L^p Bounds for the parabolic singular integral operator

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

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

Let α₁,..., α_n be fixed real numbers, α_i ≥ 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 A_λ, if for any λ > 0 and x ∈ ℝⁿ

Moreover, Ω(x) satisfies the following condition

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

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

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

Theorem A. If Ω ∈ H¹(S^n-1) satisfies (1.1) and (1.2); then the operator T_Ω is bounded on L^p(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 T_ϕ,Ω in ℝⁿ⁺¹ along Γ by

where (x, x_n+1) ∈ ℝⁿ × ℝ = ℝⁿ⁺¹.

On the other hand, we note that if α₁ = ... = α_n = 1, then ρ(x) = |x|, α = n and (ℝⁿ, ρ) = (ℝⁿ, |·|). In this case, T_ϕ,Ω is 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 L^p boundedness of the parabolic singular integral operator T_ϕ,Ω along Γ when Ω ∈ F_β (S^n-1). For a β > 0, F_β (S^n-1) denotes the set of all Ω which are integrable over S^n-1 and satisfies

Condition (1.3) was introduced by Grafakos and Stefanov [8]. The examples in [8] show that there is the following relationship between F_β (S^n-1) and H¹(S^n-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,..., α_n}. In addition, let Ω ∈ F_β (S^n-1) for some β > 0 and satisfies (1.1) and (1.2), then T_ϕ,Ω is bounded on L^p(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,..., α_n}. In addition, let and satisfies (1.1) and (1.2), then T_{ϕ, Ω} is bounded on L^p(ℝⁿ⁺¹) for 1 < p < ∞.

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 = ρ^α-1J (φ₁,..., φ_n-1)dρdσ, where , dσ is the element of area of S^n-1 and ρ^α-1J(φ₁,..., φ_n-1) is the Jacobian of the above transform. In [1], it was shown there exists a constant L ≥ 1 such that 1 ≤ J(φ₁,..., φ_n-1) ≤ L and J(φ₁,..., φ_n-1) ∈ C^∞((0, 2π)^n-2 × (0, π)). So, it is easy to see that J is also a C^∞ function in the variable y' ∈ S^n-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⁺ ↦ ℝ^d 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 ℝ₊ to ℝⁿ⁺¹. For suitable f, the maximal function associated to the homogeneous curve Γ is defined by

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

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

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

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

and

for every f ∈ L^p(ℝⁿ⁺¹).

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

For each j, we define Φ_j in ℝⁿ by

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

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 < ∞,

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

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

In short

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

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.

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 D_k = {y ∈ ℝⁿ : 2^k < ρ(y) ≤ 2^k+1} and define the family of measures σ_k on ℝⁿ⁺¹ by

and σ* f(x) = sup_k∈ℤ(|σ_k| * | f | (x).

It is easy to see that

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

For (ξ, ξ_n+1) ∈ ℝⁿ × ℝ, y' ∈ S^n-1, and λ ∈ ℤ. Let

Set Λ = {α_i : α_i is the positive integers which is less than m in {α₁,..., α_n} and . Then , where α_i ∈ Λ, and is not a subset of {α₁,..., α_n}. 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

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

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

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

Let , we have

On the other hand, it is easy to see that

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

where . Thus, by (1.3), we get

Therefore,

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

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

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

The authors declare that they have no competing interests.

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.

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.

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