , , , and . Assume that satisfies (8); then, the commutator operator is bounded on .

Corollary 20. Let , , , and . Assume that satisfies (8); then, for any given and any ball , there exists a constant independent of , , and such thatwhere .

Naturally, when , we have the following unweighted results.

Corollary 21. Let . Assume that satisfies the condition (26) with ; then, the -type Calderón–Zygmund operator is bounded on .

Corollary 22. Let . Assume that satisfies the condition (26) with ; then, the -type Calderón–Zygmund operator is bounded from into .

Corollary 23. Let and . Assume that satisfies (8) and satisfies the condition (26) with ; then, the commutator operator is bounded on .

Corollary 24. Let and . Assume that satisfies (8) and satisfies the condition (26) with ; then, for any given and any ball , there exists a constant independent of , , and such thatwhere .

Let , , be a growth function with doubling constant . If, for any fixed and , we set , thenFor the doubling constant satisfying , which means that for some , then we are able to verify that is an increasing function and satisfies the condition (26) with some .

Definition 25. Let and be a growth function in . One can denote by the generalized Morrey space of type, which is defined bywhereIn this situation, we also have .

From the definitions given above, we get , , and by the choice of . Thus, by the above unweighted results (Corollaries 21–24), we can also obtain strong-type estimate and endpoint estimate of and in the generalized Morrey spaces when and satisfies the doubling condition (23).

Corollary 26. Let . Suppose that satisfies the doubling condition (23) and ; then, the -type Calderón–Zygmund operator is bounded on .

Corollary 27. Let . Suppose that satisfies the doubling condition (23) and ; then, the -type Calderón–Zygmund operator is bounded from into .

Corollary 28. Let and . Suppose that satisfies (8) and satisfies the doubling condition (23) with ; then, the commutator operator is bounded on .

Corollary 29. Let and . Suppose that satisfies (8) and satisfies the doubling condition (23) with ; then, for any given and any ball , there exists a constant independent of , , and such thatwhere .

3. Proof of Theorems 10 and 11

Proof of Theorem 10. Let with and . For an arbitrary point , set for the ball centered at and of radius , . We represent asby the linearity of the -type Calderón–Zygmund operator , we writeBelow, we will give the estimates of and , respectively. By the weighted boundedness of (see Theorem 4), we haveMoreover, since when with , then by the condition (26) of and inequality (12), we obtainAs for the term , it is clear that when and , we get . We then decompose into a geometrically increasing sequence of concentric balls and obtain the following pointwise estimate:From this, it follows thatBy using Hölder’s inequality and condition on , we getHence,Notice that for ; then, by using the condition (26) of again, inequality (13), and the fact that , we find thatwhich gives our desired estimate . Combining the estimates above for and and then taking the supremum over all balls , we complete the proof of Theorem 10.

Proof of Theorem 11. Let with . For an arbitrary ball , we represent asthen, for any given , by the linearity of the -type Calderón–Zygmund operator , one can writeWe first consider the term . By the weighted weak (1, 1) boundedness of (see Theorem 4), we haveMoreover, since when , then we apply the condition (26) of and inequality (12) to obtain thatAs for the term , it follows directly from Chebyshev’s inequality and the pointwise estimate (51) thatAnother application of condition on gives thatConsequently,Recall that ; therefore, by using the condition (26) of again, inequality (13), and the fact that , we getwhich implies our desired estimate . Summing up the estimates above for and and then taking the supremum over all balls and all , we finish the proof of Theorem 11.

4. Proof of Theorems 12 and 16

To prove our main theorems in this section, we need the following lemma about functions.

Lemma 30. Let be a function in . Then,(i)for every ball in and for all ,(ii)for every ball in and for all with ,

Proof. For the proof of , we refer the reader to [3]. For the proof of (ii), we refer the reader to [15].

Proof of Theorem 12. Let with and . For each fixed ball , as before, we represent as , where and . By the linearity of the commutator operator , we writeSince is bounded on for and , then by the well-known boundedness criterion for the commutators of linear operators, which was obtained by Álvarez et al. in [16], we know that is also bounded on for all and , whenever . This fact together with the condition (26) of and inequality (12) impliesLet us now turn to the estimate of . By definition, for any , we haveIn the proof of Theorem 10, we have already shown that (see (51))Following the same arguments as in (51), we can also prove thatHence, from the pointwise estimates above for and , it follows thatBelow, we will give the estimates of , , and , respectively. Using of Lemma 30, Hölder’s inequality, and the condition, we obtainwhere in the last inequality we have used the estimate (55). Applying of Lemma 30, Hölder’s inequality, and the condition, we can deduce thatFor any , since when with , then by using the condition (26) of and inequality (13) together with the fact that , we thus obtainwhere the last series is convergent since the exponent is positive. This implies our desired estimate . It remains to estimate the last term . An application of Hölder’s inequality gives us thatIf we set , then we have because (see [1, 2]). Thus, it follows from of Lemma 30 and the condition thatTherefore, in view of estimate (55), we conclude thatSummarizing the estimates derived above and then taking the supremum over all balls , we complete the proof of Theorem 12.

Proof of Theorem 16. For any fixed ball in , as before, we represent as , where and . Then, for any given , by the linearity of the commutator operator , one can writeBy using Theorem 5 and the previous estimate (35), we getMoreover, since when , then by the condition (26) of and inequality (12), we havewhich is our desired estimate. We now turn to deal with the term . Recall that the following inequalityis valid. So we can further decompose asBy using the previous pointwise estimate (51) and Chebyshev’s inequality together with of Lemma 30, we deduce thatFurthermore, note that for any . It then follows from the condition and the previous estimate (34) thatwhere in the last inequality we have used estimate (63). On the other hand, applying the pointwise estimate (70) and Chebyshev’s inequality, we haveFor the term , since , by the condition and the fact that ,Furthermore, we use the generalized Hölder’s inequality with weight (31) to obtainIn the last inequality, we have used the well-known fact that (see [14])It is equivalent to the inequalitywhich is just a corollary of the well-known John–Nirenberg’s inequality (see [9]) and the comparison property of weights. Hence, by estimate (63),For the last term , we proceed as follows. Using of Lemma 30 together with the facts and , we deduce thatRecall that . We can now argue exactly as we did in the estimation of (74) to getLet us now substitute this estimate (92) into the term ; we get the desired inequalityThis completes the proof of Theorem 16.

5. Partial Results on Two-Weight Problems

In the last section, we consider related problems about two-weight, weak-type inequalities with . Let be the classical Calderón–Zygmund operator with standard kernel; that is, when with . It is well known that is a bounded operator on for all and , and, of course, is a bounded operator from into . In the two-weight context, however, the condition is “not” sufficient for the weak-type inequality for . More precisely, given a pair of weights and , , the weak-type inequalitydoes not hold if : there exists a positive constant such that, for every cube ,one can see [17, 18] for some counterexamples. Here, all cubes are assumed to have their sides parallel to the coordinate axes; will denote the cube centered at and has side length . In [17, 19], Cruz-Uribe and Pérez considered the problem of finding sufficient conditions on a pair of weights such that satisfies the weak-type inequality (94) (). They showed in [19] that if we strengthened the condition (95) by adding a “power bump” to the left-hand term, then inequality (94) holds for all . More specifically, if there exists a number such that, for every cube in ,then the classical Calderón–Zygmund operator is bounded from into . Moreover, in [17], the authors improved this result by replacing the “power bump” in (96) by a smaller “Orlicz bump.” To be more precise, they introduced the following -type condition in the scale of Orlicz spaces:where is the mean Luxemburg norm of on cube with Young function . It was shown that inequality (94) still holds under the -type condition on , and this result is sharp since it does not hold in general when .

On the other hand, the following Sharp function estimate for was established in [8]: there exists some , , and a positive constant such that, for any and ,where is the standard Hardy–Littlewood maximal operator and is the well-known Sharp maximal operator defined asHere, the supremum is taken over all the cubes containing and denotes the mean value of over ; namely, . It was pointed out in [19] (Remark ) that, by using this Sharp function estimate (98), we can also show inequality (94) is true for more general operator , under condition (96) on . Then, we obtain a sufficient condition for to be weak with .