Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces
© Zhang and Liu; licensee Springer. 2014
Received: 23 February 2014
Accepted: 30 June 2014
Published: 22 July 2014
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.
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  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 .
Later on, Al-Salman et al.  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  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.  proved that is bounded on for , provided that for fixed . Chen and Ding  extended the above result to the case . Later on, Chen and Lu  proved that is bounded on for , provided that for some . This result was recently refined by Liu and Wu , who extended the range of β to the case and the range of p to the case . When and , Al-Salman  obtained the following result.
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:
is continuous increasing function satisfying that is monotonous;
there exist constant and such that and for all .
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 ).
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 . In fact, Theorem 1.2 with extends the result of  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 [, 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  in the study of bounds of singular integrals associated to certain surfaces.
Definition 1.1 ()
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 . 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.
for arbitrary functions .
where . Thus, Lemma 2.1 follows from the standard interpolation arguments. □
By an argument which is similar to those used in [, Proposition 3.1], one can easily get the following lemma. The details are omitted here.
- (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.
which, together with (2.11), completes the proof of Lemma 2.3. □
Lemma 2.4 ([, Lemma 2.2])
where and does not depend on .
Lemma 2.5 ([, Lemma 2.2])
is bounded on for . The bound is independent of the coefficients of for all and f but depends on φ.
Lemma 2.6 ()
3 Proofs of main theorems
Then Theorem 1.1 follows from (3.7)-(3.10) and Lemma 2.3. □
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. □
Set and .
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. □
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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