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Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces
Journal of Inequalities and Applications volume 2014, Article number: 265 (2014)
Abstract
In this note we establish the boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces, which improve and extend some previous results. The main ingredient is to present a systematic treatment with several singular integral operators.
MSC:42B20, 42B15, 42B25.
1 Introduction
Let , , be the n-dimensional Euclidean space and denote the unit sphere in equipped with the induced Lebesgue measure dσ. Let () be fixed real numbers. Define the function by , . It is clear that, for each fixed , the function is a decreasing function in . We let denote the unique solution of the equation . Fabes and Rivière [1] showed that is a metric space, which is often called the mixed homogeneity space related to . For , we let be the diagonal matrix . Let and , we denote by for , where .
The change of variables related to the spaces is given by the transformation
Thus , where is the Jacobian of the above transform and , . Obviously, and there exists such that
It is easy to see that
Let Ω be integrable on and satisfy
For and a suitable function , we define the parabolic Marcinkiewicz integral operator on by
When , we denote by . Clearly, if and , the operator reduces to the classical Marcinkiewicz integral operator denoted by , which was introduced by Stein [2] and investigated by many authors (see [3–9] for example). In particular, Ding et al. [5] proved that if , then is bounded on for . Subsequently, Chen et al. [4] showed that is bounded on for if for some . Here
The functions class was introduced by Grafakos and Stefanov [10] in the study of boundedness of singular integral operator with rough kernels. It follows from [10] that for , and for any . Moreover,
and
Later on, Al-Salman et al. [11] proved that is bounded on for provided that . It is well known that and do not contain each other. When and with being a real polynomial on ℝ satisfying , Wu [12] proved that is bounded on for provided that for some . The boundedness for the Marcinkiewicz integral operator associated to polynomial mappings has also been obtained (see [6, 13]).
When (), and , we denote by . In 2008, Ding et al. [14] proved that is bounded on for , provided that for fixed . Chen and Ding [15] extended the above result to the case . Later on, Chen and Lu [16] proved that is bounded on for , provided that for some . This result was recently refined by Liu and Wu [17], who extended the range of β to the case and the range of p to the case . When and , Al-Salman [18] obtained the following result.
Theorem A Let and . Suppose that for some with satisfying (1.1)-(1.2).
-
(i)
If with P being a real polynomial on ℝ, then are bounded on for . The bounds are independent of the coefficients of P.
-
(ii)
If , then are bounded on for . Here is the set of all functions ϕ which satisfy:
-
(a)
is continuous increasing function satisfying that is monotonous;
-
(b)
there exist constant and such that and for all .
-
(a)
Remark 1.1 There are some model examples in the class , such as (), (), , real-valued polynomials P on ℝ with positive coefficients and and so on. For , there exists a constant such that (see [19]).
It is natural to ask whether Theorem A also holds if the range of β is relaxed to and the range of p is relaxed to . In this paper, we will give an affirmative answer to this question. Our main results can be stated as follows.
Theorem 1.1 Let and with being real valued polynomials on ℝ satisfying and . Suppose that for some satisfying (1.1)-(1.2). Then are bounded on for . The bounds are independent of the coefficients of for all but depend on and φ.
Theorem 1.2 Let and with and and if . Suppose that for some satisfying (1.1)-(1.2). Then are bounded on for . The bounds are independent of the coefficients of but depend on N and φ.
Remark 1.2 It is clear that Theorem 1.1 implies Theorem 1.2. When , and , Theorem 1.1 implies the result of [12]. In fact, Theorem 1.2 with extends the result of [12] to the mixed case. Comparing Theorem A with Theorem 1.2, the range of β is extended to the case and the range of p is enlarged to the case . Thus Theorem 1.2 essentially improves and generalizes the corresponding results in Theorem A. In addition, Theorem 1.2 implies the result [[17], Theorem 1.3] when .
When , we have the following result.
Theorem 1.3 Let be real analytic on . Let with and . Suppose that for some satisfying (1.1)-(1.2). Then are bounded on for . The bounds are independent of the coefficients of but depend on φ and N.
We remark that when and , the surface given as in Theorem 1.3 recovers , which was originally introduced by Al-Balushi and Al-Salman [20] in the study of bounds of singular integrals associated to certain surfaces.
The third type of surfaces we consider are polynomial compound subvarieties. To state the rest of our result, we need to recall some notations. Let be the set of polynomials on which have real coefficients and degrees not exceeding m, and let be the collection of polynomials in which are homogeneous of degree m. For , we set
Definition 1.1 ([21])
Let , and . An integrable function Ω on is said to be in the space if
It should be pointed out that the condition (1.5) was introduced by Al-Salman and Pan [21] (also see [22]) in a study of the boundedness of singular integrals with rough kernels. It is easy to check that . Moreover, it was shown in [21] that
The rest of the results can be stated as follows.
Theorem 1.4 Let with being a polynomial for . Let and . Suppose that Ω satisfies (1.1)-(1.2) and for some . Then are bounded on for . The bounds are independent of the coefficients of for all but depend on and φ.
Theorem 1.5 Let with being a polynomial for . Let and . Suppose that Ω satisfies (1.1)-(1.2) and for some . Then are bounded on for . The bounds are independent of the coefficients of for all but depend on and φ.
Remark 1.3 When (), Theorem 1.4 implies Theorem 1.2 with in [13]. Obviously, Theorem 1.5 follows from Theorem 1.4 because of (1.6).
The rest of this paper is organized as follows. After recalling some preliminary notations and lemmas in Section 2, we will prove our results in Section 3. We would like to remark that the main methods employed in this paper is a combination of ideas and arguments from [12, 21, 23]. The main ingredient in our proofs is to give a systematic treatment with these operators mentioned above.
Throughout this paper, we let satisfy . The letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence, but independent of the essential variables.
2 Preliminaries
Lemma 2.1 Let be a family of measures. Suppose that
holds for some and . Then there exists a constant such that
for arbitrary functions .
Proof By the assumption, we have
On the other hand, by the dual argument, there exists a function satisfying such that
where . Thus, Lemma 2.1 follows from the standard interpolation arguments. □
Let be a sequence of real positive numbers with satisfying . Let be a collection of functions satisfying the following conditions:
where C is independent of t and k. Let and be a linear transformation. For each , we define the multiplier operators in by
By an argument which is similar to those used in [[8], Proposition 3.1], one can easily get the following lemma. The details are omitted here.
Lemma 2.2 Let be as in (2.1) and arbitrary functions on . Then
-
(i)
for each fixed and ,
(2.2) -
(ii)
for each fixed and ,
(2.3)
The following lemma is our main ingredient in the proof of our main results.
Lemma 2.3 Let be a family of uniformly bounded Borel measures on . Let be a sequence of real numbers with satisfying . Let and be a linear transformation. Suppose that
for all . Then for and , there exists a constant such that
Proof Let be as in (2.1). Then we can write
Case 1. . It follows from (2.2) and (2.7) that
For each fixed , we set
Invoking Lemma 2.1 and the Littlewood-Paley theory imply
On the other hand, by Plancherel’s theorem and (2.4)-(2.5), we have
where . That is,
Interpolating between (2.9) and (2.10), there exists such that
For fixed and , we can choose such that . Thus
which, together with (2.8), implies
Case 2. . By (2.3) and (2.7), we have for and ,
Let
By (2.6), [[23], p.544, Lemma] and the Littlewood-Paley theory, we have, for and ,
On the other hand, by the same arguments as in (2.10), we have
where is as in (2.10). On interpolation between (2.13) and (2.14), for fixed and , we can choose and such that and
This, combined with (2.12), implies
which, together with (2.11), completes the proof of Lemma 2.3. □
Lemma 2.4 ([[13], Lemma 2.2])
Suppose and , where are real parameters, and are distinct positive (not necessarily integer) exponents. Then for any and ,
where and does not depend on .
Lemma 2.5 ([[24], Lemma 2.2])
Let with being real polynomials defined on . Suppose that . Then the operator defined by
is bounded on for . The bound is independent of the coefficients of for all and f but depends on φ.
Lemma 2.6 ([25])
Let , be real analytic on . Suppose that is linearly independent set. If for some , then
3 Proofs of main theorems
Proof of Theorem 1.1 Let . For , let . For , and , let and . Set and
Then we can write
where is the linear transformation given by
For each , and , we define the measures and by
We get from (3.1)
On the other hand, by a change of variable, we have
where
By Lemma 2.4, we have
Combining the trivial inequality with the fact that is increasing in , we have
where . This, together (3.3) with the fact that , implies
Now we can choose a function such that for and for . For , and , we define the measures by
Here we use the convention . It is easy to see that
It follows from (3.2), (3.5), and the trivial estimate that, for ,
By the definition of and (3.6), we can write
On the other hand, by a change of variable we have
where is as in Lemma 2.5 and . By Lemma 2.5 and Minkowski’s inequality, we have
This inequality, together with the definition of , yields
Then Theorem 1.1 follows from (3.7)-(3.10) and Lemma 2.3. □
Proof of Theorem 1.3 Let Φ, , φ, ϕ be as in Theorem 1.3. For , we set . Define the measures and by
Following the notation in [20], let be a maximal linearly independent subset of , where , and . Thus, for , there exist such that
This implies that there exists a linear transformation such that
where . Thus
where
By Lemma 2.4, we have
Since is increasing in for any , and with for any . We can deduce from (3.13) and the trivial estimate that
where . Invoking Lemma 2.6 and (3.14), we obtain, for ,
On the other hand, we have
Notice that
This combining with Lemma 2.5 and Minkowski’s inequality, implies
Then the rest of the proof of Theorem 1.3 follows from an argument which is similar to those in the proof of Theorem 1.1 and (3.15)-(3.17). We omit the details. □
Proof of Theorem 1.4 Let , , where is a polynomial for . Let
and
For , we let
where
Set and .
For each , and , we define the measures and by
For , let denote the number of multi-indices satisfying , and define the linear transformation by
By the change of variables, we have
where
Let
By Lemma 2.4, we have
By this inequality, together with the trivial estimate , we get
Since , and , we immediately obtain
On the other hand, we have
In addition, using Lemma 2.5, one can easily check that
Then the rest proof of Theorem 1.4 follows from similar arguments to the proof of Theorem 1.1 and (3.18)-(3.20). Details will be omitted. □
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Acknowledgements
The authors would like to thank the referees for their carefully reading and invaluable comments. This work was supported by the NNSF of China (Nos. 11101339, 11371295).
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Zhang, D., Liu, F. Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces. J Inequal Appl 2014, 265 (2014). https://doi.org/10.1186/1029-242X-2014-265
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DOI: https://doi.org/10.1186/1029-242X-2014-265