Abstract
A systematic treatment is given of singular integrals and Marcinkiewicz integrals associated with surfaces generated by polynomial compound mappings as well as related maximal functions with rough kernels in , which relates to the Grafakos-Stefanov function class. Certain boundedness and continuity for these operators on Triebel-Lizorkin spaces and Besov spaces are proved by applying some criterions of bounds and continuity for several operators on the above function spaces.
1. Introduction
Let be the -dimensional Euclidean space and denote the unit sphere in equipped with the induced Lebesgue measure . Assume that is a homogeneous function of degree zero and satisfies For a suitable function defined on , a complex number with ), and a suitable mapping , we consider the singular integral operators and parametric Marcinkiewicz integral operators in by Define the related maximal operators and by where is the set of all measurable functions with .
The primary purpose of this paper is to study the bounds and continuity of the singular integral operators and Marcinkiewicz integral operators associated with surfaces generated by polynomial compound mappings as well as related maximal functions with rough kernels in on the Triebel-Lizorkin spaces and Besov spaces. Before stating our main results, let us recall some pertinent definitions, notations, and backgrounds.
Definition 1 (function class ). For , the function class is the set of all functions which satisfy The function class was originally introduced by Fan and Sato [1]. It is closely related to the Grafakos-Stefanov function class , which was first introduced in [2] and is given by It was shown in [1, 3] that
To introduce some known results, we need to recall one more function space .
Definition 2 (function class ). For , the function class is the set of all measurable functions satisfying It is clear that for and .
When , the operators defined in (2) reduce to the classical Calderón-Zygmund operator which was originally studied by Calderón and Zygmund [4] and later investigated by many authors (see [1, 2, 5, 6], etc.). In 2009, Fan and Sato [1] first studied the bounds for with which belongs to . More precisely, the above authors established the bounds for with if for and for some . Recently, Liu and Wu [7] extended the result of [1] to the singular integrals along polynomial compound curves in the mixed homogeneity setting.
Let us recall the definitions of Triebel-Lizorkin spaces and Besov spaces.
Definition 3 (Triebel-Lizorkin spaces and Besov spaces). Let be the tempered distribution class on . For and , the homogeneous Triebel-Lizorkin spaces and Besov spaces are defined by where for and satisfies the conditions ; ; if . The inhomogeneous versions of Triebel-Lizorkin spaces and Besov spaces, which are denoted by and , respectively, are obtained by adding the term to the right hand side of (11) or (12) with replaced by , where (the Schwartz class), if .
The following properties are well-known (see [8, 9] for more details):
Recently, the investigation of the bounds for singular integrals with rough kernels in on Triebel-Lizorkin spaces and Besov spaces has received some attention of many authors (see [3, 10, 11]). Particularly, Liu et al. [10] obtained the following result.
Theorem A (see [10]). Let , where is a real-valued polynomial with and . Suppose that for some and for some satisfying (1). Then
is bounded on for and ;
is bounded on for , , and .
It should be pointed out that there is a gap in the proof of part (i) in Theorem A. To the best of my knowledge, it is unknown whether Theorem A(i) holds. However, we can obtain the following result.
Theorem 4. Let with each being a real-valued polynomial on satisfying . Suppose that for some and for some satisfying (1).
Then, for and , there exists a constant such that for all , where is independent of the coefficients of . Here is the set of all interiors of the convex hull of three squares , and , .
Then, for , , and , there exists a constant such that for all , where is independent of the coefficients of .
Applying a switched method following from [12], Theorem 4 yields the following more general result.
Theorem 5. Let with each being a real-valued polynomial on with and . Here is the set of all nonnegative (or nonpositive) and monotonic functions satisfying with , where depends only on . Suppose that for some and for some satisfying (1). Let be given as in Theorem 4.
Then, for and , there exists a constant such that for all , where is independent of the coefficients of .
Then, for , , and , there exists a constant such that for all , where is independent of the coefficients of .
Remark 6. It is not difficult to see that when and If , then and if is nonnegative and increasing, or nonpositive and decreasing; and if is nonnegative and decreasing or nonpositive and increasing (see [12]).
(iii) It follows from Theorem 5 that is bounded on for and under the same conditions of Theorem 5.
It is well known that the operators defined in (3) have their roots in the classical Marcinkiewicz integral operators , corresponding to , , and . The bounds for parametric Marcinkiewicz integrals have been extensively studied by many authors (see [13–15], etc.). In recent years, the investigation of boundedness for parametric Marcinkiewicz integral operators on the Triebel-Lizorkin spaces has also attracted the attention of many authors (see [16–19] for examples). Particularly, Yabuta [18] proved the following result.
Theorem B (see [18]). Let and with , where is the set of all functions which satisfy the following conditions:
is a positive increasing function on such that is monotonic on for some ;
there exist positive constants and such that and for all .
Suppose that for some and for some satisfying (1). Let be given as in Theorem 4. Then
is bounded on for and ;
is bounded on for , , and .
Remark 7. We notice that . There are some examples for the class , such as , , , and real-valued polynomials on with positive coefficients and . It should be pointed out that there exists such that for any (see [16]).
The second one of main results is listed as follows.
Theorem 8. Let with each being a real-valued polynomial on satisfying and . Suppose that for some and for some satisfying (1). Let be given as in Theorem 4.
Then for and , there exists a constant such that for all , where is independent of the coefficients of .
is continuous from to for and .
Then, for , , and , there exists a constant such that for all , where is independent of the coefficients of .
is continuous from to for , , and .
Remark 9. Parts (i) and (iii) in Theorem 8 extend Theorem B, which corresponds to the case . Comparing with the singular integral operators, the continuity of the singular integral operators on the Triebel-Lizorkin spaces and Besov spaces can be obtained automatically by the corresponding boundedness since the singular integral operators are linear. However, the continuity of the Marcinkiewicz integral operators on the above function spaces is nontrivial. The reason for this is twofold. First, the Marcinkiewicz integral operators are not linear. Second, can not imply and .
Remark 10. By employing the method in the proof of [20, Theorem 1.4] and applying some estimates about Fourier transforms of measures appearing in the proof of Theorem 8, one can obtain that is bounded on for if are given as in Theorem 8.
The study of integral operators in form (4) is motivated by the early work of Fefferman on singular integral operators with rough kernels multiplied by bounded radials functions [6] and was introduced by Chen and Lin [21]. Recently, the Triebel-Lizorkin space and Besov space bounds for maximal operators have also been investigated by many authors. For example, see [22, 23] for the Hardy-Littlewood maximal operator and [24, 25] for the maximal functions related to rough singular integrals and Marcinkiewicz integrals. Motivated by the above works, we shall establish the following theorem.
Theorem 11. Let with each being a real-valued polynomial on satisfying and . Suppose that for some satisfying (1).
Then, for and , there exists a constant such that for all , where is independent of the coefficients of . Here is the set of all interiors of the convex hull of two squares and .
and are continuous from to for , , and .
Then, for , , and , there exists a constant such that for all , where is independent of the coefficients of .
and are continuous from to for , , and .
Remark 12. Note that . By using the estimates of measures appearing in the proof of Theorem 11 and the arguments similar to those used in deriving [7, Theorem 1.9], we can obtain that is bounded on for under the conditions of Theorem 11.
Applying (15)-(16), Theorems 4, 5, 8, and 11, and Remarks 6, 9, 10, and 12, we can obtain the following result immediately.
Theorem 13. Under the same conditions of Theorems 4, 5, 8, and 11 and Remarks 6, 9, 10, and 12 with , these operators are bounded and continuous on and , respectively.
Due to the fact that , Theorem 13 may yield directly the following conclusion.
Theorem 14. Let and with each being a real-valued polynomial on satisfying and .
If for some and for some satisfying (1), let be given as in Theorem 4. Then
is bounded and continuous on for and . is also bounded and continuous on for , , and ;
is bounded and continuous on for and . is also bounded and continuous on for , , and .
If for some satisfying (1), then and are bounded and continuous on for , , and . and are also bounded and continuous on for , , and .
The paper is organized as follows. Section 2 contains some known results, which play key roles in the proofs of main results. In Section 3, we will present some criterions on the boundedness and continuity of several operators on Triebel-Lizorkin spaces and Besov spaces, which are the main ingredients of our proofs. The proofs of main results will be given in Section 4. We remark that the methods employed in this paper follow from a combination of ideas and arguments in [10, 12, 17, 18, 23, 24, 26, 27], among others. It should be also point out that our methods can be used to deal with other integral operators, such as singular integrals, Marcinkiewicz integrals, and related maximal functions associated with other surfaces with other rough kernels.
Throughout the paper, we denote by the conjugate index of , which satisfies . The letter or , sometimes with certain parameters, will stand for positive constants not necessarily the same one at each occurrence but are independent of the essential variables. In what follows, we set . We denote by the difference of for an arbitrary function defined on and ; that is, . We also set and .
2. Preliminary Lemmas
This section is devoted to recalling some known lemmas, which plays key roles in the proofs of main theorems. Let us begin with the following lemma of van der Corput type.
Lemma 15 (see [28]). Let and , where are real parameters, and are distinct positive (not necessarily integer) exponents. Then, for , the following holds: where and does not depend on as long as .
Applying Lemma 15 and the arguments similar to those used in the proof of [16, Lemma 2.2], we can obtain the following result.
Lemma 16. Let , where are real parameters, and are distinct positive (not necessarily integer) exponents. Suppose that satisfying is monotonic on for some . Then, for any and , the following holds: with , where is independent of , but may depend on , , and .
Proof. By the change of variables, we have where , , and . We can also write where Since is monotonic, applying Lemma 15, we obtain for , where . By (29) and the integration by parts, we have This proves Lemma 16.
Lemmas 17–19 are known and will play key roles in the proofs of Theorems 4, 8, and 11, respectively.
Lemma 17 (see [26]). Let , where are real-valued polynomials defined on and are arbitrary functions defined on . Suppose that is a homogeneous function of degree zero and for some . Define the measures by If belongs to the interior of the convex hull of three squares , , and , then, for arbitrary functions , there exists independent of such that The constant is independent of and the coefficients of .
Lemma 18 (see [17]). Let , where , are real-valued polynomials on and are arbitrary functions independent of . Define the family of measures on by Suppose that for some and . If belongs to the interior of the convex hull of three cubes , , and , then, for arbitrary functions , there exists such that The constant is independent of and the coefficients of .
Lemma 19 (see [24]). Let , where and are real-valued polynomials on , and are arbitrary functions defined on . Suppose that . Define the measures by If belongs to the interior of the convex hull of two cubes and , then, for arbitrary functions , there exists independent of and the coefficients of such that
Below is the vector-valued inequality of the Hardy-Littlewood maximal functions, which is one of the main ingredients of our proofs.
Lemma 20 (see [16]). Let be the Hardy-Littlewood maximal operator on . Then for any , .
In order to deal with Marcinkiewicz integrals and maximal functions, we need a useful characterization of Triebel-Lizorkin spaces and Besov spaces.
Lemma 21 (see [18]). Let .
If , , and , then If , , and , then
To prove Theorem 5, we need the following results.
Lemma 22 (see [12]). Let be given as in Theorem 5. Suppose that for some , then .
The following lemma is a key switched result about singular integrals associated with compound surfaces.
Lemma 23 (see [29]). Let and be given as in Theorem 5. Let be defined as in (2) and . Define the operator by If is nonnegative and increasing, then .
If is nonnegative and decreasing, then .
If is nonpositive and decreasing, then .
If is nonpositive and increasing, then .
3. Some Criterions
To prove Theorem 4, we need the following criterion on the boundedness of the convolution operators on Triebel-Lizorkin spaces, which is a variant of [15, Theorem 1.10]. This can be proved by making some minor modifications in the proof of [15, Theorem 1.10]. We omit the details.
Proposition 24. Let and be a family of measures on . For , let be some sequences of positive real numbers with satisfying for some . For , let and be linear transformations. Suppose that there exists constants and such that
for every ;
for every and ;
if for , , and ;
for , , and ;
for any and arbitrary functions , there exists a positive constant which is independent of such that for some with and .
Let be the line segment from to with and . Then there exists a positive constant such that holds for any and .
To establish the Triebel-Lizorkin space boundedness parts in Theorems 8 and 11, we will give the following lemma, which is the heart of the proofs of Theorems 8 and 11.
Proposition 25. Let and be a family of Borel measures on . Let be the total variation of . For , let be some sequences of positive real numbers with satisfying for some . For , let and be linear transformations. Suppose that there exist , , and such that the following conditions hold for , , , and :
;
;
if ;
Let be the line segment from to with and . Then there exists a positive constant such that holds for any and .
Proof. For any , let . By [5, Lemma 6.1], there are two nonsingular linear transformations and such that For and , we define the family of measures by where such that for and for . Then (50) together with assumption (i) implies that It follows that Therefore, to prove (48), it suffices to show that there exists such that for any , , and .
We now prove (53). By our assumptions (ii)-(iii), we have Let be a collection of functions on with the following properties: Define the multiplier operator on by Note that . By [16, Lemma 2.5] we obtain By Minkowski’s inequality we have Define the mixed norm for measurable functions on by For any , let Then we have By (54), Hölder’s inequality, Minkowski’s inequality, Fubini’s theorem, Plancherel’s theorem, and Lemma 21(ii), we have where with and It follows from (62) and (14) that We now prove For , let be a radial function in defined by , where and is given as in (50). Define and by where and with . It is easy to check that where . (67) together with Lemma 20 yields that for any and , . Define for . We get from (68) that, for any and , ,On the other hand, by the definition of we have It follows that By (69), (71), and assumption (iv), we have for arbitrary functions and . Then (72) together with Lemma 21(i) and (57) leads to This proves (65).
By the interpolation between (64) and (65), we obtain that, for and , there exists such that Then, (74) together with (61) yields (48) and completes the proof of Proposition 25.
The following result is a criterion on the boundedness and continuity of several operators on Besov spaces, which can be used to prove the boundedness and continuity result on Besov spaces in Theorems 8 and 11.
Proposition 26 (see [23]). Assume that for some . If for any . Then is bounded on for any and . Particularly, if satisfies for arbitrary functions defined on , then is continuous from to for any and .
To establish the Triebel-Lizorkin space continuity parts in Theorems 8 and 11, we will give the following criterion of continuity for several sublinear operators on Triebel-Lizorkin spaces.
Proposition 27. Assume that is a sublinear operator and the following conditions hold.
for some .
For all , the following holds: For arbitrary functions defined on , the following holds: There exist and such that Then is continuous from to .
Proof. Let in as . By (15) we see that in and in as . Since as , by assumptions (i) and (iii) we obtain that in as . Therefore, it suffices to show that in as .
We shall prove this claim by contradiction. Without loss of generality we may assume that there exists such that for every . Since in as , by extracting a subsequence we may assume that as for almost every . It follows that as for every and almost every . We get from assumption (ii) and the sublinearity of that for . For convenience, we set for and . It follows from Lemma 21(i) that for and . By assumption (iv) we obtain It follows that as . One can extract a subsequence such that . Define a function by One can easily check that and Since , we have that for every and almost every . (85) together with the dominated convergence theorem leads to for every and almost every . By the fact again, we have for almost every . Using (85) we obtain for almost every and . It follows from (86)–(88) and the dominated convergence theorem that for almost every . By (85) again, it holds that for almost every . By (89)-(90), the fact , and the dominated convergence theorem, we obtain This yields that as and gives a contradiction.
4. Proofs of Theorems 4, 5, and 11
In this section we shall prove Theorems 4–11. In what follows, let . For , we set . Then there are integers such that for any and for all . For and , we set for and . For , we define the linear transformation by
We now turn to prove Theorems 4, 5, and 11
Proof of Theorem 4. Define by It is clear to see that for any and . For and , we define the measures by It is clear that By the change of the variables, we have On the other hand, it is easy to check that By the change of the variables and Hölder’s inequality, we have where By the Van der Corput lemma, there exists a constant , such that When , since is increasing in , we have where . Combining (99), (102) with the fact that yields that On the other hand, Lemma 17 yields that for belonging to the interior of the convex hull of three squares , , and . Here is independent of the coefficients of .
Take . By (96)–(98), (103)-(104) and Proposition 24, we obtain for , , and all , where is given as in Theorem 4. This proves Theorem 4(i).
On the other hand, it follows from Theorem 4(i) and (13) that is bounded on for and . This together with the arguments similar to those used in deriving [30, Theorem 1.2] yields Theorem 4(ii).
Proof of Theorem 5. Theorem 5 follows from Theorem 4 and Lemmas 22 and 23.
Proof of Theorem 8. Define by Clearly, for any and . For , define the family of measures by where is defined in the same way as , but with and replaced by and , respectively. By the change of variables and Minkowski’s inequality, we have By Lemma 18, we obtain for belonging to the interior of the convex hull of three cubes , , and . Here is independent of the coefficients of .
One can easily check that It follows from (112) that By a change of variable, we have Since by Lemma 16, we have For , since is increasing in , we have where . Combining (114), (115), and (117) with the fact that yields that when . It follows from (118) that when .
Take . By Remark 7 we have that . It follows from (110)-(111), (113), (119), and Proposition 25 that holds for , any , and all . Thus (120) together with (109) yields that holds for , any , and all . Here is independent of the coefficients of . On the other hand, one can easily check that for any and for arbitrary functions and defined on . By Lemma 21(i), we have for all and . Here is independent of the coefficients of . And (122) and (124) yield Theorem 8(i). We get from Remark 10 that is bounded on for and . Note that is a sublinear operator. These facts together with (121)–(123) yield Theorem 8(ii). Theorem 8(iii) and (iv) follow from the bounds for and (122)-(123).
Proof of Theorem 11. We first consider the operator . One can easily check that By Lemma 21(i) and (125) we have for and . Therefore, to prove Theorem 11(i) for , it suffices to show that for and belonging to the set of all interiors of the convex hull of two squares and . Here is independent of the coefficients of .
Let , and be given as in the proof of Theorem 8. Define the family of measures and on by By duality we have One can easily check that On the other hand, It follows that Invoking Lemma 16 we obtain When , since is increasing in , we have where . Combining (132), (134) with the fact that yields that if . By Lemma 19 we have for belonging to the interior of the convex hull of two cubes and . Here is independent of the coefficients of .
Take . By Remark 7 we have that . By (130), (135)-(136), and Proposition 25 we obtain that for , and belonging to the interior of the convex hull of two squares and . Here is independent of the coefficients of . Equation (137) together with (129) yields (127). By arguments similar to those used in deriving and in [31], one can obtain Thus (138) together with (127) yields that for and belonging to the set of all interiors of the convex hull of two squares and . Here is independent of the coefficients of . One can easily check that By Lemma 21(i) and (140) we have for and . Then Theorem 11(i) follows from (126)-(127), (139), and (141).
It is known that both and are sublinear operators. Moreover, one can easily check that for arbitrary functions defined on . It follows from Remark 12 that for . It follows from (142)-(143), (127), (139), and Proposition 27 that Theorem 11(ii) holds. Theorem 11(iii)-(iv) follows from (125), (140), (142)-(143), and Proposition 26.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was partially supported by the NNSF of China (Grant no. 11701333) and Support Program for Outstanding Young Scientific and Technological Top-Notch Talents of College of Mathematics and Systems Science (Grant no. Sxy2016K01).