Research

# Lp Bounds for the parabolic singular integral operator

Yanping Chen1*, Feixing Wang2 and Wei Yu3

Author Affiliations

1 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

2 Department of Information and Computer Science, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

3 Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China

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Journal of Inequalities and Applications 2012, 2012:121 doi:10.1186/1029-242X-2012-121

 Received: 7 March 2012 Accepted: 30 May 2012 Published: 30 May 2012

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.

### Abstract

Let 1 < p < ∞ and n ≥ 2. The authors establish the Lp(ℝn+1) boundedness for a class of parabolic singular integral operators with rough kernels.

MR(2000) Subject Classification: 42B20; 42B25.

##### Keywords:
parabolic singular integral operator; rough kernel; surfaces of revolution

### 1 Introduction

Let α1,..., αn be fixed real numbers, αi ≥ 1. For fixed x ∈ ℝn, the function is a decreasing function in ρ > 0. We denote the unique solution of the equation F(x, ρ) = 1 by ρ(x). Fabes and Rivière [1] showed that ρ(x) is a metric on ℝn, and (ℝn, ρ) is called the mixed homogeneity space related to .

For λ > 0, let . Suppose that Ω(x) is a real valued and measurable function defined on ℝn. We say is Ω(x) is homogeneous of degree zero with respect to Aλ, if for any λ > 0 and x ∈ ℝn

(1.1)

Moreover, Ω(x) satisfies the following condition

(1.2)

where J(x') is a function defined on the unit sphere Sn-1 in ℝn, which will be defined in Section 2.

In 1966, Fabes and Rivière [1] proved that if Ω ∈ C1(Sn-1) satisfying (1.1) and (1.2), then the parabolic singular integral operator TΩ is bounded on Lp(ℝn) for 1 < p < ∞, where TΩ is defined by

In 1976, Nagel et al. [2] improved the above result. They showed TΩ is still bounded on Lp(ℝn) for 1 < p < ∞ if replacing Ω ∈ C1(Sn-1) by a weaker condition Ω ∈ L log+ L(Sn-1). Recently, Chen et al. [3] improve Theorem A, the result is

Theorem A. If Ω ∈ H1(Sn-1) satisfies (1.1) and (1.2); then the operator TΩ is bounded on Lp(ℝn) for 1 < p < ∞.

For a suitable function ϕ on [0, 1), and Γ = {(y, ϕ(ρ(y)): y ∈ ℝn}. Define the singular integral operator Tϕin ℝn+1 along Γ by

where (x, xn+1) ∈ ℝn × ℝ = ℝn+1.

On the other hand, we note that if α1 = ... = αn = 1, then ρ(x) = |x|, α = n and (ℝn, ρ) = (ℝn, |·|). In this case, Tϕis just the classical singular integral operator along surfaces of revolution, which was studied by the authors of [4-7].

The purpose of this article is to investigate the Lp boundedness of the parabolic singular integral operator Tϕalong Γ when Ω ∈ Fβ (Sn-1). For a β > 0, Fβ (Sn-1) denotes the set of all Ω which are integrable over Sn-1 and satisfies

(1.3)

Condition (1.3) was introduced by Grafakos and Stefanov [8]. The examples in [8] show that there is the following relationship between Fβ (Sn-1) and H1(Sn-1):

We shall state our main results as follows:

Theorem 1 Let m ∈ ℕ. Suppose that ϕ is a polynomial of degree m and , where are the all positive integers which is less than m in {α1,..., αn}. In addition, let Ω ∈ Fβ (Sn-1) for some β > 0 and satisfies (1.1) and (1.2), then Tϕis bounded on Lp(ℝn+1) for .

Corollary 1 Let m ∈ ℕ. Suppose that ϕ is a polynomial and , where are the all positive integers which is less than m in {α1,..., αn}. In addition, let and satisfies (1.1) and (1.2), then Tϕ, Ω is bounded on Lp(ℝn+1) for 1 < p < ∞.

### 2 Notations and lemmas

In this section, we give some notations and lemmas which will be used in the proof of Theorem 1. For any x ∈ ℝn, set

Then dx = ρα-1J (φ1,..., φn-1)dρdσ, where , is the element of area of Sn-1 and ρα-1J(φ1,..., φn-1) is the Jacobian of the above transform. In [1], it was shown there exists a constant L ≥ 1 such that 1 ≤ J(φ1,..., φn-1) ≤ L and J(φ1,..., φn-1) ∈ C((0, 2π)n-2 × (0, π)). So, it is easy to see that J is also a Cfunction in the variable y' ∈ Sn-1. For simplicity, we denote still it by J(y').

In order to prove our theorems, we need the following lemmas:

Lemma 2.1. ([9]) Let d ∈ ℕ. Suppose that γ (t): ℝ+ ↦ ℝd satisfies for a fixed matrix M, and assume γ(t) doesn't lie in an affine hyperplane. Then

Lemma 2.2. ([9]) Suppose that and are fixed real numbers, ϕ(t) is a polynomial and is a function from + to n+1. For suitable f, the maximal function associated to the homogeneous curve Γ is defined by

(2.1)

Then for 1 < p ≤ ∞, there is a constant C > 0, independent of , the coefficient of ϕ(t) and f, such that

(2.2)

Lemma 2.3. Let L : ℝn+1 → ℝn be a linear transformation. Suppose that {σk}k∈ℤ is a sequence of uniformly bounded measures on d satisfying

(2.3)

for ξ ∈ ℝn+1 and k ∈ ℤ. For any 1 < p0 < and A > 0

(2.4)

holds for arbitrary functions {gk}k∈ℤ on n+1. Then for there exists a constant Cp = C(p, n) which is independent of L such that

(2.5)

and

(2.6)

for every f Lp(ℝn+1).

Proof. The main idea of the proof is taken from [7,8], we assume that = (ξ1,..., ξn) = ζ for ξ = (ξ1,..., ξn, ξn+1) ∈ ℝn+1. Choose a such that 0· ψ ≤ 1, supp(ψ) ⊆ (1/4, 4), and

(2.7)

For each j, we define Φj in ℝn by

for ξ = (ξ1,..., ξn+1) ∈ ℝn+1. If we set

(2.8)

and let δ represent the Dirac delta on ℝ, then by (2.7), for any Schwartz function f,

where

By using (2.4) and Littlewood-Paley theory (as in [3]), one obtains that for any 1 < p0 < ∞,

(2.9)

On the other hand, by using Plancherel's theorem and (2.3), If j > 0, using the estimate we have

(2.10)

Similar to the proof of (2.10), using Plancherel's theorem and (2.3), if j < 0 we get

(2.11)

In short

(2.12)

By interpolating between (2.9) and (2.12), we obtain

(2.13)

for

and some β > 0. Thus, (2.5) follows from (2.13). One may then use a randomization argument to derive (2.6). Lemma 2.1 is proved.

### 3 Proof of Theorem 1

The main idea of the proof of Theorem 1 is taken from [10]and [11]. Let Ω satisfies (1.1), (1.2), and (1.3) for some β > 0. Let Φ(y) = (y, ϕ(ρ(y))), where , m ∈ ℕ. Let Dk = {y ∈ ℝn : 2k < ρ(y) ≤ 2k+1} and define the family of measures σk on ℝn+1 by

(3.1)

and σ* f(x) = supk∈ℤ(|σk| * | f | (x).

It is easy to see that

(3.2)

In light of (3.2) and Lemma 2.3, it suffices to show that σk satisfies (2.3) and (2.4).

For (ξ, ξn+1) ∈ ℝn × ℝ, y' ∈ Sn-1, and λ ∈ ℤ. Let

Set Λ = {αi : αi is the positive integers which is less than m in {α1,..., αn} and . Then , where αi ∈ Λ, and is not a subset of {α1,..., αn}. Therefore, we get

Without loss of generality, we may assume Λ consists of r distinct numbers and let If are all distinct, by Lemma 2.1, we get immediately

(3.3)

If {αj} only consists of s distinct numbers, we suppose that

where s is a positive integer with 1 ≤ s n, l1, l2,..., ls are positive integers such that l1 + l2 + ··· + ls = n and are distinct. Obviously,

does not lie in an affine hyperplane in ℝs+m-r. Then using Lemma 2.1 again, there exists C > 0 such that for any vector η = (η1,..., ηn) ∈ ℝn,

Let , we have

(3.3a)

On the other hand, it is easy to see that

(3.4)

From (3.3), (3.3') and (3.4), we get

where . Thus, by (1.3), we get

Therefore,

(3.5)

On the other hand, by (1.2), we can obtain

(3.6)

Clearly, (3.5) and (3.6) imply (2.3) holds. Finally, we shall show that (2.4) holds.

By Lemma 2.2, we obtain ||MΦ(f)||p C||f||p, where C > 0 is independent of k, the coefficient of ϕ(t) and f, since Ω is integrable on Sn-1, thus ||σ*(f)||p C||f||p. This shows (2.4) holds. This completes the proof of the Theorem 1.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

YC carried out the parabolic singular integral operator studies and drafted the manuscript. WY participated in the study of Littlewood-Paley theory. FW conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

### Acknowledgements

The research was supported by the NSF of China (Grant No. 10901017), NCET of China (Grant No. NCET-11-0574), and the Fundamental Research Funds for the Central Universities.

### References

1. Fabes, E, Rivière, N: Singular integrals with mixed homogeneity. Studia Math. 27, 19–38 (1966)

2. Nagel, A, Riviere, N, Wainger, S: On Hilbert transforms along curves, II. Am Math J. 98, 395–403 (1976). Publisher Full Text

3. Chen, Y, Ding, Y, Fan, D: A parabolic singular integrals with rough kernel. J Aust Math Soc. 84, 163–179 (2008)

4. Fan, D, Guo, K, Pan, Y: A note of a rough singular integral operator. Math Inequal Appl. 7381, 73–81 (1999)

5. Lu, S, Pan, Y, Yang, D: Rough singular integrals associated to surfaces of revolution. P Am Math Soc. 129, 2931–2940 (2001). Publisher Full Text

6. Kim, W, Wainger, S, Wright, J, Ziesler, S: Singular integrals and maximal functions associated to surfaces of revolution. Bull Lond Math Soc. 28, 291–296 (1996). Publisher Full Text

7. Cheng, LC, Pan, Y: Lp bounds for singular integrals associated to surfaces of revolution. J Math Anal Appl. 265, 163–169 (2002). Publisher Full Text

8. Grafakos, L, Stefanov, A: Convolution Calderón-Zygmund singular integral operators with rough kernel. Indiana Univ math J. 47, 455–469 (1998)

9. Stein, EM, Wainger, S: Problems in harmonic analysis related to curvature. Bull Am Math Soc. 84, 1239–1295 (1978). Publisher Full Text

10. Xue, Q, Ding, Y, Yabuta, K: Parabolic Littlewood-Paley g-function with rough kernels. Acta Math Sin (Engl Ser). 24, 2049–2060 (2008). Publisher Full Text

11. Duoandikoetxea, J, Rubio de Francia, JL: Maximal and singular integral operators via Fourier transform estimates. Invent Math. 84, 541–561 (1986). Publisher Full Text