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.
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.
2 On multiple singular integrals
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. □
3 On the multiple Marcinkiewicz integrals
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).
- Al-Qassem H, Al-Salman A, Cheng L, Pan Y: Marcinkiewicz integrals on product spaces. Stud. Math. 2005, 167: 227–234. 10.4064/sm167-3-4MathSciNetView ArticleGoogle Scholar
- Al-Salman A, Al-Qassem H, Pan Y: Singular integrals on product domains. Indiana Univ. Math. J. 2006, 55(1):369–387. 10.1512/iumj.2006.55.2626MathSciNetView ArticleGoogle Scholar
- Al-Salman A: Parabolic Marcinkiewicz integrals along surfaces on product domains. Acta Math. Sin. Engl. Ser. 2011, 27(1):1–18.MathSciNetView ArticleGoogle Scholar
- Chen J, Ding Y, Fan D: Certain square functions on product spaces. Math. Nachr. 2001, 230: 5–18. 10.1002/1522-2616(200110)230:1<5::AID-MANA5>3.0.CO;2-OMathSciNetView ArticleGoogle Scholar
- Chen Y, Ding Y, Fan D: A parabolic singular integral operator with rough kernel. J. Aust. Math. Soc. 2008, 84: 163–179.MathSciNetView ArticleGoogle Scholar
- Chen J, Fan D, Ying Y: The method of rotation and Marcinkiewicz integrals on product domains. Stud. Math. 2002, 153(1):41–58. 10.4064/sm153-1-4MathSciNetView ArticleGoogle Scholar
- Choi Y: Marcinkiewicz integrals with rough homogeneous kernel of degree zero on product domains. J. Math. Anal. Appl. 2001, 261: 53–60. 10.1006/jmaa.2001.7465MathSciNetView ArticleGoogle Scholar
- Chen L, Le H: Singular integrals with mixed homogeneity in product spaces. Math. Inequal. Appl. 2011, 14(1):155–172.MathSciNetGoogle Scholar
- Duoandikoetxea J, Rubio de Francia JL: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 1986, 84: 541–561. 10.1007/BF01388746MathSciNetView ArticleGoogle Scholar
- Duoandikoetxea J: Multiple singular integrals and maximal functions along hypersurfaces. Ann. Inst. Fourier 1986, 36(4):185–206. 10.5802/aif.1073MathSciNetView ArticleGoogle Scholar
- Fan D, Guo K, Pan Y: A note of a rough singular integral operator. Math. Inequal. Appl. 1999, 2(1):73–81.MathSciNetGoogle Scholar
- Fan D, Guo K, Pan Y: Singular integrals with rough kernels on product spaces. Hokkaido Math. J. 1999, 28: 435–460.MathSciNetView ArticleGoogle Scholar
- Fabes E, Reviére N: Singular integrals with mixed homogeneity. Stud. Math. 1966, 27: 19–38.Google Scholar
- Fefferman R: Singular integrals on product domains. Bull. Am. Math. Soc. 1981, 4: 195–201. 10.1090/S0273-0979-1981-14883-7MathSciNetView ArticleGoogle Scholar
- Fefferman F, Stein EM: Singular integrals on product domains. Adv. Math. 1982, 45: 117–143. 10.1016/S0001-8708(82)80001-7MathSciNetView ArticleGoogle Scholar
- Grafakos L, Stefanov A: bounds for singular integrals and maximal singular integrals with rough kernels. Indiana Univ. Math. J. 1998, 47: 455–469.MathSciNetView ArticleGoogle Scholar
- Hu G, Lu S, Yan D: boundedness for the Marcinkiewicz integrals on product spaces. Sci. China Ser. A 2003, 46(1):75–82. 10.1360/03ys9008MathSciNetView ArticleGoogle Scholar
- Nagel A, Riviére NM, Wainger S: On Hilbert transforms along curves, II. Am. J. Math. 1976, 98: 395–403. 10.2307/2373893View ArticleGoogle Scholar
- Ricci F, Stein EM: Multiparameter singular integrals and maximal functions. Ann. Inst. Fourier 1992, 42: 637–670. 10.5802/aif.1304MathSciNetView ArticleGoogle Scholar
- Ricci R, Stein EM: Harmonic analysis on nilpotent groups and singular integrals I: oscillatory integrals. J. Funct. Anal. 1987, 73: 179–184. 10.1016/0022-1236(87)90064-4MathSciNetView ArticleGoogle Scholar
- Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton; 1993.Google Scholar
- Walsh T: On the function of Marcinkiewicz. Stud. Math. 1972, 44: 203–217.Google Scholar
- Wu H: Boundedness of multiple Marcinkiewicz integral operators with rough kernels. J. Korean Math. Soc. 2006, 43(3):635–658.MathSciNetGoogle Scholar
- Wu H: A rough multiple Marcinkiewicz integral along continuous surfaces. Tohoku Math. J. 2007, 59(2):145–166. 10.2748/tmj/1182180732MathSciNetView ArticleGoogle Scholar
- Wu H: General Littlewood-Paley functions and singular integral operators on product spaces. Math. Nachr. 2006, 279(4):431–444. 10.1002/mana.200310369MathSciNetView ArticleGoogle Scholar
- Wu H, Xu J: Rough Marcinkiewicz integrals associated to surfaces of revolution on product domains. Acta Math. Sci., Ser. B 2009, 29(2):294–304.MathSciNetView ArticleGoogle Scholar
- Wu H, Yang S: On multiple singular integrals along polynomial curves with rough kernels. Acta Math. Sin. Engl. Ser. 2008, 24(2):177–184.MathSciNetView ArticleGoogle Scholar
- Ying Y: A note on singular integral operators on product domains. J. Math. Study 1999, 32(3):264–271. in ChineseMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.