Journal of Function Spaces
Volume 2018 (2018), Article ID 7052490, 12 pages
https://doi.org/10.1155/2018/7052490

Research Article

A Note on Marcinkiewicz Integrals along Submanifolds of Finite Type

Daiqing Zhang

College of Mathematics and Physics, Fujian University of Technology, Fuzhou, Fujian 350118, China

Correspondence should be addressed to Daiqing Zhang; zhangdaiqing2011@163.com

Received 28 February 2018; Accepted 10 April 2018; Published 12 June 2018

Academic Editor: Dashan Fan

Copyright © 2018 Daiqing Zhang. 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.

Abstract.

We study the parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels. The kernels of our operators are allowed to be very rough both on the unit sphere and in the radial direction. Under the rather weakened size conditions on the integral kernels, the bounds will be established for such operators. As applications, the corresponding results for parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley functions are also given.

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 that satisfies

For a suitable mapping , we define the parametric Marcinkiewicz integral operator on by

where (the Schwartz class), (), and . Here and denotes the set of all measurable functions satisfying

If and , we denote .

When , the operator reduces to the classical Marcinkiewicz integral operator , which was first introduced by Stein [1] and has been studied by many authors. For example, see [2] for the case (the Hardy space on the unit sphere (see [3, 4])), [5] for the case , [6] for the case (the Block space generated by -block), and [7] for the case (the Grafakos-Stefanov function class (see [8])). The study of parametric Marcinkiewicz integral operator has attracted the attention of many authors; we refer the readers to consult [9–12] for relevant results. For general mapping , the operator has also been investigated extensively by many authors. For example, see [7, 13–16] for polynomial curves, [13, 14, 17, 18] for polynomial mapping, [18, 19] for homogeneous mappings, [14, 20, 21] for surfaces of revolution, and [18, 22] for submanifolds of finite type. For other interesting works related to this topic we can consult [23–29].

We notice that the following inclusion relations are valid:

The main purpose of this paper is to study the boundedness of parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels as well as parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley functions. Precisely, let and be a smooth function. In what follows we always assume that is of finite type at the origin; that is, for any , there exists a multi-index such that and , where , , and the notation “” denotes the inner product in . We define the parametric Marcinkiewicz integral operator by

where and are given as in (2). When , we denote by . Ding et al. [18] proved that is bounded on for all , provided that for some . Later on, Al-Salman and Al-Qassem [22] extended the above result to the case . To the best of our knowledge, there is nothing to introduce some relevant results on with , even in the special case .

On the other hand, the investigation of singular integrals along submanifolds of finite type has also attracted the attention of many authors (see [30–32]). In particular, in order to study singular integral operator along submanifolds of finite type with rough kernel both on the unit sphere and in the radial direction

Sato [31] introduced the radial functions class , which denotes the set of all functions satisfying

Here

where

where the supremum is taken over all and such that (see [33, 34]).

We now introduce the result of [31] as follows.

Theorem A (see [31]). Let for some . Suppose that and satisfies condition (1). Then

for all .

Based on the above, we find that it is natural to ask the following question.

Question B. Is the operator bounded on under the conditions and for some ?

This question can be addressed in the affirmative by the following theorem.

Theorem 1. Let for some and satisfy condition (1).

(i)If , then

for all .

(ii)If , then

for all .

Actually, Theorem 1 can be obtained by extrapolation arguments (see [17] or [35]) and the following result.

Theorem 2. Let for some and for some . Then

for all . Here the constant is independent of and .

Applying Theorem 2 and extrapolation argument, we can obtain the following result.

Theorem 3. Let for some and satisfy condition (1).

(i)If for some , then

for all .

(ii)If for some , then

for all .

Finally, we also consider the corresponding parametric Marcinkiewicz integral operators and related to the Littlewood-Paley -function and the area integral, respectively, which are defined by

where , , and ;

where , , and are given as in (2); is of finite type.

As applications of Theorems 1 and 3, we can get the following result.

Theorem 4. Let , for some and , satisfying condition (1). Then, for and , the following inequalities are true:

Remark 5. It should be point out that the bounds for Marcinkiewicz integrals along submanifolds of finite type for or are still unknown. Another inviting possibility is the investigation of the validity of Theorems 1-4 if for some is replaced by for some .

The paper is organized as follows. Section 2 is devoted to presenting some preliminary notations and lemmas, which play key roles in the proofs of our main results. The proofs of main results will be given in Section 3. We remark that the methods employed in this paper follow from a combination of ideas and arguments in [17, 30, 31, 35], among others. Throughout this note, we let denote the conjugate index of which satisfies . The letter will stand for positive constants not necessarily the same one at each occurrence but is independent of the essential variables.

2. Notations and Lemmas

Let us begin with introducing two lemmas, which follow from [30, 31].

Lemma 6 (see [30] or [31]). Let be smooth of finite type at the origin. Define by

for and . Then, there exist constants and a mapping from to a finite set of positive integers such that

Lemma 7 (see [30] or [31]). Let be real-valued. Let and with . Suppose that is compactly supported and that

where is a positive integer. Then, there exists a positive constant depending only on and such that

for all and .

It was known that, for a function defined on an appropriate subinterval of by for fixed and , , where is as in Lemma 6 (see [31]).

Define a family of measures on by

Lemma 8. Let for some and for some . Then there exist a positive integer , a positive number , and a positive constant , such that

The constants and are independent of .

Proof. Let and and be as in Lemma 6. Without loss of generality, we assume that . Set . Set

for ,  ,  ,  , and . By the changes of variables, one finds

for any . Choose a function satisfying . Define , where . Then, if , the following holds:

We take for a suitable with and , which will be specified below. We assume and define

Then by (27) we get

Let

By Lemma 7, we have

for , where . It is easy to see that and . Therefore, applying integration by parts and Hölder’s inequality, we see that

where . Therefore, we have

where is independent of , and . We put , then by Lemma 6 we have

Therefore, if is a positive integer such that and ,

This combining with (29) implies

where . If , the conclusion of Lemma 8 follows from the estimate . This completes the proof of Lemma 8.

Applying similar arguments in Lemma 8 and [31, Lemma 2.5], we can obtain the following lemma.

Lemma 9. Let for some , and let be a real-valued polynomial on with . Suppose that , is a homogeneous function of degree zero, and for . Then there exists a constant independent of , and the coefficients of the polynomial such that

where .

For , let be as in Lemma 8 and at the origin be defined by

for . For , define by

Set . For each , let denote the number of multi-indices satisfying . Label the coordinate of by the of multi-indices satisfying . That is, . For we define the linear transformations by

where .

Let . By [36, Lemma 6.1], there exist two nonsingular linear transformations and such that

where depends only on and is a projection operator from to .

For and , we define the family of measures and the related maximal operators and on by

where is defined in the same way as , but with replaced by and replaced by , . By the definition of , it is easy to see that .

Lemma 10. Let , , and for some . Then, for and , there exists a constant such that

where for and .

Proof. By the straightforward calculations (43) holds. By Lemmas 89 we have

Similarly,

where for and . Thus (45) holds. On the other hand, by a change of variable, we have

which combining with (43) implies that

Similarly, we can obtain

This completes the proof of Lemma 10.

Motivated by the idea in [37], we have the following result, which will play a key role in the estimates on some vector-valued norm inequalities.

Lemma 11. Let be as in Lemma 8 and . Then for and any , there exists a constant such that

The constant is independent of , and .

Proof. For convenience, we set . It is easy to verify that

For , we define the family of measures and maximal operators on by

where is defined in the same way as , but with replaced by and replaced by . Thus, we have

Therefore, to prove (51), it suffices to prove that

Then by the proof of Lemma 10 and a straightforward calculation we get that, for ,

In what follows, we prove (56) by induction on .
Case 1. Without loss of generality, we may assume that is nonnegative and . It is easy to check that , which implies (56) for .
Case 2. Suppose that (56) holds for , . We shall prove (56) for . Let be supported in and for . Define the Borel measures on by

for , where are as in (41). It follows from (41) and (57)-(58) that

In addition, by (59) and a well-known result on maximal functions (see [36]), we have

where

By our assumption we have

where is independent of , and . It remains to prove that

where is as above. By a well-known property of Rademacher’s functions, (64) follows from

where with , or . Below we prove (65). Choose a sequence of nonnegative functions in such that

where are independent of , and . Define the Fourier multiplier operator by

Then

By the Littlewood-Paley theory, we have

This combining with Plancherel's theorem yields

where

We get from (41) and (60) that

where is as in Lemma 10. (72) together with (68) yields that

Thus,

which combining the Littlewood-Paley theory and (60)-(63) with the proof of Lemma in [37, p.544] implies that

Interpolating between (72) and (75) and combining with (68), we get

which leads to

Reasoning as above, (60)-(63), (77), the proof of Lemma in [37, p.544], the Littlewood-Paley theory, and interpolation imply

By using this argument repeatedly, we can obtain ultimately that

This proves (65) and completes the proof of (51).
Now, we prove (52). It suffices to prove that

for all and , is independent of , and the coefficients of for all . We shall prove (80) by induction on . When , it is easy to see that

which implies (80) for . Assuming that (80) holds for , we will prove that (80) holds for . Let be as in (57). Define the family of measures by

By Lemma 10 and (41), one can easily check that

By the definition of and a well-known result on maximal function (see [36]), we have

where

From our assumption, (83)-(85), and similar arguments to those in getting (41), we get (80) for . Thus (80) holds. Lemma 11 is proved.

Applying Lemma 11, we can obtain the following result.

Lemma 12. Let be as in Lemma 8. Then, for and any , there exists a constant such that

The constant is independent of , and .

Proof. We shall prove Lemma 12 by using the method in [17]. First we prove (87). For fixed , by duality, there exists a nonnegative function in with such that

By a change of variable and Hölder’s inequality, we obtain

Thus by (89)-(90) and Hölder’s inequality, one can check that

where . We get from (52) that

which combining with (91) implies (87). Next, we shall prove (88) for . By duality, there exist functions defined on with such that

where

Since , there exists a nonnegative function such that

By a similar argument to in (90) and (51), we have

where . Equation (88) follows from (93) and (96). This completes the proof of Lemma 12.

3. Proof of Theorem 2

This section is devoted to the proofs of our main results.

Proof of Theorem 2. Assume that and for some satisfying (1). By Minkowski’s inequality, we can write

Let be as in (57). For ,  , and , we define the family of measures by

It is clear that

Here we use the convention . By Lemma 10 and (41), we get

for . This together with a straightforward calculation yields that

It follows from (97), (99), and Minkowski’s inequality that

Choose a sequence of nonnegative functions in such that

where are independent of . Define the Fourier multiplier operator by

Then we have

where

Below we estimate the estimates for . By Lemma 12 and the definition of , we have that, for , there exists a positive constant , which is independent of , and such that

By (107)-(108) and the Littlewood-Paley theory, we have

Similarly, we can obtain

On the other hand, by (41),(101), and Plancherel’s theorem, we have

where is as in Lemma 10 and

Thus

The interpolation between (109) and (113) leads to

where . It follows from (105) and (114) that

Similarly, by (105) and interpolation between (110) and (113), we have

Using (102), (115)-(116), we get

This finishes the proof of Theorem 2.

Proof of Theorem 3. By the same arguments as in the proof of [17, Theorem 2.3 (b)], we can obtain Theorem 3. We omit the details here.

Proof of Theorem 4. By the argument similar to those used in deriving Lemma 4.2 in [38], we can obtain Theorem 4. We omit the details here.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that she has no conflicts of interest.

Acknowledgments

The author was supported by the Natural Science Foundation of Fujian University of Technology (JAT170399, GY-Z15124, and GY-Z160129).

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