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Journal of Function Spaces
Volume 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

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

Correspondence should be addressed to Daiqing Zhang; moc.361@1102gniqiadgnahz

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 satisfiesFor a suitable mapping , we define the parametric Marcinkiewicz integral operator on bywhere (the Schwartz class), (), and . Here and denotes the set of all measurable functions satisfyingIf 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 [912] for relevant results. For general mapping , the operator has also been investigated extensively by many authors. For example, see [7, 1316] 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 [2329].

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 bywhere 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 [3032]). 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 directionSato [31] introduced the radial functions class , which denotes the set of all functions satisfyingHerewherewhere 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). Thenfor 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 , thenfor all .(ii)If , thenfor 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 . Thenfor 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 , thenfor all .(ii)If for some , thenfor 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 bywhere , , 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 byfor 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 thatwhere is a positive integer. Then, there exists a positive constant depending only on and such thatfor 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 thatThe constants and are independent of .

Proof. Let and and be as in Lemma 6. Without loss of generality, we assume that . Set . Setfor ,  ,  ,  , and . By the changes of variables, one findsfor 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 defineThen by (27) we getLetBy Lemma 7, we havefor , where . It is easy to see that and . Therefore, applying integration by parts and Hölder’s inequality, we see thatwhere . Therefore, we havewhere is independent of , and . We put , then by Lemma 6 we haveTherefore, if is a positive integer such that and ,This combining with (29) implieswhere . 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 thatwhere .

For , let be as in Lemma 8 and at the origin be defined byfor . For , define bySet . 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 bywhere .

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

Proof. By the straightforward calculations (43) holds. By Lemmas 89 we haveSimilarly,where for and . Thus (45) holds. On the other hand, by a change of variable, we havewhich combining with (43) implies thatSimilarly, we can obtainThis 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 thatThe constant is independent of , and .

Proof. For convenience, we set . It is easy to verify thatFor , we define the family of measures and maximal operators on bywhere is defined in the same way as , but with replaced by and replaced by . Thus, we haveTherefore, to prove (51), it suffices to prove thatThen 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 byfor , where are as in (41). It follows from (41) and (57)-(58) thatIn addition, by (59) and a well-known result on maximal functions (see [36]), we havewhereBy our assumption we havewhere is independent of , and . It remains to prove that