L^p Bounds for the parabolic singular integral operator

Abstract

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.

1 Introduction

Let α₁,..., α _n be fixed real numbers, α _i ≥ 1. For fixed x ∈ ℝ ⁿ , the function $F\left(x,\rho \right)={\sum }_{i=1}^n\frac{{x_i}^2}{{\rho }^{2{\alpha }_i}}$ 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 ℝ ⁿ , and (ℝ ⁿ, ρ) is called the mixed homogeneity space related to ${\left\{{\alpha }_i\right\}}_{i=1}^n$.

For λ > 0, let $A_{\lambda }=\left(\begin{array}{cc}\hfill {\lambda }^{{\alpha }_1}\hfill & \hfill 0\hfill \\ \hfill \ddots \hfill \\ \hfill 0\hfill & \hfill {\lambda }^{{\alpha }_n}\hfill \end{array}\right)$. Suppose that Ω(x) is a real valued and measurable function defined on ℝ ⁿ . We say is Ω(x) is homogeneous of degree zero with respect to A _λ , if for any λ > 0 and x ∈ ℝ ⁿ

$$\Omega \left(A_{\lambda }x\right)=\Omega \left(x\right).$$

(1.1)

Moreover, Ω(x) satisfies the following condition

$$\underset{S^{n-1}}{\int }\Omega \left(x^{\prime }\right)J\left(x^{\prime }\right)d\sigma \left(x^{\prime }\right)=0,$$

(1.2)

where J(x') is a function defined on the unit sphere S^n-1in ℝ ⁿ , 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 (ℝ ⁿ ) for 1 < p < ∞, where T_Ω is defined by

$$T_{\Omega }f\left(x\right)=\mathsf{\text{p}}\mathsf{\text{.v}}\mathsf{\text{.}}\underset{{ℝ}^n}{\int }\frac{\Omega \left(y\right)}{\rho {\left(y\right)}^{\alpha }}f\left(x-y\right)dy\mathsf{\text{and}}\alpha =\sum _{i=1}^n{\alpha }_i.$$

In 1976, Nagel et al. [2] improved the above result. They showed T_Ω is still bounded on L^p (ℝ ⁿ ) 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_Ωis bounded on L^p (ℝ ⁿ ) for 1 < p < ∞.

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

$$\left(T_{\varphi ,\Omega }f\right)\left(x,x_{n+1}\right)=\mathsf{\text{p}}\mathsf{\text{.v}}\mathsf{\text{.}}\underset{{ℝ}^n}{\int }f\left(x-y,x_{n+1}-\varphi \left(\rho \left(y\right)\right)\right.\frac{\Omega \left(y\right)}{\rho {\left(y\right)}^{\alpha }}dy,$$

where (x, x_n+1) ∈ ℝ ⁿ × ℝ = ℝⁿ⁺¹.

On the other hand, we note that if α₁ = ... = α _n = 1, then ρ(x) = |x|, α = n and (ℝ ⁿ, ρ) = (ℝ ⁿ , |·|). In this case, T_ϕ,Ωis 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,Ωalong 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-1and satisfies

$$\underset{\xi \in S^{n-1}}{\mathsf{\text{sup}}}\underset{S^{n-1}}{\int }\left|\Omega \left(\theta \right)\right|{\left(\mathsf{\text{ln}}\frac{1}{|\theta \cdot \xi |}\right)}^{1+\beta }d\theta <\infty .$$

(1.3)

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):

$$\bigcap _{\beta >0}{F}_{\beta }\left(S^{n-1}\right)⊈H^1\left(S^{n-1}\right)⊈\bigcup _{\beta >0}{F}_{\beta }\left(S^{n-1}\right).$$

We shall state our main results as follows:

Theorem 1 Let m ∈ ℕ. Suppose that ϕ is a polynomial of degree m and$\frac{d^{{\alpha }_i}\varphi \left(t\right)}{d{t}^{{\alpha }_i}}{|}_{t=0}=0$, where${\alpha }_i^{\prime }s$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,Ωis bounded on L^p (ℝⁿ⁺¹) for$p\in \left(\frac{2+2\beta }{1+2\beta },2+2\beta \right)$.

Corollary 1 Let m ∈ ℕ. Suppose that ϕ is a polynomial and$\frac{d^{{\alpha }_i}\varphi \left(t\right)}{d{t}^{{\alpha }_i}}{|}_{t=0}=0$, where${\alpha }_i^{\prime }s$are the all positive integers which is less than m in {α₁,..., α _n }. In addition, let$\Omega \in {\cap }_{\beta >0}{F}_{\beta }\left(S^{n-1}\right)$and satisfies (1.1) and (1.2), then T _{ϕ, Ω} is bounded on L^p (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 ∈ ℝ ⁿ , set

$$\begin{array}{c}{x}_1={\rho }^{{\alpha }_1}\mathsf{\text{cos}}{\phi }_1\cdot \cdot \cdot \mathsf{\text{cos}}{\phi }_{n-2}\mathsf{\text{cos}}{\phi }_{n-1}\hfill \\ x_2={\rho }^{{\alpha }_2}\mathsf{\text{cos}}{\phi }_1\cdot \cdot \cdot \mathsf{\text{cos}}{\phi }_{n-2}\mathsf{\text{sin}}{\phi }_{n-1}\hfill \\ \dots \dots \dots \dots \dots \dots \dots \dots \hfill \\ x_{n-1}={\rho }^{{\alpha }_{n-1}}\mathsf{\text{cos}}{\phi }_1\mathsf{\text{sin}}{\phi }_2\hfill \\ x_n={\rho }^{{\alpha }_n}\mathsf{\text{sin}}{\phi }_1.\hfill \end{array}$$

Then dx = ρ^α-1J (φ₁,..., φ_n-1)dρdσ, where $\alpha =\sum _{i=1}^n{\alpha }_i$, dσ is the element of area of S^n-1and 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${\gamma }^{\prime }\left(t\right)=M\left(\frac{\gamma \left(t\right)}{t}\right)$for a fixed matrix M, and assume γ(t) doesn't lie in an affine hyperplane. Then

$$\underset{1}{\overset{2}{\int }}{e}^{i\gamma \left(t\right)\cdot \eta }dt\le C|\eta {|}^{1/d}.$$

Lemma 2.2. ([9]) Suppose that${\lambda }_j^{\prime }s$and${\alpha }_j^{\prime }s$are fixed real numbers, ϕ(t) is a polynomial and$\Gamma \left(t\right)=\left({\lambda }_1{t}^{{\alpha }_1},\dots ,{\lambda }_n{t}^{{\alpha }_n},\varphi \left(t\right)\right)$is a function from ℝ₊to ℝⁿ⁺¹. For suitable f, the maximal function associated to the homogeneous curve Γ is defined by

$$M_{\Gamma }\left(f\right)\left(x\right)=\underset{h}{\mathsf{\text{sup}}}\frac{1}{h}\underset{0}{\overset{h}{\int }}\left|f\left(x-\Gamma \left(t\right)\right)\right|dt,h>0.$$

(2.1)

Then for 1 < p ≤ ∞, there is a constant C > 0, independent of${\lambda }_j^{\prime }s$, the coefficient of ϕ(t) and f, such that

$$\left|\right|M_{\Gamma }\left(f\right)|{|}_{L^p}\le C|\left|f\right|{|}_{L^p}.$$

(2.2)

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

$$\left|{\hat{\sigma }}_k\left(\xi \right)\right|\le C\mathsf{\text{ min}}\left\{\left|A_{2^k}L\xi \right|,{\left(\mathsf{\text{ln}}\left(\left|A_{2^k}L\xi \right|\right)\right)}^{-1-\beta }\right\}$$

(2.3)

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

$${∥{\left(\sum _{k\in \mathbb{Z}}{|{\sigma }_k*g_k|}^2\right)}^{1/2}∥}_{L^{p_0}}\le A{∥{\left(\sum _{k\in \mathbb{Z}}{\left|g_k\right|}^2\right)}^{1/2}∥}_{L^{p_0}}$$

(2.4)

holds for arbitrary functions {g _k }_k∈ℤon ℝⁿ⁺¹. Then for $p\in \left(\frac{2+2\beta }{1+2\beta },2+2\beta \right)$there exists a constant C _p = C(p, n) which is independent of L such that

$${∥\sum _{k\in \mathbb{Z}}{\sigma }_k*f∥}_{L^p}\le C_p{∥f∥}_{L^p}$$

(2.5)

and

$${∥{\left(\sum _{k\in \mathbb{Z}}{|{\sigma }_k*f|}^2\right)}^{1/2}∥}_{L^p}\le C{\left|\right|f\left|\right|}_{L^p}$$

(2.6)

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 $\psi \in C_0^{\infty }\left(ℝ\right)$ such that 0· ≤ ψ ≤ 1, supp(ψ) ⊆ (1/4, 4), and

$$\sum _{j\in \mathbb{Z}}{\left[\psi \left(2^{j}t\right)\right]}^2\equiv 1$$

(2.7)

For each j, we define Φ _j in ℝⁿby

$$\widehat{{\Phi }_j}\left(\zeta \right)=\psi \left(2^j\rho \left(\zeta \right)\right)\hat{f}\left(\zeta \right)$$

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

$$Tf=\sum _{k\in \mathbb{Z}}{\sigma }_k*f,$$

(2.8)

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

$$Tf=\sum _{j\in \mathbb{Z}}{T}_{j}f,$$

where

$$T_{j}f=\sum _{k\in \mathbb{Z}}\left({\Phi }_{j+k}\otimes \delta \right)*{\sigma }_k*\left({\Phi }_{j+k}\otimes \delta \right)*f.$$

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

$${\left|\right|T_{j}f\left|\right|}_{L^{p_0}\left({ℝ}^{n+1}\right)}\le C_{p_0}{\left|\right|f\left|\right|}_{L^{p_0}\left({ℝ}^{n+1}\right)}.$$

(2.9)

On the other hand, by using Plancherel's theorem and (2.3), If j > 0, using the estimate $\left|\widehat{{\sigma }_k}\left(\xi \right)\right|\le C\left|A_{2^k}\zeta \right|$ we have

$$\begin{array}{cc}\hfill {∥T_j\left(f\right)∥}_{L^2\left({ℝ}^{n+1}\right)}& \le \sum _k\underset{2^{-j-k-1}\le \rho \left(\zeta \right)\le 2^{-j-k+1}}{\int }\left|\hat{f}\left(\xi \right){|}^2\right|A_{2^k}\zeta {|}^{2}d\xi \hfill \\ =\sum _k\underset{{}_{2^{-j-k-1}\le \rho \left(\zeta \right)\le 2^{-j-k+1}}}{\int }|\hat{f}\left(\xi \right){|}^2\hfill \\ \left(2^{2k{\alpha }_1}\rho {\left(\zeta \right)}^{2{\alpha }_1}{\left({{\zeta }^{\prime }}_1\right)}^2+\dots +2^{2k{\alpha }_n}\rho {\left(\zeta \right)}^{2{\alpha }_n}{\left({\zeta }_n^{\prime }\right)}^2\right)d\xi \hfill \\ \le 2^{-2j\mathsf{\text{ min}}\left\{{\alpha }_j\right\}}\sum _k\underset{2^{-j-k-1}\le \rho \left(\zeta \right)\le 2^{-j-k+1}}{\int }|\hat{f}\left(\xi \right){|}^2\hfill \\ \left({\left({\zeta }_1^{\prime }\right)}^2+\dots +{\left({\zeta }_n^{\prime }\right)}^2\right)d\xi \hfill \\ \le 2^{-2j}\sum _k\underset{2^{-j-k-1}\le \rho \left(\zeta \right)\le 2^{-j-k+1}}{\int }|\hat{f}\left(\xi \right){|}^{2}d\xi \hfill \\ =C{2}^{-2j}\left|\right|f|{|}_{L^2\left({ℝ}^{n+1}\right)}.\hfill \end{array}$$

(2.10)

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

$${\left|\right|T_{j}f\left|\right|}_{L^2\left({ℝ}^{n+1}\right)}\le C{\left(1+\left|j\right|\right)}^{-\left(1+\beta \right)}{\left|\right|f\left|\right|}_{L^2\left({ℝ}^{n+1}\right)}.$$

(2.11)

In short

$${\left|\right|T_{j}f\left|\right|}_{L^2\left({ℝ}^{n+1}\right)}\le C{\left(1+\left|j\right|\right)}^{-\left(1+\beta \right)}{\left|\right|f\left|\right|}_{L^2\left({ℝ}^{n+1}\right)},forj\in \mathbb{Z}.$$

(2.12)

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

$${\left|\right|T_{j}f\left|\right|}_{L^p\left({ℝ}^{n+1}\right)}\le C{\left(1+\left|j\right|\right)}^{-\left(1+\beta \right)}{\left|\right|f\left|\right|}_{L^p\left({ℝ}^{n+1}\right)}.$$

(2.13)

for

$$p\in \left(\frac{2+2\beta }{1+2\beta },2+2\beta \right)$$

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 $\varphi \left(t\right)={\sum }_{j=0}^m{a}_j{t}^j$, m ∈ ℕ. Let D _k = {y ∈ ℝ ⁿ : 2 ^k < ρ(y) ≤ 2^k+1} and define the family of measures σ _k on ℝⁿ⁺¹by

$$\underset{{ℝ}^{n+1}}{\int }f\left(y,y_{n+1}\right)d{\sigma }_k=\underset{D_k}{\int }f\left(y,\varphi \left(\rho \left(y\right)\right)\right)\frac{\Omega \left(y\right)}{\rho {\left(y\right)}^{\alpha }}dy,$$

(3.1)

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

It is easy to see that

$$\left|\right|{\sigma }_k\left|\right|=\underset{D_k}{\int }\frac{\left|\Omega \left(y^{\prime }\right)\right|}{\rho {\left(y\right)}^{\alpha }}dy=\underset{S_{n-1}}{\int }\underset{2^k}{\overset{2^{k+1}}{\int }}\left|\Omega \left(y^{\prime }\right)\right|J\left(y^{\prime }\right)|\frac{d\rho }{\rho }d\sigma \left(y^{\prime }\right)\le C.$$

(3.2)

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) ∈ ℝ ⁿ × ℝ, y' ∈ S^n-1, and λ ∈ ℤ. Let

$$I_{\lambda }\left(\xi ,{\xi }_{n+1},y^{\prime }\right)=\underset{1}{\overset{2}{\int }}{e}^{i\left[A_{\lambda \rho }\xi \cdot y^{\prime }+{\xi }_{n+1}\varphi \left(\lambda \rho \right)\right]}d\rho .$$

Set Λ = {α _i : α _i is the positive integers which is less than m in {α₁,..., α _n } and $\overline{\Lambda }=\left\{1,2,...,m\right\}\backslash \Lambda$. Then $\frac{d^{{\alpha }_i}\varphi \left(t\right)}{d{t}^{{\alpha }_i}}{|}_{t=0}=0$, where α _i ∈ Λ, and $\overline{\Lambda }$ is not a subset of {α₁,..., α _n }. Therefore, we get

$$A_{\lambda \rho }\xi \cdot y^{\prime }+{\xi }_{n+1}\varphi \left(\lambda \rho \right)={\rho }^{{\alpha }_1}{\lambda }^{{\alpha }_1}{\xi }_1{y}_1^{\prime }+...+{\rho }^{{\alpha }_n}{\lambda }^{{\alpha }_n}{\xi }_n{y}_n^{\prime }+{\xi }_{n+1}\sum _{j\in \overline{\Lambda }}{a}_j{\left(\lambda \rho \right)}^j.$$

Without loss of generality, we may assume Λ consists of r distinct numbers and let $\overline{\Lambda }=\left\{i_1,i_2,...,i_{m-r}\right\}$ If ${\alpha }_j^{\prime }s$ are all distinct, by Lemma 2.1, we get immediately

$$\begin{array}{cc}\hfill \left|I_{\lambda }\left(\xi ,{\xi }_{n+1},y^{\prime }\right)\right|& \le {\left(\left|{\lambda }^{{\alpha }_1}{\xi }_1{y}_1^{\prime }\right|+\cdots +\left|{\lambda }^{{\alpha }_n}{\xi }_n{y}_n^{\prime }\right|+\left(m--r\right)\left|\lambda {\xi }_{n+1}\right|\right)}^{-1/\left(n+m-r\right)}\hfill \\ \le {\left(|{\lambda }^{{\alpha }_1}{\xi }_1{y}_1^{\prime }+\cdots +{\lambda }^{{\alpha }_n}{\xi }_n{y}_n^{\prime }|\right)}^{-1/\left(n+m-r\right)}=|A_{\lambda }\xi \cdot y^{\prime }{|}^{-1/\left(n+m-r\right)}.\hfill \end{array}$$

(3.3)

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

$$\begin{array}{c}{\alpha }_{\mathsf{\text{1}}}={\alpha }_{\mathsf{\text{2}}}=\cdots ={\alpha }_{l_1},\hfill \\ {\alpha }_{l_1+\mathsf{\text{1}}}=\cdots ={\alpha }_{l_1+l_{\mathsf{\text{2}}}},\hfill \\ \dots \hfill \\ {\alpha }_{l_1+\cdots +{l_s}_{-\mathsf{\text{1}}}+1}=\cdots ={\alpha }_n,\hfill \end{array}$$

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 ${\alpha }_1,{\alpha }_{l_1+l_2},\cdots ,{\alpha }_{l_1+\cdots +l_{s-1},}{\alpha }_n$ are distinct. Obviously,

$$\gamma \left(t\right)=\left(t^{{\alpha }_1},t^{{\alpha }_{l_1+l_2}},\dots ,t^{{\alpha }_{l_1+\cdots +l_{s-1}}},t^{{\alpha }_n},t^{i_1},t^{i_2},\dots ,t^{i_{m-r}}\right)$$

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 ) ∈ ℝ ⁿ ,

$$\begin{array}{c}\underset{1}{\overset{2}{\int }}{e}^{2i\left({\eta }_1+\cdots +{\eta }_{l_1}\right)t^{{\alpha }_{l_1}}+\left({\eta }_{l_1+1}+\cdots +{\eta }_{l_1+l_2}\right)t^{{\alpha }_{l_1+l_2}}+\cdots +\left({\eta }_{l_1+\cdots +l_{s-1}+1}+\cdots +{\eta }_n\right)t^{{\alpha }_n}+\lambda {\xi }_{n+1}{\sum }_{j\in \overline{\Lambda }}{t}^j}dt\hfill \\ \le C\left({|{\eta }_1+\cdots +{\eta }_{l_1}|}^2+{|{\eta }_{l_1+1}+\cdots +{\eta }_{l_1+l_2}|}^2+\cdots \right.\hfill \\ {\left.+{|{\eta }_{l_1+\cdots +l_{s-1}+1}+\cdots +{\eta }_n|}^2+\left(m-r\right){\left|\lambda {\xi }_{n+1}\right|}^2\right)}^{-1/2\left(s+m-r\right)}\hfill \\ \le C{\left(|{\eta }_1+\cdots +{\eta }_{l_1}|+|{\eta }_{l_1+1}+\cdots +{\eta }_{l_1+l_2}|+\cdots +|{\eta }_{l_1+\cdots +l_{s-1}+1}+\cdots +{\eta }_n|\right)}^{-1/\left(s+m-r\right)}\hfill \\ \le C{\left|\sum _{j=1}^n{\eta }_j\right|}^{-1/\left(s+m-r\right)}.\hfill \end{array}$$

Let ${\eta }_j={\lambda }^{{\alpha }_j}{\xi }_j{y}_j^{\prime }$, we have

$$\begin{array}{cc}\hfill \left|I_{\lambda }\left(\xi ,{\xi }_{n+1},y\right)\right|& \le {\left(\left|{\lambda }^{{\alpha }_1}{\xi }_1{y}_1^{\prime }\right|+\cdots +\left|{\lambda }^{{\alpha }_n}{\xi }_n{y}_n^{\prime }\right|\right)}^{-1/\left(s+m-r\right)}\hfill \\ \le {\left(|{\lambda }^{{\alpha }_1}{\xi }_1{y}_1^{\prime }+\cdots +{\lambda }^{{\alpha }_n}{\xi }_n{y}_n^{\prime }|\right)}^{-1/\left(s+m-r\right)}=|A_{\lambda }\xi \cdot y^{\prime }{|}^{-1/\left(s+m-r\right)}.\hfill \end{array}$$

(3.3a)

On the other hand, it is easy to see that

$$\left|I_{\lambda }\left(\xi ,{\xi }_{n+1},y^{\prime }\right)\right|\le 1.$$

(3.4)

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

$$\left|I_{\lambda }\left(\xi ,{\xi }_{n+1},y^{\prime }\right)\right|\le \frac{C{\left[\mathsf{\text{ln}}\left(1/|{\eta }^{\prime }\cdot y^{\prime }|\right)\right]}^{1+\beta }}{{\left(\mathsf{\text{ln}}\left|A_{\lambda }\xi \right|\right)}^{1+\beta }},\mathsf{\text{for}}\left|A_{\lambda }\xi \right|\ge 2,$$

where ${\eta }^{\prime }=\frac{A_{\lambda }\xi }{\left|A_{\lambda }\xi \right|}$. Thus, by (1.3), we get

$$\underset{S^{n-1}}{\int }\left|I_{\lambda }\left(\xi ,{\xi }_{n+1},y^{\prime }\right)\Omega \left(y^{\prime }\right)\right|d\sigma \left(y^{\prime }\right)\le C{\left(\mathsf{\text{ln}}\left|A_{\lambda }\xi \right|\right)}^{-\left(1+\beta \right)}.$$

Therefore,

$$\begin{array}{c}\left|\widehat{{\sigma }_k}\left(\xi ,{\xi }_{n+1}\right)\right|\hfill \\ =\left|\underset{D_k}{\int }{e}^{i\left(\xi \cdot y+{\xi }_{n+1}\varphi \left(\rho \left(y\right)\right)\right)}\frac{\Omega \left(y\right)}{\rho {\left(y\right)}^{\alpha }}dy\right|\hfill \\ =\left|\underset{S^{n-1}}{\int }\underset{2^k}{\overset{2^{k+1}}{\int }}{e}^{i\left(\xi \cdot A_{\rho }{y}^{\prime }+{\xi }_{n+1}\varphi \left(\rho \right)\right)}\Omega \left(y^{\prime }\right)J\left(y^{\prime }\right)\frac{d\rho }{\rho }d\sigma \left(y^{\prime }\right)\right|\hfill \\ =\left|{\int }_{S^{n-1}}\underset{1}{\overset{2}{\int }}{e}^{i\left(\xi \cdot A_{2^k\rho }{y}^{\prime }+{\xi }_{n+1}\varphi \left(2^k\rho \right)\right)}\Omega \left(y^{\prime }\right)J\left(y^{\prime }\right)\frac{d\rho }{\rho }d\sigma \left(y^{\prime }\right)\right|\hfill \\ \le C\underset{S^{n-1}}{\int }\left|I_{2^k}\left(\xi ,{\xi }_{n+1},y^{\prime }\right)\right|\left|\Omega \left(y^{\prime }\right)\right|d\sigma \left(y^{\prime }\right)\hfill \\ \le C{\left(\mathsf{\text{ln}}\left|A_{2^k}\xi \right|\right)}^{-\left(1+\beta \right)}.\hfill \end{array}$$

(3.5)

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

$$\begin{array}{c}\left|\widehat{{\sigma }_k}\left(\xi ,{\xi }_{n+1}\right)\right|\hfill \\ =\left|\underset{D_k}{\int }{e}^{i\left(\xi \cdot y+{\xi }_{n+1}\varphi \left(\rho \left(y\right)\right)\right)}\frac{\Omega \left(y\right)}{\rho {\left(y\right)}^{\alpha }}dy\right|\hfill \\ =\left|\underset{2^k}{\overset{2^{k+1}}{\int }}\underset{S^{n-1}}{\int }{e}^{i\left(\xi \cdot A_{\rho }{y}^{\prime }+{\xi }_{n+1}\varphi \left(\rho \right)\right)}\Omega \left(y^{\prime }\right)J\left(y^{\prime }\right)d\sigma \left(y^{\prime }\right)\frac{d\rho }{\rho }\right|\hfill \\ =\left|\underset{2^k}{\overset{2^{k+1}}{\int }}\underset{S^{n-1}}{\int }\left(e^{i\left(\xi \cdot A_{\rho }{y}^{\prime }+{\xi }_{n+1}\varphi \left(\rho \right)\right)}-e^{i{\xi }_{n+1}\varphi \left(\rho \right)}\right)\Omega \left(y^{\prime }\right)J\left(y^{\prime }\right)d\sigma \left(y^{\prime }\right)\frac{d\rho }{\rho }\right|\hfill \\ \le C\underset{2^k}{\overset{2^{k+1}}{\int }}\underset{S^{n-1}}{\int }|e^{i\left(\xi \cdot A_{\rho }{y}^{\prime }+{\xi }_{n+1}\varphi \left(\rho \right)\right)}-e^{i{\xi }_{n+1}\varphi \left(\rho \right)}|\left|\Omega \left(y^{\prime }\right)\right|\left|J\left(y^{\prime }\right)\right|d\sigma \left(y^{\prime }\right)\frac{d\rho }{\rho }\hfill \\ \le C\underset{2^k}{\overset{2^{k+1}}{\int }}\underset{S^{n-1}}{\int }|\xi \cdot A_{\rho }{y}^{\prime }|\left|\Omega \left(y^{\prime }\right)\right|\left|J\left(y^{\prime }\right)\right|d\sigma \left(y^{\prime }\right)\frac{d\rho }{\rho }\hfill \\ \le C\underset{2^k}{\overset{2^{k+1}}{\int }}\underset{S^{n-1}}{\int }|A_{2^k}\xi \cdot y^{\prime }|\left|\Omega \left(y^{\prime }\right)\right|\left|J\left(y^{\prime }\right)\right|d\sigma \left(y\right)\frac{d\rho }{\rho }\hfill \\ \le C\left|A_{2^k}\xi \right|\underset{2^k}{\overset{2^{k+1}}{\int }}\underset{S^{n-1}}{\int }\left|\Omega \left(y^{\prime }\right)\right|\left|J\left(y^{\prime }\right)\right|\left|\frac{A_{2^{k+1}}\xi }{A_{2^{k+1}}\xi }\cdot y^{\prime }\right|d\sigma \left(y^{\prime }\right)d\rho \hfill \\ \le C\left|A_{2^k}\xi \right|.\hfill \end{array}$$

(3.6)

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

$$\begin{array}{c}{\sigma }_k^{*}\left(f\right)\left(x\right)\hfill \\ =\underset{k\in ℝ}{\mathsf{\text{sup}}}\left(\left|{\sigma }_k\right|*\left|f\right|\right)\left(x\right)\hfill \\ =\underset{D_k}{\int }\left|f\left(x-\Phi \left(y\right)\right)\right|\frac{\left|\Omega \left(y\right)\right|}{\rho {\left(y\right)}^{\alpha }}dy\hfill \\ =\underset{S^{n-1}}{\int }\underset{2^k}{\overset{2^{k+1}}{\int }}\left|f\left(x-\Phi \left(A_{\rho }{y}^{\prime }\right)\right)\right|\left|\Omega \left(y^{\prime }\right)\right|\frac{d\rho }{\rho }d\sigma \left(y^{\prime }\right)\hfill \\ \le \frac{1}{2^k}\underset{S^{n-1}}{\int }\left|\Omega \left(y^{\prime }\right)\right|\left(\underset{2^k}{\overset{2^{k+1}}{\int }}\left|f\left(x-\Phi \left(A_{\rho }{y}^{\prime }\right)\right)\right|d\rho \right)d\sigma \left(y^{\prime }\right)\hfill \\ \le C\underset{S^{n-1}}{\int }\left|\Omega \left(y^{\prime }\right)\right|M_{\Phi }\left(f\right)\left(x\right)d\sigma \left(y^{\prime }\right).\hfill \end{array}$$

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.