- Open Access
Multiple singular integrals and Marcinkiewicz integrals with mixed homogeneity along surfaces
© Liu and Wu; licensee Springer. 2012
- Received: 18 February 2012
- Accepted: 17 August 2012
- Published: 31 August 2012
This paper is devoted to studying the singular integrals and Marcinkiewicz integrals with mixed homogeneity along surfaces, which contain many classical surfaces as model examples, on the product domains (). Under rather weak size conditions of the kernels, the -boundedness for such operators is established. These results essentially extend certain previous results.
- singular integrals
- Marcinkiewicz integrals
- maximal operators
- mixed homogeneity
- product domains
Let (), , be the d-dimensional Euclidean space and be the unit sphere in equipped with the induced Lebesgue measure . 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 Riviére  showed that is a metric space, which is often called the mixed homogeneity space related to . For , we let be the diagonal matrix . Let , we denote by for , where .
As is well known, a singular integral operator of the type (1.1) originally arose from the study on the existence and regularity results of the heat equation and the more general parabolic differential operator with constant coefficients. In 1966, Fabes and Riviére  showed that T is bounded on for if . Subsequently, Nagel, Riviére and Wainger  weakened the regularity condition on Ω to the case . Recently, Chen, Ding and Fan  extended further the condition to the case .
Based on the above, a natural question is as follows.
Question 1.1 For the general case () and (), is bounded on under the condition (1.5) for some ?
Wu and Yang  proved that if with , then is bounded on for . In this paper, we will extend the result above as follows.
is continuous strictly increasing and satisfying that is monotonous;
there exist constants such that and for all .
Suppose that Ω satisfies (1.2)-(1.3) and for some . Then defined as in (1.7) is bounded on for . The bound is independent of the coefficients of (), but depends on φ, ψ, , , m, n and β.
Taking , this is the desired constant.
Remark 1.2 We remark that the model examples for functions are (), , and real-valued polynomials P on with positive coefficients and (see ). Theorem 1.1 extends the result of , which is the multiple-parameter generalization of the result in [11, 16], to the mixed homogeneity setting, even in the special case . Also, by (1.6), Theorem 1.1 is distinct from the result of , even in the special case .
When , (; ), we denote by , which is the classical Marcinkiewicz integral on the product domains and is studied extensively by many authors (see [1, 3, 4, 6, 7, 17, 23–26]et al.). In particular, Al-Qassem, Al-Salman, Cheng and Pan  showed that if , then is bounded on for ; Hu, Lu and Yan  (also see [23, 26]) proved that if for , then is bounded on for . For the general operator , when () and , Al-Salman  showed that is bounded on for provided that .
A natural question which arises from the above is the following:
Question 1.2 Under the condition (1.5) with , is also bounded on for ?
This question will be addressed by our next theorem.
Theorem 1.2 Let (), φ, ψ be as in Theorem 1.1. Suppose for some and satisfies (1.2)-(1.3). Then defined as (1.9) is bounded on for . The bound is independent of the coefficients of () but depends on φ, ψ, , , m, n and β.
Remark 1.3 Theorem 1.2 extends the result of  to the mixed homogeneity setting, even for the special case . And by (1.6), Theorem 1.2 is distinct from the result of , even in the special case .
The rest of this paper is organized as follows. After recalling some notation and establishing some preliminary lemmas, we will prove Theorem 1.1 in Section 2. And the proof of Theorem 1.2 will be given in Section 3. We remark that our some ideas in the proofs of our main results are taken from [3, 9, 11, 17], but our methods and technique are more delicate and complex than those used in [3, 9, 11, 17].
Throughout this paper, the letter C or c, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence but independent of the essential variables.
Lemma 2.1 (cf. , pp.476-478])
is bounded on for . The bound is independent of the coefficients of () and f.
is bounded on for . The bound is independent of the coefficients of () and f, but depends on ϕ.
This implies that . Then Lemma 2.2 follows from Lemma 2.1. □
is bounded on for . The bound is independent of the coefficients of () and f, but depends on φ, ψ, , , m, n.
which completes the proof of Lemma 2.3. □
Lemma 2.4 (cf. , p.186, Corollary])
with and C does not depend on as long as .
The constant is independent of the coefficients of but depends on φ; and is independent of the coefficients of but depends on ψ.
This proves Lemma 2.5. □
The constant C is independent of the coefficients of and .
This completes the proof of Lemma 2.6. □
- (ii)if , then(2.14)
- (iii)if , then(2.15)
- (iv)if and , then(2.16)
Here and below, () is as in Remark 1.1, the constant C is independent of the coefficients of ().
Then (2.14) follows from (2.6)-(2.7) with (2.19). Similarly, we get (2.15). Finally, (2.16) follows from (2.8), (2.17), (2.20) and (2.21). This completes the proof of Lemma 2.7. □
By Lemma 2.3 and the definition of , it is easy to verify the following lemma.
for . The constant C is independent of the coefficients of and .
Applying Lemma 2.8 and , p.544, Lemma], we can obtain
for and any arbitrary functions . The constant C is independent of the coefficients of and .
Now we are in the position of proving Theorem 1.1.
, , ;
, where C is a constant.
This together with (2.22) and (2.25) completes the proof of Theorem 1.1. □
The constant C is independent of the coefficients of and .
This completes the proof of Lemma 3.1. □
Applying Lemma 3.1 and the same arguments as in proving Lemma 2.7, we have
- (ii)if , then
- (iii)if , then
- (iv)if and , then
Here the constant C is independent of the coefficients of ().
By Lemma 2.2 and the same arguments as in proving Lemma 2.3, we have the following lemma:
is bounded on for . The bound is independent of the coefficients of () but depends on φ, ψ, , , m, n.
Applying Lemma 3.3, we have
The constant C is independent of the coefficients of ().
Furthermore, applying Lemma 3.4 and , p.544, Lemma], we can obtain
where the constant C is independent of the coefficients of ().
where are arbitrary functions defined on . The constant is independent of the coefficients of ().
which implies .
which implies . Then Lemma 3.6 follows from the standard interpolation arguments. □
By the arguments similar to those used in , pp.78-81], we easily establish this lemma. The details are omitted.
Now we turn to prove Theorem 1.2.
This together with (3.21) completes the proof of Theorem 1.2. □
The authors would like to thanks the referees for their carefully reading and invaluable comments. This work was Supported by the NNSF of China (11071200) and the NSF of Fujian Province of China (No. 2010J01013).
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