Boundedness for maximal operators
over hypersurfaces in $\mathbb{R}^{3}$ ^†^†thanks: This work is supported by the National Key R&D Program of China (No.2023YFA1010800); the Natural Science Foundation of China (No.12271435, No.12301113).

Wenjuan Li
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China
Huiju Wang
School of Mathematics and Statistics, Henan University, Kaifeng, 475000, China Corresponding author’s email address: [email protected]

Abstract: In this article, we study maximal functions related to hypersurfaces with vanishing Gaussian curvature in $\mathbb{R}^{3}$ . Firstly, we characterize the $L^{p}\rightarrow L^{q}$ boundedness of local maximal operators along homogeneous hypersurfaces. Moreover, weighted $L^{p}$ -estimates are obtained for the corresponding global operators. Secondly, for a class of hypersurfaces that lack a homogeneous structure and pass through the origin, we attempt to look for other geometric properties instead of height of hypersurfaces to characterize the optimal $L^{p}$ -boundedness of the corresponding global maximal operators.

Keywords: Maximal function; height of hypersurfaces; vanishing Gaussian curvature.

Mathematics Subject Classification: 42B20, 42B25.

1 Introduction

Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ and $\eta\in C_{0}^{\infty}(S)$ be a smooth non-negative function with compact support. Then the associated averaging operator is defined by

A_{t}f(y)=\int_{S}f(y-tx)\eta(x)d\sigma(x),\quad t>0,

(1.1)

where $d\sigma$ denotes the surface measure on $S$ . A fundamental and still largely open problem is to characterize the $L^{p}(\mathbb{R}^{n})$ -boundedness of the global maximal operator associated to the hypersurface where the Gaussian curvature at some points is allowed to vanish. Iosevich [8] gave a confirmed answer for the $2$ -dimensional case, i.e. $S$ is a curve of finite type in $\mathbb{R}^{2}$ . For higher dimension case, one can see the works by Buschenhenke-Dendrinos-Ikromov-Müller [1], Buschenhenke-Ikromov-Müller [2], Greenleaf [3], Ikromov-Müller [4, 5], Ikromov-Kempe-Müller [6, 7], Iosevich-Sawyer [9, 10], Li [11], Zimmermann [16] etc.

However, much less is known about the $L^{p}\rightarrow L^{q}$ -boundedness $(p\leq q)$ for local maximal operator $\sup_{t\in[1,2]}|A_{t}|$ associated with hypersurface $S$ whose Gaussian curvature vanishes at some points, especially when $S\subset\mathbb{R}^{n}$ , $n\geq 3$ . Once we obtain the $L^{p}\rightarrow L^{q}$ -boundedness for local maximal operators $\sup_{t\in[1,2]}|A_{t}|$ , then the weighted estimates will follow for the corresponding global maximal operators by the methodology of sparse domination (see Section 1.2 in [12]). In this paper, we mainly concentrate on two kinds of maximal operators, specifically, in Section 1.1 we study maximal operators associated with homogeneous hypersurfaces in $\mathbb{R}^{3}$ ; in Section 1.2, we investigate other maximal operators along a class of hypersurfaces that lack a homogeneous structure and pass through the origin.

1.1 Maximal operators associated with homogeneous hypersurfaces

We first consider the maximal operators along the hypersurfaces $S:=\{(x_{1},x_{2},\Phi(x_{1},x_{2})+c):(x_{1},x_{2})\in U\}$ , where $c\in\mathbb{R}$ , $U$ is a small neighborhood of the origin in $\mathbb{R}^{2}$ . $\Phi$ is a homogeneous polynomial that satisfies $\Phi(\lambda^{1/m}\cdot)=\lambda\Phi(\cdot)$ for all $\lambda>0$ . We always assume that $m\geq 2$ in this paper. In order to describe the structure of $\Phi$ , we introduce the following proposition, which appears in the Proposition 2.2 in [4].

Proposition 1.1.

Let $P$ be a $(\kappa_{1},\kappa_{2})$ -homogeneous polynomial of degree one, i.e. $P(\lambda^{\kappa_{1}\cdot},\lambda^{\kappa_{2}}\cdot)=\lambda P(\cdot,\cdot)$ for each $\lambda>0$ , and assume that $P$ is not of the form $cx_{{}_{1}}^{\nu_{1}}x_{2}^{\nu_{2}}$ . Then $\kappa_{1},\kappa_{2}$ are uniquely determined by $P$ , and $\kappa_{1},\kappa_{2}\in\mathbb{Q}$ . We assume that $\kappa_{1}\leq\kappa_{2}$ , and write

\kappa_{1}=\frac{q}{m},\kappa_{2}=\frac{p}{m},\quad\langle p,q,m\rangle=1,

so that in particular $q\leq p$ . Then $\langle p,q\rangle=1$ , and there exist non-negative integers $\alpha_{1},\alpha_{2}$ and a $(1,1)$ -homogeneous polynomial $Q$ such that the polynomial $P$ can be written as

P(x_{1},x_{2})=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}Q(x_{1}^{p},x_{2}^{q}).

More precisely, $P$ can be written in the form

P(x_{1},x_{2})=c_{\Phi}x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}\Pi_{h=1}^{N}(x_{2}^{q}-\lambda_{h}x_{1}^{p})^{n_{h}}

with $N\geq 1$ , distinct $\lambda_{h}\in\mathbb{C}\backslash\{0\}$ and multiplicities $n_{h}\in\mathbb{N}\backslash\{0\}$ with $\nu_{1},\nu_{2}\in\mathbb{N}$ (possibly different from $\alpha_{1},\alpha_{2}$ ).

Applying Proposition 1.1 with $\kappa_{1}=\kappa_{2}=1/m$ , then we write $\Phi$ as

\Phi(x_{1},x_{2})=c_{\Phi}x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}\Pi_{h=1}^{N}(x_{2}-\lambda_{h}x_{1})^{n_{h}}.

(1.2)

It is clear that $\nu_{1}+\nu_{2}+\sum_{h=1}^{N}n_{h}=m$ . When one of $\nu_{1}$ , $\nu_{2}$ , $n_{h}$ , $1\leq h\leq N$ equals $m$ , then after some linear change of coordinates, the hypersurface $S$ transforms to $\{(y_{1},y_{2},y_{1}^{m}+c):(y_{1},y_{2})\in U\}$ , and the related maximal operators have been studied in [7, 16, 11, 12]. In this paper, we concentrate on the case when $0\leq\nu_{1}<m$ , $0\leq\nu_{2}<m$ and $0\leq n_{h}<m$ .

We also introduce the definition of the height of $\Phi$ which is original from [6]. The height of $\Phi$ is defined by

h_{\Phi}:=\max\{\frac{m}{2},\textsf{ordd}\hskip 2.84544pt\Phi\},

where $\textsf{ordd}\hskip 2.84544pt\Phi=\sup_{x\in S^{1}}\textsf{ordd}\hskip 2.84544pt\Phi(x)$ , $S^{1}$ is the unit circle in $\mathbb{R}^{2}$ and $\textsf{ordd}\hskip 2.84544pt\Phi(x)$ is the smallest non-negative integer $j$ such that the $j$ -th order total derivative of $\Phi$ at $x$ is non-zero.

Now we are ready to state our main results. Let $Z_{\Phi}$ be the zero set of $\Phi$ , and $Z_{H\Phi}$ be the zero set of the determinant of the Hessian matrix of $\Phi$ , and $\Delta_{0}$ be constructed by the interior of the quadrilateral with vertices $P_{1}=(0,0)$ , $P_{2}=(2/3,2/3)$ , $P_{3}=(2/3,1/3)$ , $P_{4}=(3/5,1/5)$ , and the segment $P_{1}P_{2}$ with $P_{1}$ included and $P_{2}$ excluded. We first consider the local maximal operator $\sup_{t\in[1,2]}|A_{t}|$ when $Z_{H\Phi}\subset Z_{\Phi}$ .

Theorem 1.2.

Assuming that $Z_{H\Phi}\subset Z_{\Phi}$ , then we have
(1) when $c=0$ , there exists a constant $C_{p,q}>0$ such that $\|sup_{t\in[1,2]}|A_{t}|\|_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq C_{p,q}$ for

(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{1}}=\{(\frac{1}{p},\frac{1}{q})\in\Delta_{0}:\frac{h_{\Phi}+1}{p}-\frac{h_{\Phi}+1}{q}-1<0\};

(2) when $c\neq 0$ , there exists a constant $C_{p,q}>0$ such that $\|sup_{t\in[1,2]}|A_{t}|\|_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq C_{p,q}$ for

(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{2}}=\{(\frac{1}{p},\frac{1}{q})\in\Delta_{0}:\frac{h_{\Phi}+1}{p}-\frac{1}{q}-1<0\}.

Theorem 1.2 is sharp up to the endpoints. For example, let $\Phi(x_{1},x_{2})=x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}$ , where $\nu_{1}$ and $\nu_{2}$ are positive integers such that $\nu_{1}>\nu_{2}\geq 2$ . Then it is clear that $Z_{H\Phi}\subset Z_{\Phi}$ and $h_{\Phi}=\nu_{1}$ . By a similar argument as in the proof of Theorem 2.7 in [12], when $c=0$ , it can be obtained that $\sup_{t\in[1,2]}|A_{t}|$ cannot be $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded unless $\frac{\nu_{1}+1}{p}-\frac{\nu_{1}+1}{q}-1\leq 0$ , and when $c\neq 0$ , $\frac{\nu_{1}+1}{p}-\frac{1}{q}-1\leq 0$ is necessary for the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ -boundedness of $\sup_{t\in[1,2]}|A_{t}|$ . Using a method similar to that of the spherical maximal operator, the requirement that $(\frac{1}{p},\frac{1}{q})$ belongs to the closure of $\Delta_{0}$ is necessary. Proof of the necessary conditions is provided in the appendix. The proof of the positive results in Theorem 1.2 is left to Section 4 below.

Figure 1: The range of $\widetilde{\Delta_{1}}$ when the height $h_{\Phi}$ changes.

Figure 2: The range of $\widetilde{\Delta_{2}}$ when the height $h_{\Phi}$ changes.

Notice when $h_{\Phi}=3/2$ , the results in Theorem 1.2 (1) coincide with that of the local spherical maximal operators (see the article [14]), and when $h_{\Phi}>3/2$ , the results in Theorem 1.2 depends heavily on the height of $\Phi$ , see Figure 1 and Figure 2. However, when $Z_{H\Phi}\cap Z_{\Phi}\subsetneqq Z_{H\Phi}$ , the level set $\{(x_{1},x_{2}):\Phi(x_{1},x_{2})=1\}$ will also play an essential role in the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ . In what follows, we assume that the level set $\{(x_{1},x_{2}):\Phi(x_{1},x_{2})=1\}$ determines a curve of finite type $M$ near each straight line that is contained in $Z_{H\Phi}\backslash Z_{\Phi}$ , $2\leq M\leq m$ . When $M=2$ , we define $\Delta_{M}:=\Delta_{0}$ , and when $M\geq 3$ we define $\Delta_{M}$ in the following cases:

(1) $M\geq 6$ , $\Delta_{M}$ is constructed by the interior of the quadrilateral with vertices $P_{1}=(0,0)$ , $P_{2}=(\frac{M+1}{2M},\frac{M+1}{2M})$ , $P_{3}=(\frac{4}{M+2},\frac{2}{M+2})$ , $P_{4}=(\frac{3}{M+2},\frac{1}{M+2})$ , and the segment $P_{1}P_{2}$ with $P_{1}$ included and $P_{2}$ excluded;

(2) $4\leq M\leq 5$ , $\Delta_{M}$ is constructed by the interior of the quadrilateral with vertices $P_{1}=(0,0)$ , $P_{2}=(\frac{M+1}{2M},\frac{M+1}{2M})$ , $P_{3}=(\frac{2M+4}{5M+2},\frac{M+2}{5M+2})$ , $P_{4}=(\frac{3}{M+2},\frac{1}{M+2})$ , and the segment $P_{1}P_{2}$ with $P_{1}$ included and $P_{2}$ excluded;

(3) $M=3$ , $\Delta_{M}$ is constructed by the interior of the quadrilateral with vertices $P_{1}=(0,0)$ , $P_{2}=(\frac{M+1}{2M},\frac{M+1}{2M})$ , $P_{3}=(\frac{2M+4}{5M+2},\frac{M+2}{5M+2})$ , $P_{4}=(\frac{3M+6}{8M+4},\frac{M+2}{8M+4})$ , and the segment $P_{1}P_{2}$ with $P_{1}$ included and $P_{2}$ excluded.

Figure 3: The range of $\Delta_{M}$ when $M$ changes.

Theorem 1.3.

Suppose that $Z_{H\Phi}\cap Z_{\Phi}\subsetneqq Z_{H\Phi}$ , we have the following results.
(1) If $c=0$ , then there exists a constant $C_{p,q}>0$ such that $\|sup_{t\in[1,2]}|A_{t}|\|_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq C_{p,q}$ for

(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{3}}=\{(\frac{1}{p},\frac{1}{q})\in\Delta_{M}:\frac{h_{\Phi}+1}{p}-\frac{h_{\Phi}+1}{q}-1<0\};

(2) If $c\neq 0$ , then there exists a constant $C_{p,q}>0$ such that $\|sup_{t\in[1,2]}|A_{t}|\|_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq C_{p,q}$ for

(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{4}}=\{(\frac{1}{p},\frac{1}{q})\in\Delta_{M}:\frac{h_{\Phi}+1}{p}-\frac{1}{q}-1<0\}.

Theorem 1.3 cannot be improved in some cases. For example, we choose $\Phi(x_{1},x_{2})=x_{1}^{M}+x_{2}^{M}$ , and consider the local maximal operator $\sup_{t\in[1,2]}|A_{t}|$ related to the hypersurfaces $\{(x_{1},x_{2},x_{1}^{M}+x_{2}^{M}+c):|x_{1}|<\epsilon|x_{2}|\}$ , $M\geq 6$ , $\epsilon\ll 1$ . It is clear that $h_{\Phi}=\frac{M}{2}$ , the level set $\{(x_{1},x_{2}):x_{1}^{M}+x_{2}^{M}=1\}$ determines a curve which is finite type $M$ at $x_{1}=0$ . When $c\neq 0$ , Theorem 1.3 (2) implies that $\sup_{t\in[1,2]}|A_{t}|$ is $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded if $(\frac{1}{p},\frac{1}{q})$ satisfies $\frac{1}{q}\leq\frac{1}{p}<\frac{3}{q}$ and $\frac{M/2+1}{p}-\frac{1}{q}-1<0$ . This result is sharp up to the endpoints, and the proof of the sharpness can be found in the appendix. When $c=0$ , it follows from Theorem 1.3 (1) that $\sup_{t\in[1,2]}|A_{t}|$ is $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded if $(\frac{1}{p},\frac{1}{q})\in\Delta_{M}$ . Moreover, it can be proved that the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ cannot be true unless $\frac{1}{p}\leq\frac{2M}{M+1}$ , $\frac{1}{p}-\frac{1}{q}-\frac{2}{M+2}\leq 0$ and $\frac{1}{q}\leq\frac{1}{p}\leq\frac{3}{q}$ . Comparing with the results obtained by Theorem 1.3 (1), only the necessity of the segment connecting $P_{2}$ and $P_{3}$ is unknown. Readers may refer to the appendix for the proof of the necessary conditions.

In a small neighborhood of the straight line mentioned before that is contained in $Z_{H\Phi}\backslash Z_{\Phi}$ , we can reduce the proof of Theorem 1.3 to investigating the local maximal operator related to the hypersurfaces $\{(r(1+s^{M}g(s)),rs,r^{m}+c):r\in[1,2],s\in(0,\epsilon_{0})\},$ $2\leq M\leq m$ , whose $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ -boundedness will be obtained in Theorem 1.4 below. In fact, Theorem 1.4 will refer to more general maximal operators. Section 2 and Section 3 are dedicated to proving Theorem 1.4. Theorem 1.3 will be proved in Section 4.

Theorem 1.4.

Let

\widehat{d\mu}(\xi)=\int_{\mathbb{R}^{2}}e^{-ir(1+s^{M}g(s))\xi_{1}-irs\xi_{2}-ir^{d}\xi_{3}}\psi(s)\rho(r)dsdr,

where $g$ is a smooth function with $g(0)\neq 0$ , $\rho$ and $\psi$ are smooth cutoff functions, supp $\rho\subset[1,2]$ and supp $\psi\subset(0,\epsilon_{0})$ with $\epsilon_{0}$ sufficiently small, $d\geq 2$ and $M\geq 2$ are positive integers. Given the averaging operator

A_{t}f(y)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc\xi_{3}}\widehat{d\mu}(t\xi)\hat{f}(\xi)d\xi

(1.3)

with some constant $c\geq 0$ , then we have the following estimate

\biggl{\|}\sup_{t\in[1,2]}|A_{t}f|\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\max\{1,c^{\frac{1}{q}}\}\|f\|_{L^{p}(\mathbb{R}^{3})},

(1.4)

provided that $(\frac{1}{p},\frac{1}{q})$ belongs to $\Delta_{M}$ .

The following weighted estimates for the global maximal operator $\sup_{t>0}|A_{t}|$ follow from Theorem 1.2, Theorem 1.3 and the methodology of sparse domination. One can see [12] and the appendix in this paper for more details.

Theorem 1.5.

Let $0<p<r<q$ , $\omega\in A_{\frac{r}{p}}\cap RH_{\left(\frac{q}{r}\right)^{\prime}}$ , $\alpha=\max\left(\frac{1}{r-p},\frac{q-1}{q-r}\right)$ . Then there holds the weighted estimate

\biggl{\|}\sup_{t>0}|A_{t}|\biggl{\|}_{L^{r}(\omega)\rightarrow L^{r}(\omega)}\leq C\left([\omega]_{A_{\frac{r}{p}}}[\omega]_{RH_{\left(\frac{q}{r}\right)^{\prime}}}\right)^{\alpha}

in each of the following cases:
(1) $A_{t}$ is defined by (1.1) with $c=0$ , $Z_{H\Phi}\subset Z_{\Phi}$ , and $(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{1}}$ ;
(2) $A_{t}$ is defined by (1.1) with $c\neq 0$ , $Z_{H\Phi}\subset Z_{\Phi}$ , and $(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{2}}$ ;
(3) $A_{t}$ is defined by (1.1) with $c=0$ , $Z_{H\Phi}\cap Z_{\Phi}\subsetneqq Z_{H\Phi}$ , and $(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{3}}$ ;
(4) $A_{t}$ is defined by (1.1) with $c\neq 0$ , $Z_{H\Phi}\cap Z_{\Phi}\subsetneqq Z_{H\Phi}$ , and $(\frac{1}{p},\frac{1}{q})\in\widetilde{\Delta_{4}}$ .

We notice that the unweighted $L^{p}$ -estimates for the global maximal operators associated with mixed homogeneous hypersurfaces in $\mathbb{R}^{3}$ have been studied in [6], where the hypersurface is given as a graph of $\Phi+c$ , where $c\in\mathbb{R}$ and $\Phi$ is a mixed homogeneous function. They proved that the global maximal operator is $L^{p}(\mathbb{R}^{3})$ -bounded provided that $p>h_{\Phi}\geq 2$ . Their results are sharp when $c\neq 0$ , but the sharp results for $h_{\Phi}<2$ still remain open. When $\omega=1$ and $c\neq 0$ in Theorem 1.5, our results coincide with that of [6] for the homogeneous function $\Phi$ . Indeed, Theorem 1.5 also includes some results when $h_{\Phi}<2$ or $c=0$ .

Moreover, the method we used to show Theorem 1.4 and Theorem 1.5 can also be applied to get the sharp results on the boundedness of the maximal operators along a class of hypersurfaces which do not have a homogeneous structure and pass through the origin, see Section 1.2 below.

1.2 Other maximal operators over hypersurfaces

Let $\mathcal{M}$ be the global maximal operators defined by the averaging operator (1.1) along the hypersurfaces $S=\{(x_{1},x_{2},\Psi(x_{1},x_{2})+c):(x_{1},x_{2})\in U\}$ , where $\Psi$ is a smooth function on $U$ with $\Psi(0,0)=0$ and $\nabla\Psi(0,0)=(0,0)$ , $U$ is a small neighborhood of the origin $(0,0)$ , and $c\in\mathbb{R}$ . When $c\not=0$ , and $\Psi$ is an analytic function, according to [7, Corollary 1.4], $\mathcal{M}$ is $L^{p}(\mathbb{R}^{3})$ bounded if and only if $p>h_{\Psi}$ when $h_{\Psi}\geq 2$ . If $h_{\Psi}<2$ , Buschenhenke, Dendrinos, Ikoromov and Müller still work on this topic, see recent articles [1, 2].

When $c=0$ , the article [6] gave a remark that one can get better result than $p>h_{\Psi}$ . If $\Psi$ is an analytic function, it follows from [16, Theorem 1.3.1] that $\mathcal{M}$ is $L^{p}(\mathbb{R}^{3})$ bounded if $p>2$ . However, how to characterize the sharp exponents of $L^{p}$ boundedness for $\mathcal{M}$ from geometric properties for the hypersurfaces passing through the origin? In this section, we make an attempt to answer this question.

Now we consider the maximal operators along the hypersurfaces $S=\{(x_{1},x_{2},(x_{2}-x_{1}^{d}\phi(x_{1}))^{m}):(x_{1},x_{2})\in U\}$ , $d\geq 2$ and $m\geq 2$ are positive integers, $U$ is a small neighborhood of the origin $(0,0)$ , the smooth function $\phi$ satisfies $\phi(0)\neq 0$ , $\phi^{(j)}(0)=0$ for $1\leq j\leq n-1$ and $\phi^{(n)}(0)\not=0$ , $n\geq 2$ . Let $\Psi(x_{1},x_{2})=(x_{2}-x_{1}^{d}\phi(x_{1}))^{m}$ . We have the following result.

Theorem 1.6.

Let $A_{t}$ be defined by equality (1.1). Then for each $p>\max\{3/2,2d/(d+1)\}$ , there exists $C_{p}>0$ such that $\|\sup_{t>0}|A_{t}|\|_{L^{p}(\mathbb{R}^{3})\rightarrow L^{p}(\mathbb{R}^{3})}\leq C_{p}$ . Moreover, $\sup_{t>0}|A_{t}|$ cannot be $L^{p}(\mathbb{R}^{3})$ -bounded if $p<\max\{3/2,2d/(d+1)\}$ .

Theorem 1.6 is sharp up to the endpoints, it will be proved in Section 5 below. From Theorem 1.6, we observe that the bounded exponents for such maximal operators depend heavily on the type of the curve determined by the level set $\{(x_{1},x_{2}):\Psi(x_{1},x_{2})=1\}$ . If we replace the term $x_{1}^{d}\phi(x_{1})$ in $\Psi(x_{1},x_{2})=(x_{2}-x_{1}^{d}\phi(x_{1}))^{m}$ by $\gamma(x_{1})$ , which is smooth and satisfies

\gamma^{(j)}(0)=0\text{ for each }j\in\mathbb{N},

(1.5)

then the level set $\{(x_{1},x_{2}):\Psi(x_{1},x_{2})=1\}$ determines a flat curve near $x_{1}=0$ . We can get the following theorem, whose proof can also be found in Section 5.

Theorem 1.7.

There exists a real-valued smooth function $\gamma$ defined on $\mathbb{R}$ that satisfies (1.5) such that the maximal operator $\sup_{t>0}|A_{t}|$ is not $L^{p}(\mathbb{R}^{3})$ -bounded if $p<2$ for any neighborhood of the origin, where $A_{t}$ is given by

A_{t}f(y):=\int_{U}f\bigl{(}y-t(x_{1},x_{2},(x_{2}-\gamma(x_{1}))^{m})\bigl{)}dx_{1}dx_{2}.

(1.6)

Conventions: Throughout this article, we shall use the notation $A\ll B$ , which means that there is a sufficiently large constant $G$ , which is much larger than $1$ and does not depend on the relevant parameters arising in the context in which the quantities $A$ and $B$ appear, such that $GA\leq B$ . By $A\lesssim B$ we mean that $A\leq CB$ for some constant $C$ independent of the parameters related to $A$ and $B$ . Given $\mathbb{R}^{n}$ , we write $B(0,1)$ instead of the unit ball $B^{n}(0,1)$ in $\mathbb{R}^{n}$ centered at the origin for short, and the same notation is valid for $B(x_{0},r)$ .

2 Proof of Theorem 1.4

It suffices to establish the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ -bounded estimate of $\sup_{t\in[1,2]}|A_{t,j}f|$ when $j\geq 1$ , where

A_{t,j}f(y)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc\xi_{3}}\widehat{d\mu}(t\xi)\beta(2^{-j}|\xi|)\hat{f}(\xi)d\xi,\quad\quad j\geq 1,

(2.1)

where the cutoff function $\beta$ is supported in $[1,2]$ .

We choose a smooth cutoff function $\tilde{\chi}$ which is supported in $[1/2,4]$ , and $\tilde{\chi}=1$ on $[1,2]$ , and decompose

A_{t,j}f(y)=\widetilde{A_{t,j}}f(y)+R_{t,j}f(y),

(2.2)

where

\widetilde{A_{t,j}}f(y)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc\xi_{3}}\widehat{d\mu}(t\xi)\beta(2^{-j}|\xi|)\tilde{\chi}\biggl{(}\frac{|\xi_{1}|+|\xi_{2}|}{|\xi_{3}|}\biggl{)}\hat{f}(\xi)d\xi.

We first consider the remainder term $R_{t,j}f$ . Denote

\phi_{t,\xi}(r,s)=tr(1+s^{M}g(s))\xi_{1}+trs\xi_{2}+tr^{d}\xi_{3}.

Then

\nabla_{r,s}\phi_{t,\xi}(r,s)=(ts\xi_{2}+t(1+s^{M}g(s))\xi_{1}+dtr^{d-1}\xi_{3},rt\xi_{2}+rt(Ms^{M-1}g(s)+s^{M}g^{\prime}(s))\xi_{1}).

It is clear that $|\nabla_{r,s}\phi_{t,\xi}(r,s)|\geq|\xi|$ provided that $|\xi_{3}|\gg|\xi_{1}|+|\xi_{2}|$ or $|\xi_{3}|\ll|\xi_{1}|+|\xi_{2}|$ , and there holds

|\nabla_{\xi}^{\alpha}\widehat{d\mu}(t\xi)|\leq\frac{C_{N,\alpha}}{(1+t|\xi|)^{N+|\alpha|}}

for any multi-index $\alpha$ . Then integrating by parts implies that

|R_{t,j}f(y)|\leq 2^{-Nj}\int_{\mathbb{R}^{3}}\frac{C_{N}}{(1+2^{j}|y-z-(0,0,ct)|)^{N}}|f(z)|dz

for any positive integer $N\geq 1$ . Therefore, Sobolev embedding and Young’s inequality yield that

\biggl{\|}\sup_{t\in[1,2]}|R_{t,j}f|\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\max\{1,c^{\frac{1}{q}}\}2^{-jN}\|f\|_{L^{p}(\mathbb{R}^{3})}

(2.3)

whenever $q\geq p\geq 1$ . So we are reduced to investigating the main contribution term $\widetilde{A_{t,j}}f$ .

A dyadic decomposition of the $s$ -variable and a standard scaling argument imply that

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j}}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq\sum_{l\geq\log{\frac{1}{\epsilon_{0}}}}2^{-l}2^{(M-1)l(\frac{1}{q}-\frac{1}{p})}\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j,l}}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})},

(2.4)

where

\widetilde{A_{t,j,l}}f(y)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc\xi_{3}}\widehat{d\mu_{M}}(t\xi)\beta\biggl{(}\frac{|\xi_{1}|+2^{-(M-1)l}|\xi_{2}|+|\xi_{3}|}{2^{j}}\biggl{)}\tilde{\chi}\biggl{(}\frac{|\xi_{1}|+2^{-(M-1)l}|\xi_{2}|}{|\xi_{3}|}\biggl{)}\hat{f}(\xi)d\xi,

and

\widehat{d\mu_{M}}(t\xi)=\int_{\mathbb{R}}e^{-itr^{d}\xi_{3}-itr\xi_{1}}\int_{\mathbb{R}}e^{-itr2^{-Ml}[s^{M}g(2^{-l}s)\xi_{1}+s\xi_{2}]}\tilde{\psi}(s)ds\rho(r)dr,

the cutoff function $\tilde{\psi}(s):=\tilde{\chi}(s)\psi(2^{-l}s)$ is supported in $[1,2]$ .

When $j\leq Ml$ and $|\xi_{2}|>2^{Ml}$ , integration by parts with respect to the variable $s$ in the inner integral of $\widehat{d\mu_{M}}(t\xi)$ implies that

\bigl{|}\nabla_{\xi}^{\alpha}\widehat{d\mu_{M}}(t\xi)\bigl{|}\lesssim\frac{C_{N,\alpha}}{(1+2^{-Ml}|\xi_{2}|)^{N+|\alpha|}},

for each non-negative multi-index $\alpha$ and non-negative integer $N$ , then the method of handling the remainder term $R_{t,j}$ can be applied here to complete the proof. Next we consider the case when $j\leq Ml$ and $|\xi_{2}|\leq 2^{Ml}$ . Notice that in this case, the inner integral in $\widehat{d\mu_{M}}(t\xi)$ can be put into the amplitude. If $|\xi_{3}|\gg|\xi_{1}|$ , for any $t$ and $r$ that belong to the interval $[1,2]$ , there holds

|\partial_{r}(tr^{d}\xi_{3}+tr\xi_{1})|=t|\xi_{3}|\biggl{|}dr^{d-1}+\frac{\xi_{1}}{\xi_{3}}\biggl{|}\geq\frac{d}{2}|\xi_{3}|.

By performing integration by parts on the variable $r$ , we obtain

|\nabla_{\xi}^{\alpha}\widehat{d\mu_{M}}(t\xi)|\lesssim\frac{C_{N,\alpha}}{(1+|\xi_{3}|)^{N+|\alpha|}}

(2.5)

for any positive integer $N$ and multi-index $\alpha$ . By employing the method of handling the remainder term $R_{t,j}$ , we can complete the proof when $|\xi_{3}|\gg|\xi_{1}|$ . When $|\xi_{3}|\sim|\xi_{1}|$ , there exists a solution $r_{c}=(-\frac{\xi_{1}}{d\xi_{3}})^{1/(d-1)}$ for the equation $\partial_{r}(r^{d}+r\frac{\xi_{1}}{\xi_{3}})=0$ provided that $-\frac{\xi_{1}}{\xi_{3}}\in[d,d2^{d-1}]$ . Then by the method of stationary phase, we only need to consider the main contribution term given by

\widetilde{A_{t,j,l,1}}f(y)=2^{-\frac{j}{2}}\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc\xi_{3}+itc_{d}\xi_{3}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)}}a_{j,l,1}(t,\xi)\hat{f}(\xi)\beta_{0}(2^{-Ml}|\xi_{2}|)d\xi,

where $\beta_{0}$ is a smooth cutoff function supported in $[-1,1]$ ,

a_{j,l,1}(t,\xi)=\frac{2^{j/2}}{(1+t|\xi_{3}|)^{1/2}}\beta\biggl{(}\frac{|\xi_{1}|+2^{-(M-1)l}|\xi_{2}|+|\xi_{3}|}{2^{j}}\biggl{)}\tilde{\chi}\biggl{(}\frac{|\xi_{1}|+2^{-(M-1)l}|\xi_{2}|}{|\xi_{3}|}\biggl{)}\tilde{\chi}(-\frac{\xi_{1}}{(d-1)\xi_{3}}),

which is a symbol of order zero in $\xi$ . We will deal with this main contribution term later.

Next, we will consider the case when $j>Ml$ . According to the definition of $\widetilde{A_{t,j,l}}$ , we have $|\xi_{1}|+2^{-(M-1)l}|\xi_{2}|\sim|\xi_{3}|$ . If $|\xi_{3}|\gg|\xi_{1}|$ , for any $t$ , $r$ and $s$ that belong to the interval $[1,2]$ , we have

\biggl{|}\partial_{r}[tr^{d}\xi_{3}+tr\xi_{1}+t2^{-Ml}r(s^{M}g(2^{-l}x)\xi_{1}+s\xi_{2})]\biggl{|}\geq\frac{d}{2}|\xi_{3}|.

Therefore, we still maintain the validity of inequality (2.5). Furthermore, if $|\xi_{3}|\sim|\xi_{1}|\gg|\xi_{2}|$ or $|\xi_{3}|\sim|\xi_{1}|\ll|\xi_{2}|$ , there holds the uniform estimate

\biggl{|}\partial_{s}[2^{-Ml}tr(s^{M}g(2^{-l}s)\xi_{1}+s\xi_{2})]\biggl{|}\gtrsim_{M}2^{-Ml}|\xi_{3}|

for every $t$ , $r$ and $s$ that belong to the interval $[1,2]$ . By utilizing integration by parts in the inner integral of $\widehat{d\mu_{M}}(t\xi)$ , we get

|\nabla_{\xi}^{\alpha}\widehat{d\mu_{M}}(t\xi)|\lesssim\frac{C_{N,\alpha}}{(1+2^{-Ml}|\xi_{3}|)^{N+|\alpha|}}

(2.6)

for any positive integer $N$ and multi-index $\alpha$ . Since $j>Ml$ , we can also finish the proof when $|\xi_{3}|\sim|\xi_{1}|\gg|\xi_{2}|$ or $|\xi_{3}|\sim|\xi_{1}|\ll|\xi_{2}|$ as in the similar proof for the remainder term $R_{t,j}$ .

Now we turn to consider the subcase when $|\xi_{3}|\sim|\xi_{1}|\sim|\xi_{2}|$ . Firstly, we treat the inner integral of $\widehat{d\mu_{M}}(t\xi)$ . Since $l$ is sufficiently large and $g(0)\neq 0$ , then the equation $\partial_{s}[s^{M}g(2^{-l}s)+s\frac{\xi_{2}}{\xi_{1}}]=0$ has a unique solution according to the implicit function theorem. Applying the method of stationary phase to the $s$ -integration, the inner integral in $\widehat{d\mu_{M}}(t\xi)$ can be replaced by its leading term, whose phase function can be written as $-i2^{-Ml}tr\xi_{1}\widetilde{\Psi}_{2}(\frac{\xi_{2}}{\xi_{1}})$ . Here,

\widetilde{\Psi}_{2}(\frac{\xi_{2}}{\xi_{1}})=c_{M}\biggl{(}-\frac{\xi_{2}}{\xi_{1}}\biggl{)}^{M/(M-1)}+2^{-l}\tilde{R}\biggl{(}\frac{\xi_{2}}{\xi_{1}}\biggl{)},

which is a small perturbation of $c_{M}(-\xi_{2}/\xi_{1})^{M/(M-1)}$ . Secondly, we apply the method of stationary phase to the $r$ -integration, the phase function of which is now given by

-itr^{d}\xi_{3}-itr\xi_{1}(1+2^{-Ml}\widetilde{\Psi}_{2}(\frac{\xi_{2}}{\xi_{1}})).

The equation

\partial_{r}\biggl{[}r^{d}+r\frac{\xi_{1}}{\xi_{3}}\biggl{(}1+2^{-Ml}\widetilde{\Psi}_{2}(\frac{\xi_{2}}{\xi_{1}})\biggl{)}\biggl{]}=0

has a unique solution since $l$ is sufficiently large, and by Taylor’s expansion, the phase function can be written as

-it\xi_{3}\biggl{[}\Psi_{1}(\frac{\xi_{1}}{\xi_{3}})+2^{-Ml}\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})\biggl{]},

where $\Psi_{1}(\frac{\xi_{1}}{\xi_{3}})=c_{d,M,l}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)},$ and

\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})=c_{d,M,l}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)}(-\frac{\xi_{2}}{\xi_{1}})^{M/(M-1)}+2^{-l}R(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}}).

Indeed, $\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})$ is a small perturbation of $c_{d,M,l}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)}(-\frac{\xi_{2}}{\xi_{1}})^{M/(M-1)}$ .

Therefore, when $j>Ml$ , we reduce our problem to estimating the main contribution term

\displaystyle\widetilde{A_{t,j,l,2}}f(y)=2^{-j+\frac{Ml}{2}}\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc\xi_{3}+it\xi_{3}\Psi_{1}(\frac{\xi_{1}}{\xi_{3}})+it2^{-Ml}\xi_{3}\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})}a_{j,l,2}(t,\xi)\hat{f}(\xi)d\xi,

with

	$\displaystyle a_{j,l,2}(t,\xi)$	$\displaystyle=\frac{2^{j-\frac{Ml}{2}}}{(1+t\|\xi_{3}\|)^{1/2}(1+t2^{-Ml}\|\xi_{2}\|)^{1/2}}\beta\biggl{(}\frac{\|\xi_{1}\|+2^{-(M-1)l}\|\xi_{2}\|+\|\xi_{3}\|}{2^{j}}\biggl{)}\tilde{\chi}\biggl{(}\frac{\|\xi_{1}\|+2^{-(M-1)l}\|\xi_{2}\|}{\|\xi_{3}\|}\biggl{)}$
		$\displaystyle\times\widetilde{\chi_{d,l,M}}(-\frac{\xi_{1}}{\xi_{3}})\widetilde{\chi_{l,M}}(-\frac{\xi_{2}}{\xi_{1}}).$		(2.7)

Here, the support of the cutoff functions $\widetilde{\chi_{d,l,M}}$ and $\widetilde{\chi_{l,M}}$ are uniformly contained in $[1,2]$ for each $d,l,M$ .

Based on the above stationary phase method, by Sobolev embedding, there holds

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j,l,1}}f|\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\max\{1,c^{\frac{1}{q}}\}2^{\frac{j}{q}-\frac{j}{2}}\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{q}(\mathbb{R}^{3}\times[\frac{1}{2},4])},

(2.8)

where the bump function $\tilde{\rho}$ is supported in $[1/2,4]$ and the Fourier integral operator

F_{j,l,1}(y,t)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc_{d}\xi_{3}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)}}a_{j,l,1}(t,\xi)\hat{f}(\xi)\beta_{0}(2^{-Ml}|\xi_{2}|)d\xi.

For the same reason, we have

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j,l,2}}f|\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\max\{1,c^{\frac{1}{q}}\}2^{\frac{j}{q}-j+\frac{Ml}{2}}\biggl{\|}\tilde{\rho}(t)F_{j,l,2}f(y,t)\biggl{\|}_{L^{q}(\mathbb{R}^{3}\times[\frac{1}{2},4])}

(2.9)

with the Fourier integral operator

\displaystyle F_{j,l,2}f(y,t)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+it\xi_{3}\Psi_{1}(\frac{\xi_{1}}{\xi_{3}})+it2^{-Ml}\xi_{3}\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})}a_{j,l,2}(t,\xi)\hat{f}(\xi)d\xi.

(2.10)

Theorem 1.4 can be proved by the following $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3}\times[1/2,4])$ estimates for $F_{j,l,n}$ , $n=1,2$ .

Theorem 2.1.

There holds the following estimates for $F_{j,l,n}$ , $n=1,2$ .
(E1) For each $j\geq 1$ , $n=1,2$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,n}f(y,t)\biggl{\|}_{L^{2}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim\|f\|_{L^{2}(\mathbb{R}^{3})}.

(2.11)

(E2) For each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}f(y,t)\biggl{\|}_{L^{6}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{j}{2}+\frac{Ml}{3}}\|f\|_{L^{2}(\mathbb{R}^{3})},

(2.12)

and for each $j\geq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}f(y,t)\biggl{\|}_{L^{6}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{5j}{6}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(2.13)

(E3) For each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{4}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{3j}{8}+\frac{Ml}{4}}\|f\|_{L^{2}(\mathbb{R}^{3})},

(2.14)

and for each $j>Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\|}_{L^{4}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{j}{2}+\frac{Ml}{8}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(2.15)

(E4) For each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{\infty}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{j}{2}}\|f\|_{L^{\infty}(\mathbb{R}^{3})},

(2.16)

and for each $j>Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\|}_{L^{\infty}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{j-\frac{Ml}{2}}\|f\|_{L^{\infty}(\mathbb{R}^{3})}.

(2.17)

(E5) For each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{1}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{j}{2}}\|f\|_{L^{1}(\mathbb{R}^{3})},

(2.18)

and for each $j>Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\|}_{L^{1}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{j-\frac{Ml}{2}}\|f\|_{L^{1}(\mathbb{R}^{3})}.

(2.19)

(E6) For each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{\infty}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{3j}{2}+Ml}\|f\|_{L^{1}(\mathbb{R}^{3})},

(2.20)

and for each $j>Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\|}_{L^{\infty}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{2j+\frac{Ml}{2}}\|f\|_{L^{1}(\mathbb{R}^{3})}.

(2.21)

Now we apply Theorem 2.1 to prove Theorem 1.4, and leave the proof of Theorem 2.1 to the next section. According to inequalities (2.4), (2.8) and (2.9), there holds

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{q}(\mathbb{R}^{3})}$	$\displaystyle\leq\max\{1,c^{\frac{1}{q}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{(M-1)l(\frac{1}{q}-\frac{1}{p})-l}2^{\frac{j}{q}-\frac{j}{2}}\biggl{\\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\\|}_{L^{q}(\mathbb{R}^{3}\times[\frac{1}{2},4])}$
		$\displaystyle\quad\quad+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{(M-1)l(\frac{1}{q}-\frac{1}{p})-l}2^{\frac{j}{q}-j+\frac{Ml}{2}}\biggl{\\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\\|}_{L^{q}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\biggl{)}.$		(2.22)

Firstly, we show the following diagonal results for $M\geq 3$ . By the estimate (E1) in Theorem 2.1, there holds

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j}}f|\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\max\{1,c^{\frac{1}{2}}\}2^{-\frac{j}{M}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(2.23)

By the estimates (E4) in Theorem 2.1, we have

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j}}f|\biggl{\|}_{L^{\infty}(\mathbb{R}^{3})}\lesssim\|f\|_{L^{\infty}(\mathbb{R}^{3})}.

Interpolation between the estimates in (E1) and (E5) implies

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{2M}{M+1}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{M+1}{2M}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{\frac{(M+1)j}{2M}-\frac{j}{2}+\frac{j}{2M}}+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{\frac{(M+1)j}{2M}-j+\frac{Ml}{2}}2^{\frac{j}{M}-\frac{l}{2}}\biggl{)}\\|f\\|_{L^{\frac{2M}{M+1}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{M+1}{2M}}\}\\|f\\|_{L^{\frac{2M}{M+1}}(\mathbb{R}^{3})}.$

Then we give the following off-diagonal results for $(\frac{1}{p},\frac{1}{q})$ satisfying $\frac{1}{p}=\frac{2}{q}$ . When $M\geq 6$ , we interpolate the estimates in (E4) with the estimates in (E3) to get

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{M+2}{2}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{2}{M+2}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{2}{M+2}j-\frac{j}{2}}2^{\frac{8}{M+2}(\frac{3j}{8}+\frac{Ml}{4})}2^{\frac{M-6}{2(M+2)}j}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{2}{M+2}j-j+\frac{Ml}{2}}2^{\frac{4}{M+2}j}2^{\frac{Ml}{M+2}}2^{\frac{(M-6)j}{M+2}-\frac{(M-6)Ml}{2(M+2)}}\biggl{)}\\|f\\|_{L^{\frac{M+2}{4}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{2}{M+2}}\}\\|f\\|_{L^{\frac{M+2}{4}}(\mathbb{R}^{3})}.$

When $3\leq M\leq 5$ , the following estimates can be easily obtained by interpolation, for each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{3}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{j}{2}+\frac{Ml}{3}}\|f\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})},

(2.24)

and for each $j>Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\|}_{L^{3}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{2j}{3}+\frac{Ml}{6}}\|f\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})},

(2.25)

then combining the estimate in (E3) and inequality (2), there holds

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{5M+2}{M+2}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{M+2}{5M+2}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{(M+2)(M-1)}{5M+2}l}2^{\frac{M+2}{5M+2}j-\frac{j}{2}}2^{\frac{3M+6}{5M+2}(\frac{j}{2}+\frac{Ml}{3})}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l\leq\frac{j}{M}}2^{-l}2^{-\frac{(M+2)(M-1)}{5M+2}l}2^{\frac{M+2}{5M+2}j-j+\frac{Ml}{2}}2^{\frac{2M+4}{5M+2}j}2^{\frac{(\frac{M}{2}+1)Ml}{5M+2}}\biggl{)}\\|f\\|_{L^{\frac{5M+2}{2M+4}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{M+2}{5M+2}}\}\\|f\\|_{L^{\frac{5M+2}{2M+4}}(\mathbb{R}^{3})}.$

Furthermore, we show the following off-diagonal estimates for $(\frac{1}{p},\frac{1}{q})$ satisfying $\frac{1}{p}=\frac{3}{q}$ . When $M\geq 4$ , by interpolating the estimates in (E2) with the estimates in (E4), we get

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{M+2}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{1}{M+2}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{j}{M+2}-\frac{j}{2}}2^{\frac{6}{M+2}(\frac{j}{2}+\frac{Ml}{3})}2^{\frac{M-4}{2(M+2)}j}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{j}{M+2}-j+\frac{Ml}{2}}2^{\frac{5}{M+2}j}2^{\frac{(M-4)j}{M+2}-\frac{(M-4)Ml}{2(M+2)}}\biggl{)}\\|f\\|_{L^{\frac{M+2}{3}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{1}{M+2}}\}j\\|f\\|_{L^{\frac{M+2}{3}}(\mathbb{R}^{3})}.$

In order to obtain the results for the case $M=3$ , we give the following easy estimates by interpolation, for each $1\leq j\leq Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\|}_{L^{5}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{3j}{5}+\frac{2Ml}{5}}\|f\|_{L^{\frac{5}{3}}(\mathbb{R}^{3})},

(2.26)

and for each $j>Ml$ ,

\biggl{\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\|}_{L^{5}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\lesssim 2^{\frac{4j}{5}+\frac{Ml}{5}}\|f\|_{L^{\frac{5}{3}}(\mathbb{R}^{3})},

(2.27)

then combining the estimates from (E2) and inequality (2), we obtain

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{8m+4}{M+2}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{M+2}{8M+4}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{(2M+4)(M-1)}{8M+4}l}2^{\frac{M+2}{8M+4}j-\frac{j}{2}}2^{\frac{5M+10}{8M+4}(\frac{3j}{5}+\frac{2Ml}{5})}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{-\frac{(2M+4)(M-1)}{8M+4}l}2^{\frac{M+2}{8M+4}j-j+\frac{Ml}{2}}2^{\frac{7M+2}{8M+4}j}2^{\frac{Ml(4-M)}{4M+2}}\biggl{)}\\|f\\|_{L^{\frac{8M+4}{3M+6}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{M+2}{8M+4}}\}j\\|f\\|_{L^{\frac{8M+4}{3M+6}}(\mathbb{R}^{3})}.$

Finally, by interpolation, it is easy to see that for each $(\frac{1}{p},\frac{1}{q})\in\Delta_{M}$ , $M\geq 3$ , there exists $\epsilon(p,q)>0$ such that

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,j}}f|\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\max\{1,c^{\frac{1}{q}}\}2^{-\epsilon(p,q)j}\|f\|_{L^{p}(\mathbb{R}^{3})}.

This completes the proof when $M\geq 3$ . For the case $M=2$ , since the curve $(s,s^{2}g(s)+1)$ is non-degenerate near $s=0$ , after conducting stationary phase analysis, we are reduced to considering Fourier integral operators related to hypersurfaces with two non-vanishing principal curvatures. Hence, we can get the estimates using a similar argument as in [14]. Then we finish the proof of Theorem 1.4.

3 Proof of Theorem 2.1

It is clear that the $L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3}\times[1/2,4])$ estimates in (E1) follows directly from Plancherel’s theorem. So we first consider the $L^{2}(\mathbb{R}^{3})\rightarrow L^{6}(\mathbb{R}^{3}\times[1/2,4])$ estimates in (E2). Notice that for each $t\in[\frac{1}{2},4]$ , the hypersurface $\{(\eta_{1},\eta_{3},c_{d}\eta_{3}(-\frac{\eta_{1}}{\eta_{3}})^{d/(d-1)}):2^{j}(\eta_{1},\eta_{3})\in\textmd{supp}\hskip 2.84544pta_{j,l,1}(\cdot,\xi_{2},t)\}$ and the hypersurface

\{(\eta,\eta_{3}\Psi_{1}(\frac{\eta_{1}}{\eta_{3}})+2^{-Ml}\eta_{3}\Psi_{2}(\frac{\eta_{1}}{\eta_{3}},\frac{\eta_{2}}{\eta_{1}}):2^{j}\eta\in\textmd{supp}\hskip 2.84544pta_{j,l,2}(\cdot,t)\}

for $l$ large enough have at leas one non-vanishing principal curvature. Then according to [[13], Proposition 3.4], there holds the inequalities (2.12), (2.13) for $F_{j,l,1}$ and $F_{j,l,2}$ , respectively.

Now we explain how to establish estimates for the operator $F_{j,l,1}$ in (E3)-(E5). Interpolating between $L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3}\times[1/2,4])$ and $L^{2}(\mathbb{R}^{3})\rightarrow L^{6}(\mathbb{R}^{3}\times[1/2,4])$ estimates for $F_{j,l,1}$ , then we arrive at inequality (2.14). It is noted that

F_{j,l,1}f(y,t)=K_{j,l,1}(\cdot,t)*f(y),

where

K_{j,l,1}(y,t)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+itc_{d}\xi_{3}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)}}a_{j,l,1}(t,\xi)\beta_{0}(2^{-Ml}|\xi_{2}|)d\xi.

We rewrite $K_{j,l,1}(y,t)$ as

K_{j,l,1}(y,t)=\int_{\mathbb{R}}e^{iy_{2}\xi_{2}}\beta_{0}(2^{-Ml}|\xi_{2}|)\int_{\mathbb{R}^{2}}e^{iy_{1}\xi_{1}+iy_{3}\xi_{3}+itc_{d}\xi_{3}(-\frac{\xi_{1}}{\xi_{3}})^{d/(d-1)}}a_{j,l,1}(t,\xi)d\xi_{1}d\xi_{3}d\xi_{2}.

By employing the angular decomposition of the variable $\xi_{1}/\xi_{3}$ as utilized in [Lemma 2.7, [11]] and [Lemma 2.6, [12]], see also the proof in Lemma 3.2 below, we can obtain that for each $t\in[1,2]$ and positive integer $N$ , there holds

\displaystyle|K_{j,l,1}(y,t)|\lesssim C_{N}2^{\frac{3j}{2}+Ml}\sum_{\kappa\in 2^{-\frac{j}{2}}\mathbb{Z}\cap[1,2]}(1+2^{\frac{j}{2}}|y_{1}+c_{d}t\kappa^{1/(d-1)}|+2^{Ml}|y_{2}|+2^{j}|y_{3}+\kappa y_{1}+c_{d}t\kappa^{d/(d-1)}|)^{-N}.

It follows that

\|K_{j,l,1}(y,t)\|_{L^{1}(\mathbb{R}^{3})}\lesssim 2^{\frac{j}{2}}

and

\|K_{j,l,1}(y,t)\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim 2^{\frac{3j}{2}+Ml}

for each $t\in[1,2]$ . Then inequalities (2.16), (2.18), (2.20) can be proved by Young’s inequality, we omit their proofs here.

Next, we proceed to establish the estimates for the operator $F_{j,l,2}$ in (E4)-(E5). We rewrite the operator $F_{j,l,2}$ as

F_{j,l,2}f(y,t)=K_{j,l,2}(\cdot,t)*f(y),

where

K_{j,l,2}(y,t)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+it\xi_{3}\Psi_{1}(\frac{\xi_{1}}{\xi_{3}})+it2^{-Ml}\xi_{3}\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})}a_{j,l,2}(t,\xi)d\xi.

Then ineqalities (2.17) and (2.19) follow from Young’s inequality and part (2) of Theorem 3.1 below. Additionally, the estimate (2.21) follows directly from part (1) of Theorem 3.1.

Finally, we are left with building the desired $L^{2}(\mathbb{R}^{3})\rightarrow L^{4}(\mathbb{R}^{3}\times[1/2,4])$ estimate for the operator $F_{j,l,2}$ in (E3). We may replace $\hat{f}(\xi)$ in $F_{j,l,2}f$ by $f(\xi)$ according to Plancherel’s theorem, then by the dual method as in the proof of [Proposition 3.4, [13]], inequality (2.15) will follow if we can obtain for arbitrary $G\in L^{4/3}(\mathbb{R}^{4})$ ,

\biggl{\|}F_{j,l,2}F^{*}_{j,l,2}G(z,t^{\prime})\biggl{\|}_{L_{z,t^{\prime}}^{4}(\mathbb{R}^{4})}=\biggl{\|}\int_{\mathbb{R}}\widetilde{K_{j,l,2}}(\cdot,t,t^{\prime})*_{y}G(\cdot,t)(z)dt\biggl{\|}_{L_{z,t^{\prime}}^{4}(\mathbb{R}^{4})}\lesssim 2^{j+\frac{Ml}{4}}\|G\|_{L^{4/3}(\mathbb{R}^{4})},

(3.1)

where $*_{y}$ denotes convolution with respect to the variable $y$ , and

\widetilde{K_{j,l,2}}(y,t,t^{\prime})=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+i(t^{\prime}-t)\xi_{3}\Psi_{1}(\frac{\xi_{1}}{\xi_{3}})+i(t^{\prime}-t)2^{-Ml}\xi_{3}\Psi_{2}(\frac{\xi_{1}}{\xi_{3}},\frac{\xi_{2}}{\xi_{1}})}\overline{a_{j,l,2}}(t,\xi)a_{j,l,2}(t^{\prime},\xi)d\xi.

For fixed $t$ , $t^{\prime}$ , by Plancherel’s theorem, there holds

\biggl{\|}\widetilde{K_{j,l,2}}(\cdot,t,t^{\prime})*_{y}G(\cdot,t)(z)\biggl{\|}_{L_{z}^{2}(\mathbb{R}^{3})}\lesssim\|G(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}.

(3.2)

Theorem 3.1 below implies that for each $t\neq t^{\prime}$ , the following estimate holds true

\biggl{\|}\widetilde{K_{j,l,2}}(\cdot,t,t^{\prime})*_{y}G(\cdot,t)(z)\biggl{\|}_{L_{z}^{\infty}(\mathbb{R}^{3})}\lesssim 2^{2j+\frac{Ml}{2}}|t-t^{\prime}|^{-1}\|G(\cdot,t)\|_{L^{1}(\mathbb{R}^{3})}.

(3.3)

Interpolation between (3.2) and (3.3) yields

\biggl{\|}\widetilde{K_{j,l,2}}(\cdot,t,t^{\prime})*_{y}G(\cdot,t)(z)\biggl{\|}_{L_{z}^{4}(\mathbb{R}^{3})}\lesssim 2^{j+\frac{Ml}{4}}|t-t^{\prime}|^{-\frac{1}{2}}\|G(\cdot,t)\|_{L^{\frac{4}{3}}(\mathbb{R}^{3})}.

(3.4)

By combining (3.4) and the Hardy-Littlewood-Sobolev inequality, we obtain

	$\displaystyle\biggl{\\|}\int_{\mathbb{R}}\widetilde{K_{j,l,2}}(\cdot,t,t^{\prime})*_{y}G(\cdot,t)(z)dt\biggl{\\|}_{L_{z,t^{\prime}}^{4}(\mathbb{R}^{4})}$	$\displaystyle\leq\biggl{\\|}\int_{\mathbb{R}}\biggl{\\|}\widetilde{K_{j,l,2}}(\cdot,t,t^{\prime})*_{y}G(\cdot,t)(z)\biggl{\\|}_{L_{z}^{4}(\mathbb{R}^{3})}dt\biggl{\\|}_{L_{t^{\prime}}^{4}(\mathbb{R})}$
		$\displaystyle\lesssim 2^{j+\frac{Ml}{4}}\biggl{\\|}\int_{\mathbb{R}}\|t-t^{\prime}\|^{-\frac{1}{2}}\\|G(\cdot,t)\\|_{L^{\frac{4}{3}}(\mathbb{R}^{3})}\biggl{\\|}_{L_{t^{\prime}}^{4}(\mathbb{R})}$
		$\displaystyle\lesssim 2^{j+\frac{Ml}{4}}\\|G(y,t)\\|_{L_{y,t}^{4/3}(\mathbb{R}^{4})}.$

Then we arrive at inequality (3.1).

Therefore, in order to complete the proof of Theorem 2.1, we are left to investigate the estimate for the kernel $K_{j,l,2}$ related to the Fourier integral operator $F_{j,l,2}$ .

Theorem 3.1.

We have the following estimates for $K_{j,l,2}$ , which also hold true for $\widetilde{K_{j,l,2}}$ .
(1) For fixed $y\in\mathbb{R}^{3}$ and $t\in(0,+\infty)$ , there holds the pointwise estimate

|K_{j,l,2}(y,t)|\lesssim 2^{2j+\frac{Ml}{2}}t^{-1},

(3.5)

where the implied constant is independent of $y$ and $t$ .
(2) For each $t\in[1,2]$ , we have the uniform estimate

\|K_{j,l,2}(\cdot,t)\|_{L^{1}(\mathbb{R}^{3})}\lesssim 2^{j-\frac{Ml}{2}}.

(3.6)

Theorem 3.1 follows from Lemma 3.2 below.

Lemma 3.2.

There holds the following pointwise estimates for $K_{j,l,2}$ .
(1) If $t\in(0,2^{-j}]$ , then for any positive integer $N$ , there holds

|K_{j,l,2}(y,t)|\lesssim C_{N}2^{2j}t^{-1}(1+2^{j}|y|)^{-N}.

(3.7)

(2) If $t\in(2^{-j},2^{-j+Ml}]$ , we have

		$\displaystyle\|K_{j,l,2}(y,t)\|\lesssim C_{N}2^{2j+\frac{Ml}{2}}t^{-1}\times$
		$\displaystyle\sum_{\kappa\in\mathbb{I}_{1}}(1+2^{\frac{j}{2}}t^{-\frac{1}{2}}\|y_{1}+c_{d,M,l}t\kappa^{1/(d-1)}\|+2^{j}\|y_{2}\|+2^{j}\|y_{3}+\kappa y_{1}+c_{d,M,l}t\kappa^{d/(d-1)}\|)^{-N},$		(3.8)

where $\mathbb{I}_{1}:=(2^{j}t)^{-\frac{1}{2}}\mathbb{Z}\cap[1,2]$ .
(3) If $t\in(2^{-j+Ml},+\infty)$ , we obtain

		$\displaystyle\|K_{j,l,2}(y,t)\|\lesssim C_{N}2^{2j+\frac{Ml}{2}}t^{-1}\times$
		$\displaystyle\sum_{\kappa\in\mathbb{I}_{1}}\sum_{\tilde{\kappa}\in\mathbb{I}_{2}}(1+2^{\frac{j}{2}}t^{-\frac{1}{2}}\|y_{1}+c_{1,t}(\kappa,\tilde{\kappa})\|+2^{\frac{j+Ml}{2}}t^{-\frac{1}{2}}\|y_{2}+c_{2,t}(\kappa,\tilde{\kappa})\|+2^{j}\|y_{3}+c_{3,t}(y_{1},y_{2},\kappa,\tilde{\kappa})\|)^{-N},$		(3.9)

where $\mathbb{I}_{2}:=(2^{j-Ml}t)^{-\frac{1}{2}}\mathbb{Z}\cap[1,2]$ ,

c_{1,t}(\kappa,\tilde{\kappa})=c_{d,M,l}t\kappa^{\frac{1}{d-1}}+c_{d,M,l}2^{-Ml}t\kappa^{\frac{d}{d-1}-\frac{M}{M-1}-1}\tilde{\kappa}^{\frac{M}{M-1}}+2^{-(M+1)l}t\partial_{1}R(\kappa,\frac{\tilde{\kappa}}{\kappa})-2^{-(M+1)l}t\frac{\tilde{\kappa}}{\kappa^{2}}\partial_{2}R(\kappa,\frac{\tilde{\kappa}}{\kappa}),

c_{2,t}(\kappa,\tilde{\kappa})=c_{d,M,l}2^{-Ml}t\kappa^{\frac{d}{d-1}-\frac{M}{M-1}}\tilde{\kappa}^{\frac{1}{M-1}}+t2^{-(M+1)l}\frac{1}{\kappa}\partial_{2}R(\kappa,\frac{\tilde{\kappa}}{\kappa}),

c_{3,t}(y_{1},y_{2},\kappa,\tilde{\kappa})=\kappa y_{1}+\tilde{\kappa}y_{2}+t\Psi_{1}(\kappa)+t2^{-Ml}\Psi_{2}(\kappa,\frac{\tilde{\kappa}}{\kappa})+t2^{-(M+1)l}R(\kappa,\frac{\tilde{\kappa}}{\kappa}).

We firstly give the proof of Lemma 3.2, and show the proof how Theorem 3.1 can be deduced by Lemma 3.2 at the end of Section 3.

Proof. We first prove the part (1). By rescaling, we have

K_{j,l,2}(y,t)=2^{3j}\int_{\mathbb{R}^{3}}e^{i2^{j}y\cdot\eta+it2^{j}\eta_{3}\Psi_{1}(\frac{\eta_{1}}{\eta_{3}})+it2^{j-Ml}\eta_{3}\Psi_{2}(\frac{\eta_{1}}{\eta_{3}},\frac{\eta_{2}}{\eta_{1}})}a_{j,l,2}(t,2^{j}\eta)d\eta.

(3.10)

Since $t\in(0,2^{-j})$ , integrating by parts implies that

|K_{j,l,2}(y,t)|\leq\frac{C_{N}2^{3j}}{(1+2^{j}|y|)^{N}}\leq\frac{C_{N}2^{2j}}{t(1+2^{j}|y|)^{N}}.

To prove the part (2), we recall the definition of $a_{j,l,2}$ in equality (2), and decompose $K_{j,l,2}$ given by equality (3.10) as $K_{j,l,2}(y,t)=\sum_{\kappa\in\mathbb{I}_{1}}K_{j,l,2,\kappa}(y,t)$ , where

K_{j,l,2,\kappa}(y,t)=2^{3j}\int_{\mathbb{R}^{3}}e^{i2^{j}y\cdot\eta+it2^{j}\eta_{3}\Psi_{1}(\frac{\eta_{1}}{\eta_{3}})+it2^{j-Ml}\eta_{3}\Psi_{2}(\frac{\eta_{1}}{\eta_{3}},\frac{\eta_{2}}{\eta_{1}})}a_{j,l,2,\kappa}(t,2^{j}\eta)d\eta,

and $a_{j,l,2,\kappa}(t,2^{j}\eta)=a_{j,l,2}(t,2^{j}\eta)\tilde{\chi}\biggl{(}(2^{j}t)^{\frac{1}{2}}(\frac{\eta_{1}}{\eta_{3}}-\kappa)\biggl{)}$ . For fixed $\kappa$ , changing variables $\eta_{1}\rightarrow(2^{j}t)^{-\frac{1}{2}}\eta_{1}+\kappa\eta_{3}$ , its phase function becomes

	$\displaystyle 2^{\frac{j}{2}}t^{-\frac{1}{2}}y_{1}\eta_{1}+2^{j}y_{2}\eta_{2}+2^{j}(y_{3}+\kappa y_{1})\eta_{3}$
	$\displaystyle+t2^{j}\eta_{3}\Psi_{1}\biggl{(}\frac{(2^{j}t)^{-\frac{1}{2}}\eta_{1}}{\eta_{3}}+\kappa\biggl{)}+t2^{j-Ml}\eta_{3}\Psi_{2}\biggl{(}\frac{(2^{j}t)^{-\frac{1}{2}}\eta_{1}}{\eta_{3}}+\kappa,\frac{\eta_{2}}{(2^{j}t)^{-\frac{1}{2}}\eta_{1}+\kappa\eta_{3}}\biggl{)}.$

By Taylor’s expansion of $\Psi_{1}\biggl{(}\frac{(2^{j}t)^{-\frac{1}{2}}\eta_{1}}{\eta_{3}}+\kappa\biggl{)}$ at $\eta_{1}=0$ , the phase function can be written as

\displaystyle 2^{\frac{j}{2}}t^{-\frac{1}{2}}(y_{1}+t\Psi_{1}^{\prime}(\kappa))\eta_{1}+2^{j}y_{2}\eta_{2}+2^{j}(y_{3}+\kappa y_{1}+t\Psi_{1}(\kappa))\eta_{3}+\frac{\eta_{1}^{2}}{2\eta_{3}}\Psi_{1}^{\prime\prime}(\kappa)+R_{j,l,1}(\eta,t,\kappa),

where we have put all the remainder terms into $R_{j,l,1}$ . Then integration by parts yields

|K_{j,l,2,\kappa}(y,t)|\leq\frac{C_{N}2^{\frac{5j}{2}}t^{-\frac{1}{2}}}{(1+2^{\frac{j}{2}}t^{-\frac{1}{2}}|y_{1}+c_{d,M,l}t\kappa^{\frac{1}{d-1}}|+2^{j}|y_{2}|+2^{j}|y_{3}+\kappa y_{1}+c_{d,M,l}t\kappa^{d/(d-1)}|)^{N}}.

This implies the inequality (3.10) since $t\leq 2^{-j+Ml}$ .

Similarly, we can get the part (3). We decompose $K_{j,l,2}(y,t)=\sum_{\kappa\in\mathbb{I}_{1}}\sum_{\tilde{\kappa}\in\mathbb{I}_{2}}K_{j,l,2,\kappa,\tilde{\kappa}}(y,t)$ , where

K_{j,l,2,\kappa,\tilde{\kappa}}(y,t)=2^{3j}\int_{\mathbb{R}^{3}}e^{i2^{j}y\cdot\eta+it2^{j}\eta_{3}\Psi_{1}(\frac{\eta_{1}}{\eta_{3}})+it2^{j-Ml}\eta_{3}\Psi_{2}(\frac{\eta_{1}}{\eta_{3}},\frac{\eta_{2}}{\eta_{1}})}a_{j,l,2,\kappa,\tilde{\kappa}}(t,2^{j}\eta)d\eta,

and $a_{j,l,2,\kappa,\tilde{\kappa}}(t,2^{j}\eta)=a_{j,l,2}(t,2^{j}\eta)\tilde{\chi}\biggl{(}(2^{j}t)^{\frac{1}{2}}(\frac{\eta_{1}}{\eta_{3}}-\kappa)\biggl{)}\tilde{\chi}\biggl{(}(2^{j-Ml}t)^{\frac{1}{2}}(\frac{\eta_{2}}{\eta_{3}}-\tilde{\kappa})\biggl{)}$ . Then we change variables $\eta_{1}\rightarrow(2^{j}t)^{-\frac{1}{2}}\eta_{1}+\kappa\eta_{3}$ , $\eta_{2}\rightarrow(2^{j-Ml}t)^{-\frac{1}{2}}\eta_{2}+\tilde{\kappa}\eta_{3}$ , $\eta_{3}\rightarrow\eta_{3}$ . We get the phase function in $K_{j,l,2,\kappa,\tilde{\kappa}}$ as follows

	$\displaystyle 2^{\frac{j}{2}}t^{-\frac{1}{2}}y_{1}\eta_{1}+2^{\frac{j+Ml}{2}}t^{-\frac{1}{2}}y_{2}\eta_{2}+2^{j}(y_{3}+\kappa y_{1}+\tilde{\kappa}y_{2})\eta_{3}$
	$\displaystyle+t2^{j}\eta_{3}\Psi_{1}\biggl{(}\frac{(2^{j}t)^{-\frac{1}{2}}\eta_{1}}{\eta_{3}}+\kappa\biggl{)}+t2^{j-Ml}\eta_{3}\Psi_{2}\biggl{(}\frac{(2^{j}t)^{-\frac{1}{2}}\eta_{1}+\kappa\eta_{3}}{\eta_{3}},\frac{(2^{j-Ml}t)^{-\frac{1}{2}}\eta_{2}+\tilde{\kappa}\eta_{3}}{(2^{j}t)^{-\frac{1}{2}}\eta_{1}+\kappa\eta_{3}}\biggl{)}.$

Then by Taylor’s expansion, we get the phase function

	$\displaystyle 2^{\frac{j}{2}}t^{-\frac{1}{2}}\biggl{(}y_{1}+t\Psi_{1}^{\prime}(\kappa)+t2^{-Ml}\partial_{1}\Psi_{2}(\kappa,\frac{\tilde{\kappa}}{\kappa})-t2^{-Ml}\frac{\tilde{\kappa}}{\kappa^{2}}\partial_{2}\Psi_{2}(\kappa,\frac{\tilde{\kappa}}{\kappa})\biggl{)}\eta_{1}$
	$\displaystyle+2^{\frac{j+Ml}{2}}t^{-\frac{1}{2}}\biggl{(}y_{2}+t2^{-Ml}\frac{1}{\kappa}\partial_{2}\Psi_{2}(\kappa,\frac{\tilde{\kappa}}{\kappa})\biggl{)}\eta_{2}+2^{j}\biggl{(}y_{3}+\kappa y_{1}+\tilde{\kappa}y_{2}+t\Psi_{1}(\kappa)+t2^{-Ml}\Psi_{2}(\kappa,\frac{\tilde{\kappa}}{\kappa})\biggl{)}\eta_{3}$
	$\displaystyle+\frac{\eta_{1}^{2}}{2\eta_{3}}\Psi_{1}^{\prime\prime}(\kappa)+\frac{\eta_{2}^{2}}{2\eta_{3}}\frac{1}{\kappa^{2}}\partial_{22}\Psi_{2}(\kappa,\frac{\tilde{\kappa}}{\kappa})+R_{j,l,2}(\eta,t,\kappa,\tilde{\kappa}),$

where all the remainder terms are included in $R_{j,l,2}$ . Then the estimate (3) follows by integration by parts. $\Box$

Proof of Theorem 3.1: (1) When $t\in(0,2^{-j+Ml})$ , according to the estimates in (1) and (2) of Lemma 3.2, it is easy to obtain inequality (3.5). Next we will show that (3.5) holds for $t\in[2^{-j+Ml},\infty)$ . Observing the part (3) in Lemma 3.2, for fixed $t$ , $\kappa$ and $\tilde{\kappa}$ , the term

(1+2^{\frac{j}{2}}t^{-\frac{1}{2}}|y_{1}+c_{1,t}(\kappa,\tilde{\kappa})|+2^{\frac{j+Ml}{2}}t^{-\frac{1}{2}}|y_{2}+c_{2,t}(\kappa,\tilde{\kappa})|+2^{j}|y_{3}+c_{3,t}(y_{1},y_{2},\kappa,\tilde{\kappa})|)^{-N}

can be viewed as a characteristic function over the set whose projection on the $(y_{1},y_{2})$ -plane is a rectangle centered at $(c_{1,t}(\kappa,\tilde{\kappa}),c_{2,t}(\kappa,\tilde{\kappa}))$ with side length $2^{-j/2}t^{1/2}\times 2^{-\frac{j+Ml}{2}}t^{1/2}$ . Hence, in order to obtain (3.5), it suffices to prove that if $(\kappa,\tilde{\kappa})\neq(\kappa^{\prime},\widetilde{\kappa}^{\prime})$ , then we have either $|c_{1,t}(\kappa,\tilde{\kappa})-c_{1,t}(\kappa^{\prime},\widetilde{\kappa}^{\prime})|\geq 2^{-\frac{j}{2}}t^{\frac{1}{2}}$ or $|c_{2,t}(\kappa,\widetilde{\kappa})-c_{2,t}(\kappa^{\prime},\widetilde{\kappa}^{\prime})|\geq 2^{-\frac{j+Ml}{2}}t^{\frac{1}{2}}$ . Indeed, if $|\kappa-\kappa^{\prime}|\geq 2^{-Ml+2}|\tilde{\kappa}-\tilde{\kappa}^{\prime}|$ , there holds

|c_{1,t}(\kappa,\kappa^{\prime})-c_{1,t}(\tilde{\kappa},\widetilde{\kappa}^{\prime})|\sim t|\kappa-\kappa^{\prime}|\geq 2^{-\frac{j}{2}}t^{\frac{1}{2}}.

Conversely, if $|\kappa-\kappa^{\prime}|<2^{-Ml+2}|\tilde{\kappa}-\widetilde{\kappa}^{\prime}|$ , we have

|c_{2,t}(\kappa,\kappa^{\prime})-c_{2,t}(\tilde{\kappa},\widetilde{\kappa}^{\prime})|\sim 2^{-Ml}t|\tilde{\kappa}-\widetilde{\kappa}^{\prime}|\geq 2^{-\frac{j+Ml}{2}}t^{\frac{1}{2}}.

Then we obtain inequality (3.5).

(2) For each $t\in[1,2]$ , inequality (3.6) follows from part (3) of Lemma 3.2, we omit the proof here.

4 Proofs of Theorem 1.2 and Theorem 1.3

Firstly, we introduce some necessary notations in our proof. For $\Phi$ which is given by (1.2), we denote

\tilde{E}_{\Phi}=\biggl{\{}n_{h}:n_{h}\geq 1,\lambda_{h}\in\mathbb{R}\text{ and }\{(x_{1},x_{2}):x_{2}=\lambda_{h}x_{1}\}\subset Z_{H\Phi}\biggl{\}}.

Notice that since $n_{h}\geq 1$ , for $n_{h}\in\tilde{E}_{\Phi}$ , there holds $\{(x_{1},x_{2}):x_{2}=\lambda_{h}x_{1}\}\subset Z_{\Phi}\bigcap Z_{H\Phi}.$ Next we define $E_{\Phi}$ according to the following four cases:

(C1) if $Z_{\Phi}\bigcap Z_{H\Phi}$ contains $\{(x_{1},x_{2}):x_{1}=0\}$ but does not contain $\{(x_{1},x_{2}):x_{2}=0\}$ , then we define

E_{\Phi}=\tilde{E}_{\Phi}\cup\{\nu_{1},m/2\};

(C2) if $Z_{\Phi}\bigcap Z_{H\Phi}$ contains $\{(x_{1},x_{2}):x_{2}=0\}$ but does not contain $\{(x_{1},x_{2}):x_{1}=0\}$ , then we define

E_{\Phi}=\tilde{E}_{\Phi}\cup\{\nu_{2},m/2\};

(C3) if $Z_{\Phi}\bigcap Z_{H\Phi}$ contains both $\{(x_{1},x_{2}):x_{2}=0\}$ and $\{(x_{1},x_{2}):x_{1}=0\}$ , then we define

E_{\Phi}=\tilde{E}_{\Phi}\cup\{\nu_{1},\nu_{2},m/2\};

(C4) if $Z_{\Phi}\bigcap Z_{H\Phi}$ contains neither $\{(x_{1},x_{2}):x_{2}=0\}$ nor $\{(x_{1},x_{2}):x_{1}=0\}$ , then we define

E_{\Phi}=\tilde{E}_{\Phi}\cup\{m/2\}.

The following lemma will show that the height $h_{\Phi}$ of the homogeneous polynomial $\Phi$ is the maximal element of $E_{\Phi}$ . To be convenient, let us denote $\mu_{\Phi}$ by the maximal element of $E_{\Phi}$ .

Lemma 4.1.

For the homogeneous polynomial $\Phi$ given by (1.2), we have $\mu_{\Phi}=h_{\Phi}$ .

We have the following observations before the proof of Lemma 4.1. For fixed $x\in S^{1}$ , if $\Phi(x)\neq 0$ , it is clear that $\textsf{ordd}\Phi(x)=0$ . Otherwise, without loss of generality, we may assume that $x\in S^{1}\cap\{x_{2}=\lambda_{h}x_{1}\}$ , where $x_{2}=\lambda_{h}x_{1}$ corresponds to the factor $(x_{2}-\lambda_{h}x_{1})^{n_{h}}$ in (1.2) with $\lambda_{h}\in\mathbb{R}$ , $n_{h}\geq 1$ . A simple caculation implies that $\textsf{ordd}\Phi(x)=n_{h}$ . Moreover, for a homogeneous polynomial $\Phi$ , the supremum in the definition of $\textsf{ordd}\Phi$ can always be achieved at some $x\in S^{1}$ . Conversely, for fixed straight line passing through the origin which is contained in $Z_{\Phi}$ , without loss of generality, we may again consider $x_{2}=\lambda_{h}x_{1}$ corresponding to the factor $(x_{2}-\lambda_{h}x_{1})^{n_{h}}$ in (1.2), there is an $x\in S^{1}\cap\{x_{2}=\lambda_{h}x_{1}\}$ such that $\textsf{ordd}\Phi(x)=n_{h}$ . In particular, if $E_{\Phi}\backslash\{m/2\}\neq\emptyset$ , then for each $\kappa\in E_{\Phi}\backslash\{m/2\}$ , there exists an $x_{\kappa}\in S^{1}$ such that $\textsf{ordd}\Phi(x_{\kappa})=\kappa$ .

Proof. We first consider the case when $h_{\Phi}=m/2$ . If $E_{\Phi}=\{\frac{m}{2}\}$ , then it is clear that $\mu_{\Phi}=h_{\Phi}=\frac{m}{2}$ ; if $E_{\Phi}\backslash\{\frac{m}{2}\}\neq\emptyset$ , there holds

\frac{m}{2}\geq\sup_{x\in S^{1}}\textsf{ordd}\Phi(x)\geq\sup_{\kappa\in E_{\Phi}\backslash\{m/2\}}\textsf{ordd}\Phi(x_{\kappa})=\sup_{\kappa\in E_{\Phi}\backslash\{m/2\}}\kappa,

then it follows that $\mu_{\Phi}=\frac{m}{2}=h_{\Phi}$ .

In the case when $h_{\Phi}>m/2$ , there exists an $x\in S^{1}\cap Z_{\Phi}$ such that $\textsf{ordd}\Phi=\textsf{ordd}\Phi(x)>m/2$ , without loss of generality, we may assume that $x\in S^{1}\cap\{x_{2}=\lambda_{h}x_{1}\}$ , where $x_{2}=\lambda_{h}x_{1}$ corresponds to the factor $(x_{2}-\lambda_{h}x_{1})^{n_{h}}$ in (1.2). Then we write $\Phi$ as

\Phi(x_{1},x_{2})=(x_{2}-\lambda_{h}x_{1})^{n_{h}}Q(x_{1},x_{2}),

where $Q$ is a homogeneous polynomial such that $Q(x_{1},\lambda_{h}x_{1})\neq 0$ for $x_{1}\neq 0$ . It is easy to claim that when $n_{h}\geq 3$ , $H\Phi(x_{1},\lambda_{h}x_{1})=0$ .

When $n_{h}>\frac{m}{2}$ and $m\geq 4$ , then we must have $n_{h}\geq 3$ . Therefore, when $m\geq 4$ , we have $n_{h}\in E_{\Phi}$ and

n_{h}=\sup_{x\in S^{1}}\textsf{ordd}\Phi(x)\geq\sup_{\kappa\in E_{\Phi}\backslash\{m/2\}}\textsf{ordd}\Phi(x_{\kappa})=\sup_{\kappa\in E_{\Phi}\backslash\{m/2\}}\kappa.

Hence there holds $\mu_{\Phi}=h_{\Phi}=n_{h}$ .

Now we are left with the case when $m=3$ or $m=2$ . When $m=3$ and $n_{h}=3$ , the conclusion follows directly by the previous claim. When $m=3$ and $n_{h}=2$ , we may write

\Phi(x_{1},x_{2})=(x_{2}-\lambda_{h}x_{1})^{2}Q(x_{1},x_{2}),

where $Q$ is a linear polynomial that is homogeneous of degree one. A direct calculation implies that $H\Phi(x_{1},\lambda_{h}x_{1})=0$ , then we have $\mu_{\Phi}=h_{\Phi}=2$ . When $m=2$ , we must have $n_{h}=2$ by the previous observation, then

\Phi(x_{1},x_{2})=c_{\Phi}(x_{2}-\lambda_{h}x_{1})^{2}.

It is obvious that $H\Phi\equiv 0$ , so $n_{h}\in E_{\Phi}$ and $\mu_{\Phi}=h_{\Phi}=2$ . This finishes the proof of Lemma 4.1. $\Box$

The proof of Theorem 1.2 is based on the following Lemma 4.2.

Lemma 4.2.

If $H\Phi$ vanishes along the straight line $x_{2}=\lambda_{h_{0}}x_{1}$ for some $\lambda_{h_{0}}\in\mathbb{R}$ and $1\leq h_{0}\leq N$ such that $n_{h_{0}}\in\tilde{E}_{\Phi}$ , we define the local maximal operator

		$\displaystyle\sup_{t\in[1,2]}\biggl{\|}\widetilde{A^{\lambda_{h_{0}}}_{t,k}}f(y)\biggl{\|}$
		$\displaystyle=\sup_{t\in[1,2]}\biggl{\|}\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(\Phi(x_{1},x_{2})+c2^{mk}))\eta_{0}(\frac{x_{2}-\lambda_{h_{0}}x_{1}}{\epsilon x_{1}})\widetilde{\eta}(x_{1},x_{2})dx_{1}dx_{2}\biggl{\|},$		(4.1)

where $k\gg 1$ , supp $\tilde{\eta}\subset\{x\in\mathbb{R}^{2}:|x|\sim 1\}$ , supp $\eta_{0}\subset B(0,1)$ , and $\epsilon>0$ is sufficiently small, such that $H\Phi$ only vanishes along $x_{2}=\lambda_{h_{0}}x_{1}$ in the angle $|x_{2}-\lambda_{h_{0}}x_{1}|\leq\epsilon|x_{1}|$ . It follows that when $c=0$ ,

\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim 1

(4.2)

holds for $(\frac{1}{p},\frac{1}{q})\in\Delta_{0}$ and $\frac{n_{h_{0}}+1}{p}-\frac{n_{h_{0}}+1}{q}-1<0$ . When $c\neq 0$ , we get

\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim 2^{\frac{mk}{q}}

(4.3)

provided that $(\frac{1}{p},\frac{1}{q})\in\Delta_{0}$ and $\frac{n_{h_{0}}+1}{p}-\frac{1}{q}-1<0$ .

Proof. After some linear transformation (which does not change the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ norm of the maximal operator), we are reduced to estimating the maximal operator defined by

\sup_{t\in[1,2]}\biggl{|}\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(\Phi(x_{1},x_{2}+\lambda_{h_{0}}x_{1})+c2^{mk}))\eta_{0}(\frac{x_{2}}{\epsilon x_{1}})\widetilde{\eta}(x_{1},x_{2}+\lambda_{h_{0}}x_{1})dx_{1}dx_{2}\biggl{|}.

Notice that

$\displaystyle\Phi(x_{1},x_{2}+\lambda_{h_{0}}x_{1})$	$\displaystyle=c_{\Phi}x_{1}^{\nu_{1}}(x_{2}+\lambda_{h_{0}}x_{1})^{\nu_{2}}\Pi_{h=1}^{N}(x_{2}+\lambda_{h_{0}}x_{1}-\lambda_{h}x_{1})^{n_{h}}$
	$\displaystyle=c_{\Phi}x_{1}^{\nu_{1}+\nu_{2}+\sum_{h\neq h_{0}}n_{h}}x_{2}^{n_{h_{0}}}\biggl{(}\frac{x_{2}}{x_{1}}+\lambda_{h_{0}}\biggl{)}^{\nu_{2}}\Pi_{h\neq h_{0}}(\frac{x_{2}}{x_{1}}+\lambda_{h_{0}}-\lambda_{h})^{n_{h}}$
	$\displaystyle=x_{1}^{\nu_{1}+\nu_{2}+\sum_{h\neq h_{0}}n_{h}}x_{2}^{n_{h_{0}}}U\biggl{(}\frac{x_{2}}{x_{1}}\biggl{)},$	(4.4)

where

U\biggl{(}\frac{x_{2}}{x_{1}}\biggl{)}:=c_{\Phi}\biggl{(}\frac{x_{2}}{x_{1}}+\lambda_{h_{0}}\biggl{)}^{\nu_{2}}\Pi_{h\neq h_{0}}(\frac{x_{2}}{x_{1}}+\lambda_{h_{0}}-\lambda_{h})^{n_{h}}

and $|U|\sim 1$ since $|x_{2}/x_{1}|<\epsilon$ and $\lambda_{h_{0}}\neq 0$ . Moreover, $1\leq n_{h_{0}}<m$ , $1\leq\nu_{1}+\nu_{2}+\sum_{h\neq h_{0}}n_{h}<m$ .

By isometric transform, we obtain

\displaystyle\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq\sum_{l\geq\log\frac{1}{\epsilon}}2^{\frac{(n_{h_{0}}+1)l}{p}-\frac{(n_{h_{0}}+1)l}{q}-l}\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})},

where $\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}$ denotes the averaging operator along the hypersurface

S^{h_{0}}_{k,l}=\biggl{\{}\biggl{(}x_{1},x_{2},x_{1}^{\nu_{1}+\nu_{2}+\sum_{h\neq h_{0}}n_{h}}x_{2}^{n_{h_{0}}}U\biggl{(}2^{-l}\frac{x_{2}}{x_{1}}\biggl{)}+2^{mk+n_{h_{0}}l}c\biggl{)}:|x_{1}|\sim|x_{2}|\sim 1\biggl{\}}.

(4.5)

Now it is clear that the surface $S^{h_{0}}_{k,l}$ has non-vanishing Garssian curvature, then

\displaystyle\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim(c2^{\frac{mk}{q}}2^{\frac{n_{h_{0}l}}{q}}+1)

(4.6)

provided that $(\frac{1}{p},\frac{1}{q})\in\Delta_{0}$ .

The estimate (4.6) arises from a general conclusion. Let $U$ be a compact set in $\mathbb{R}^{2}$ , $\Psi$ be a function with the non-degenerate Heissian matrix, $S=\{(x_{1},x_{2},\Psi(x_{1},x_{1})+c):(x_{1},x_{2})\in U\}$ be a smooth hypersurface, $c\in\mathbb{R}$ . Then the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ norm of the local maximal operator along $S$ is bounded by $\max\{1,c^{1/q}\}$ provided that $(\frac{1}{p},\frac{1}{q})\in\Delta_{0}$ . Here we outline the main idea behind the proof of this conclusion, which will be directly referenced in the rest of this paper. Upon Littlewood-Paley decomposition, when $\xi$ is constrained to $\{\xi:|\xi|\sim 2^{j}\}$ , $j\geq 1$ , if $|\xi_{1}|+|\xi_{2}|\gg|\xi_{3}|$ , it can be proven that the Fourier transform along $S$ decays rapidly to complete the corresponding maximal function estimate. Next, we present the application of the method of the stationary phase in

\widehat{d\mu_{S}}(t\xi)=\int_{\mathbb{R}^{2}}e^{-it[x_{1}\xi_{1}+x_{2}\xi_{2}+\Phi(x_{1},x_{2})\xi_{3}]}\eta(x_{1},x_{2})dx_{1}dx_{2},\quad|\xi_{1}|+|\xi_{2}|\sim 2^{j},|\xi_{1}|+|\xi_{2}|\lesssim|\xi_{3}|,

where $t\in[1,2]$ , $\eta$ is a smooth cutoff function supported in $U$ . According to Theorem 1.2.1 in [15], $\widehat{d\mu_{S}}(t\xi)$ can be expressed as the sum of a finite number of terms of the following form

e^{-it\tilde{\Phi}(\xi)}\frac{a(t,\xi)}{(1+t|\xi|)},

where the function $\widetilde{\Phi}(\xi)$ is homogeneous of degree one and its Heissian matrix has a rank of $2$ , $a(t,\xi)$ is a symbol of order zero in $\xi$ . Through Sobolev embedding, we transform the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ estimate of the maximal operator into the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3}\times[1,2])$ estimate of the following Fourier integral operator

\int_{\mathbb{R}^{3}}e^{iy\cdot\xi-it\tilde{\Phi}(\xi)}\frac{a(t,\xi)}{(1+t|\xi|)}\hat{f}(\xi)d\xi.

It is worth noting that the surface $\{(\eta,\tilde{\Phi}(\eta)):\eta\in suppa(t,\cdot)\}$ has two non-vanishing principal curvatures, which implies that the method of estimate for the corresponding Fourier integral operator is consistent with the proof described in Section 4 of reference [14].

Now let’s proceed to the proof of Lemma 4.2. Inequality (4.6) yields

\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq\sum_{l\geq\log\frac{1}{\epsilon}}2^{\frac{(n_{h_{0}}+1)l}{p}-\frac{(n_{h_{0}}+1)l}{q}-l}(c2^{\frac{mk}{q}}2^{\frac{n_{h_{0}}l}{q}}+1).

(4.7)

It follows that when $c=0$ , the right side of (4.7) is convergent provided that $\frac{n_{h_{0}}+1}{p}-\frac{n_{h_{0}}+1}{q}-1<0$ . Then inequality (4.2) holds true for $(\frac{1}{p},\frac{1}{q})\in\Delta_{0}$ and $\frac{n_{h_{0}}+1}{p}-\frac{n_{h_{0}}+1}{q}-1<0$ . When $c\neq 0$ , the right side of (4.7) converges to $c2^{\frac{mk}{q}}$ if $\frac{n_{h_{0}}+1}{p}-\frac{1}{q}-1<0$ . Hence, there holds inequality (4.3) provided that $(\frac{1}{p},\frac{1}{q})\in\Delta_{0}$ and $\frac{n_{h_{0}}+1}{p}-\frac{1}{q}-1<0$ . $\Box$

Remark 4.3.

It is clear that the similar arguments remain valid for each $\{x_{2}=\lambda_{h}x_{1}\}\subset E_{\Phi}\cap Z_{H\Phi}$ , or $\{x_{i}=0\}\subset Z_{\Phi}\cap Z_{H\Phi}$ , $i=1,2$ , whose corresponding local maximal operator can be defined in the spirit of (4.2).

Proof of Theorem 1.2: Let $A_{t}$ be the operator defined by (1.1). A dyadic decomposition and scaling argument imply

\biggl{\|}\sup_{t\in[1,2]}|A_{t}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq\sum_{k\geq 1}2^{\frac{(m+2)k}{p}-{\frac{(m+2)k}{q}}-2k}\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k}}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})},

where

\widetilde{A_{t,k}}f(y)=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(\Phi(x_{1},x_{2})+c2^{mk}))\widetilde{\eta}(x_{1},x_{2})dx_{1}dx_{2}

(4.8)

with supp $\tilde{\eta}\subset\{x\in\mathbb{R}^{2}:|x|\sim 1\}$ . Notice that since $H\Phi$ is a homogeneous function, $Z_{H\Phi}$ either contains the origin only, or contain some straight lines passing through the origin. In the first case, our results follows directly since $H\Phi$ does not vanish on the support of $\tilde{\eta}$ . In the later case, by the assumption of Theorem 1.2, we are reduced to considering the maximal operators which are localized near the straight lines containing in $Z_{\Phi}\cap Z_{H\Phi}$ , then Lemma 4.2 and Remark 4.3 can be applied to get the following conclusions.

When $c=0$ , for

	$\displaystyle(\frac{1}{p},\frac{1}{q})$	$\displaystyle\in\bigcap_{\mu\in E_{\Phi}\backslash\{m/2\}}\biggl{\{}(\frac{1}{p},\frac{1}{q})\in\Delta_{0}:\frac{\mu+1}{p}-\frac{\mu+1}{q}-1<0,\frac{m/2+1}{p}-\frac{m/2+1}{q}-1<0\biggl{\}}$
		$\displaystyle=\biggl{\{}(\frac{1}{p},\frac{1}{q})\in\Delta_{0}:\frac{h_{\Phi}+1}{p}-\frac{h_{\Phi}+1}{q}-1<0\biggl{\}},$		(4.9)

there holds

\sum_{k\geq 1}\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim\sum_{k\geq 1}2^{\frac{(m+2)k}{p}-{\frac{(m+2)k}{q}}-2k}\lesssim 1.

(4.10)

When $c\neq 0$ , for

	$\displaystyle(\frac{1}{p},\frac{1}{q})$	$\displaystyle\in\bigcap_{\mu\in E_{\Phi}\backslash\{m/2\}}\biggl{\{}(\frac{1}{p},\frac{1}{q})\in\Delta_{0}:\frac{\mu+1}{p}-\frac{1}{q}-1<0,\frac{m/2+1}{p}-\frac{1}{q}-1<0\biggl{\}}$
		$\displaystyle=\biggl{\{}(\frac{1}{p},\frac{1}{q})\in\Delta_{0}:\frac{h_{\Phi}+1}{p}-\frac{1}{q}-1<0\biggl{\}},$		(4.11)

we get

\sum_{k\geq 1}\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim\sum_{k\geq 1}2^{\frac{(m+2)k}{p}-{\frac{2k}{q}}-2k}\lesssim 1.

(4.12)

This finishes the proof of Theorem 1.2.

Now we are left with the proof of Theorem 1.3, which follows from Lemma 4.4 below.

Lemma 4.4.

If $H\Phi$ vanishes along $x_{2}=\tau x_{1}$ for some non-zero real number $\tau$ such that $\tau\neq\lambda_{h}$ for any $1\leq h\leq N$ . We consider the maximal operator

\sup_{t\in[1,2]}\biggl{|}\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(\Phi(x_{1},x_{2})+c2^{mk}))\eta_{0}(\frac{x_{2}-\tau x_{1}}{\epsilon x_{1}})\widetilde{\eta}(x_{1},x_{2})dx_{1}dx_{2}\biggl{|},

where $\epsilon>0$ is sufficiently small, such that $\Phi$ does not vanish in the angle $|x_{2}-\tau x_{1}|\leq\epsilon|x_{1}|$ . Then there holds

\biggl{\|}\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\tau}_{t,k}}\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim c2^{\frac{mk}{q}}+1

(4.13)

for $(\frac{1}{p},\frac{1}{q})\in\Delta_{M}$ .

Proof. As in Lemma 4.2, we are reduced to estimating the maximal operator defined by

\sup_{t\in[1,2]}\biggl{|}\widetilde{A^{\tau}_{t,k}}f(y)\biggl{|}=\sup_{t\in[1,2]}\biggl{|}\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(\Phi(x_{1},x_{2}+\tau x_{1})+c2^{mk}))\eta_{0}(\frac{x_{2}}{\epsilon x_{1}})\widetilde{\eta}(x_{1},x_{2}+\tau x_{1})dx_{1}dx_{2}\biggl{|}.

Here

\displaystyle\Phi(x_{1},x_{2}+\tau x_{1})

\displaystyle=c_{\Phi}x_{1}^{m}\biggl{(}\frac{x_{2}}{x_{1}}+\tau\biggl{)}^{\nu_{2}}\Pi_{h=1}^{N}(\frac{x_{2}}{x_{1}}+\tau-\lambda_{h})^{n_{h}}=x_{1}^{m}V\biggl{(}\frac{x_{2}}{x_{1}}\biggl{)},

(4.14)

where

V\biggl{(}\frac{x_{2}}{x_{1}}\biggl{)}:=c_{\Phi}\biggl{(}\frac{x_{2}}{x_{1}}+\tau\biggl{)}^{\nu_{2}}\Pi_{h=1}^{N}(\frac{x_{2}}{x_{1}}+\tau-\lambda_{h})^{n_{h}}

and $|V|\sim 1$ since $|x_{2}/x_{1}|\leq\epsilon$ and $\tau\neq 0$ . It is clear that when $|x_{2}|\leq\epsilon|x_{1}|$ and $|x_{1}|\sim 1$ ,

\Phi(x_{1},x_{2}+\tau x_{1})\neq 0,\quad\partial_{x_{1}}\Phi(x_{1},x_{2}+\tau x_{1})\neq 0.

Hence under some transformations (which does not change the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ norm of $\sup_{t\in[1,2]}|\widetilde{A_{t,k}^{\tau}}|$ up to a constant), it is sufficient to consider the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ estimate of the maximal operator along the hypersurface

\tilde{S}_{k}=\{(r(1+s^{M}g(s)),rs,r^{m}+c2^{mk}):r\in[1,2],s\in(0,\epsilon_{0})\},

(4.15)

where $2\leq M\leq m$ , $g$ is a smooth function and $g(0)\neq 0$ , $\epsilon_{0}$ is a small constant. Here we applied the assumption that the level set $\{(x_{1},x_{2}):\Phi(x_{1},x_{2})=1\}$ determines a curve of finite type $M$ near the straight line $x_{2}=\tau x_{1}$ . Finally, the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|\widetilde{A_{t,k}^{\tau}}|$ follows from Theorem 1.4 directly. $\Box$

Remark 4.5.

Notice that the results in Lemma 4.4 are independent of $\tau$ , and remain valid if $H\Phi$ vanishes along $x_{1}=0$ or $x_{2}=0$ , but $\Phi$ dose not vanish.

Proof of Theorem 1.3: As in the beginning of the proof of Theorem 1.2, then we are reduced to estimating the maximal operator $\sup_{t\in[1,2]}|\widetilde{A_{t,k}}|$ , where $\widetilde{A_{t,k}}$ is given by equality (4.8). Since $Z_{H\Phi}\cap Z_{\Phi}\subsetneqq Z_{H\Phi}$ , there exists at least one straight line passing through the origin which is contained in $Z_{H\Phi}$ but not contained in $Z_{\Phi}$ . Then Lemma 4.4 and Remark 4.5 can be applied.

When $c=0$ , for

(\frac{1}{p},\frac{1}{q})\in\biggl{\{}(\frac{1}{p},\frac{1}{q})\in\Delta_{M}:\frac{\ h_{\Phi}+1}{p}-\frac{h_{\Phi}+1}{q}-1<0\biggl{\}},

(4.16)

we obtain

\sum_{k\geq 1}\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k}}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim\sum_{k\geq 1}2^{\frac{(m+2)k}{p}-{\frac{(m+2)k}{q}}-2k}\lesssim 1.

(4.17)

When $c\neq 0$ , for

(\frac{1}{p},\frac{1}{q})\in\biggl{\{}(\frac{1}{p},\frac{1}{q})\in\Delta_{M}:\frac{h_{\Phi}+1}{p}-\frac{1}{q}-1<0\biggl{\}},

(4.18)

there holds

\sum_{k\geq 1}\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k}}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim\sum_{k\geq 1}2^{\frac{(m+2)k}{p}-{\frac{2k}{q}}-2k}\lesssim 1.

(4.19)

Then we finish the proof of Theorem 1.3.

5 Proofs of Theorem 1.6 and Theorem 1.7

Proof of Theorem 1.6: A simple calculation implies that $H\Psi$ may vanish along the curve $x_{2}=x_{1}^{d}\phi(x_{1})$ and the straight line $x_{1}=0$ . Hence we decompose the averaging operator $A_{t}$ as follows

A_{t}f(y)=\sum_{i=1}^{3}A_{t}^{i}f(y).

Here, we define the operators $A_{t}^{i}$ , $i=1,2,3$ respectively as (without of generality, we may assume that $x_{1}>0$ ):

A_{t}^{1}f(y):=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(x_{2}-x_{1}^{d}\phi(x_{1}))^{m})\biggl{(}1-\chi_{0}\biggl{(}2^{-N}\frac{x_{2}}{x_{1}^{d}\phi(x_{1})}\biggl{)}\biggl{)}\eta(x_{1},x_{2})dx_{1}dx_{2},

A_{t}^{2}f(y):=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(x_{2}-x_{1}^{d}\phi(x_{1}))^{m})\chi_{0}\biggl{(}2^{N}\frac{x_{2}-x_{1}^{d}\phi(x_{1})}{x_{1}^{d}\phi(x_{1})}\biggl{)}\eta(x_{1},x_{2})dx_{1}dx_{2},

and

	$\displaystyle A_{t}^{3}f(y):$	$\displaystyle=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(x_{2}-x_{1}^{d}\phi(x_{1}))^{m})\biggl{(}\chi_{0}\biggl{(}2^{-N}\frac{x_{2}}{x_{1}^{d}\phi(x_{1})}\biggl{)}-\chi_{0}\biggl{(}2^{N}\frac{x_{2}-x_{1}^{d}\phi(x_{1})}{x_{1}^{d}\phi(x_{1})}\biggl{)}\biggl{)}$
		$\displaystyle\quad\quad\times\eta(x_{1},x_{2})dx_{1}dx_{2},$

where $0\leq\chi_{0}\leq 1$ is a smooth cutoff function, $\chi_{0}=1$ on [-1,1] and supp $\chi_{0}\subset[-2,2]$ , and the positive integer $N$ can be sufficiently large. Next we will demonstrate that the maximal operators $\sup_{t>0}|A_{t}^{i}|$ are $L^{p}(\mathbb{R}^{3})$ -bounded for $p>\max\{\frac{2d}{d+1},\frac{3}{2}\}$ , $i=1,2,3$ .

We first consider the maximal operator $\sup_{t>0}|A_{t}^{3}|$ . By using the dyadic decomposition and isometric transformation, we obtain

\biggl{\|}\sup_{t>0}|A_{t}^{3}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{p}(\mathbb{R}^{3})}\leq\sum_{k\geq 1}2^{-(d+1)k}\biggl{\|}\sup_{t>0}|A_{t,k}^{3}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{p}(\mathbb{R}^{3})},

(5.1)

where the averaging operator $A_{t,k}^{3}$ is given by

	$\displaystyle A_{t,k}^{3}f(y):$	$\displaystyle=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(x_{2}-x_{1}^{d}\phi(2^{-k}x_{1}))^{m})\tilde{\rho}(x_{1},x_{2})$
		$\displaystyle\quad\quad\times\biggl{(}\chi_{0}\biggl{(}2^{-N}\frac{x_{2}}{x_{1}^{d}\phi(2^{-k}x_{1})}\biggl{)}-\chi_{0}\biggl{(}2^{N}\frac{x_{2}-x_{1}^{d}\phi(2^{-k}x_{1})}{x_{1}^{d}\phi(2^{-k}x_{1})}\biggl{)}\biggl{)}\eta(2^{-k}x_{1},2^{-dk}x_{2})dx_{1}dx_{2},$

and $\tilde{\rho}$ is a smooth cutoff function supported in $\{(x_{1},x_{2})\in\mathbb{R}^{2}:1/2<|(x_{1},x_{2})|<2\}$ . It is easy to see that the Heissian matrix of the function $(x_{2}-x_{1}^{d}\phi(2^{-k}x_{1}))^{m}$ degenerates only along the curve $x_{2}=x_{1}^{d}\phi(2^{-k}x_{1})$ and the straight line $x_{1}=0$ . Therefore, the hypersurface

\{(x_{1},x_{2},(x_{2}-x_{1}^{d}\phi(2^{-k}x_{1}))^{m}):1/2<|(x_{1},x_{2})|<2\}

possesses a non-vanishing Gaussian curvature everywhere when $(x_{1},x_{2})$ is restricted to the support of the function

\chi_{0}\biggl{(}2^{-N}\frac{x_{2}}{x_{1}^{d}\phi(2^{-k}x_{1})}\biggl{)}-\chi_{0}\biggl{(}2^{N}\frac{x_{2}-x_{1}^{d}\phi(2^{-k}x_{1})}{x_{1}^{d}\phi(2^{-k}x_{1})}\biggl{)}.

It follows that the maximal operator $\sup_{t>0}|A_{t,k}^{3}|$ is $L^{p}(\mathbb{R}^{3})$ -bounded if $p>3/2$ . According to (5.1), the result also holds for the maximal operator $\sup_{t>0}|A_{t}^{3}|$ .

So it is sufficient to estimate the maximal operators $\sup_{t>0}|A_{t}^{1}|$ and $\sup_{t>0}|A_{t}^{2}|$ . Theorem 1.6 follows from the two lemmas below.

Lemma 5.1.

For each $p>\max\{3/2,2d/(d+1)\}$ and $i=1,2$ , there exists a constant $C_{p}>0$ such that

\biggl{\|}\sup_{t\in[1,2]}|A_{t}^{i}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{p}(\mathbb{R}^{3})}\leq C_{p}.

(5.2)

Lemma 5.2.

For each $i=1,2$ , there exists $q_{0}>p_{0}\geq 1$ and a constant $C_{p_{0},q_{0}}>0$ such that

\biggl{\|}\sup_{t\in[1,2]}|A_{t}^{i}|\biggl{\|}_{L^{p_{0}}(\mathbb{R}^{3})\rightarrow L^{q_{0}}(\mathbb{R}^{3})}\leq C_{p_{0},q_{0}}.

(5.3)

Combining the above lemmas with the similar arguments as in [12], we can get the weighted results for the global maximal operators $\sup_{t>0}|A^{i}_{t}|$ . More details can be found in the appendix of this paper. In what follows, we first prove Lemma 5.1 and Lemma 5.2 for $i=1$ . By dyadic decomposition, isometric transform and a simple changing coordinates, for each $q\geq p\geq 1$ , there holds

\biggl{\|}\sup_{t\in[1,2]}|A_{t}^{1}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq\sum_{l\geq 1}\sum_{1\leq k\leq dl-N}2^{[(m+1)k+l](\frac{1}{p}-\frac{1}{q})}2^{-l-k}\biggl{\|}\sup_{t\in[1,2]}|A_{t,k,l}^{1}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})},

(5.4)

where

A_{t,k,l}^{1}f(y):=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-t(x_{2}+2^{k-dl}x_{1}^{d}\phi(2^{-l}x_{1})),y_{3}-tx_{2}^{m})\eta_{k,l}^{1}(x_{1},x_{2})dx_{1}dx_{2},

and

	$\displaystyle\eta_{k,l}^{1}(x_{1},x_{2})$	$\displaystyle=\tilde{\rho}(x_{1})\tilde{\rho}(x_{2}+2^{k-dl}x_{1}^{d}\phi(2^{-l}x_{1}))\biggl{(}1-\chi_{0}\biggl{(}\frac{x_{2}+2^{k-dl}x_{1}^{d}\phi(2^{-l}x_{1})}{2^{k-dl+N}x_{1}^{d}\phi(2^{-l}x_{1})}\biggl{)}\biggl{)}$
		$\displaystyle\quad\quad\times\eta(2^{-l}x_{1},2^{-k}x_{2}+2^{-dl}x_{1}^{d}\phi(2^{-l}x_{1})),$

and $\tilde{\rho}$ is a smooth function supported in $[-2,-1/2]\cup[1/2,2]$ .

By Littlewood-Paley decomposition, we define $A_{t,k,l,j}^{1}$ as in equality (2.1), and we are reduced to considering the maximal operator $\sup_{t>0}|A_{t,k,l,j}^{1}|$ , $j\geq 1$ . Let

\widehat{d\mu_{k,l}}(\xi)=\int_{\mathbb{R}^{2}}e^{-ix_{1}\xi_{1}-i[x_{2}+2^{k-dl}x_{1}^{d}\phi(2^{-l}x_{1})]\xi_{2}-ix_{2}^{m}\xi_{3}}\eta_{k,l}^{1}(x_{1},x_{2})dx_{1}dx_{2}.

When $|\xi_{1}|+|\xi_{2}|\gg|\xi_{3}|$ , we can obtain

\biggl{|}\nabla_{x_{1},x_{2}}[x_{1}\xi_{1}+[x_{2}+2^{k-dl}x_{1}^{d}\phi(2^{-l}x_{1})]\xi_{2}+x_{2}^{m}\xi_{3}]\biggl{|}\geq\biggl{|}|\xi_{1}|-2^{k-dl}|\xi_{2}|\biggl{|}+\biggl{|}|\xi_{2}|-|\xi_{3}|\biggl{|}\gtrsim|\xi|,

and by performing integration by parts, we achieve rapid decay of $\widehat{d\mu_{k,l}}(\xi)$ . Therefore, we only need to consider the main contribution term where $|\xi_{1}|+|\xi_{2}|\lesssim|\xi_{3}|$ . By an abuse of notation, we define the main contribution term by

A_{t,k,l,j}^{1}f(y)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\widehat{d\mu_{k,l}}(t\xi)\chi_{0}\biggl{(}\frac{|\xi_{1}|+|\xi_{2}|}{|\xi_{3}|}\biggl{)}\beta(2^{-j}|\xi|)\hat{f}(\xi)d\xi.

By scaling, there holds

\biggl{\|}\sup_{t\in[1,2]}|A_{t,k,l,j}^{1}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\lesssim 2^{(dl-k)(\frac{1}{q}-\frac{1}{p})}\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}.

(5.5)

Here,

\widetilde{A_{t,k,l,j}^{1}}f(y)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\widehat{d\mu_{k,l}}(t\delta_{2^{k-dl}}\xi)\chi_{0}\biggl{(}\frac{2^{k-dl}|\xi_{1}|+|\xi_{2}|}{|\xi_{3}|}\biggl{)}\beta(2^{-j}|\delta_{2^{k-dl}}\xi|)\hat{f}(\xi)d\xi,

and $\delta_{2^{k-dl}}\xi=(2^{k-dl}\xi_{1},\xi_{2},\xi_{3})$ . By the stationary phase method and Sobolev embedding, it suffices to estimate the corresponding Fourier integral operators. More concretely, when $1\leq j\leq dl-k$ ,

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq 2^{\frac{j}{q}-\frac{j}{2}}\biggl{\|}\rho(t)F_{j,k,l}^{1,1}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3}\times[1/2,4])},

(5.6)

where the bump function $\rho$ is supported in $[1/2,4]$ and the Fourier integral operator $F_{k,l,j}^{1,1}$ is defined by

	$\displaystyle F_{k,l,j}^{1,1}f(y,t)$	$\displaystyle=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+it\xi_{3}\Psi_{2}(\xi_{2}/\xi_{3})}a_{k,l,j,1}(t\xi)\tilde{\chi}\biggl{(}\frac{\xi_{2}}{\xi_{3}}\biggl{)}\chi_{0}\biggl{(}\frac{2^{k-dl}\|\xi_{1}\|+\|\xi_{2}\|}{\|\xi_{3}\|}\biggl{)}$
		$\displaystyle\quad\times\beta(2^{-j}\|\delta_{2^{k-dl}}\xi\|)\beta_{0}(2^{k-dl}\|\xi_{1}\|)\hat{f}(\xi)d\xi,$

$\Psi_{2}(\xi_{2}/\xi_{3})=c_{m}(-\xi_{2}/\xi_{3})^{m/(m-1)}$ and $a_{k,l,j,1}$ is a symbol of order zero. When $j>dl-k$ ,

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq 2^{\frac{j}{q}-j+\frac{dl-k}{2}}\biggl{\|}\rho(t)F_{j,k,l}^{1,2}\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3}\times[1/2,4])},

(5.7)

where

	$\displaystyle F_{k,l,j}^{1,2}f(y,t)$	$\displaystyle=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi+it2^{k-dl}\xi_{2}\Psi_{1}(\xi_{1}/\xi_{2})+it\xi_{3}\Psi_{2}(\xi_{2}/\xi_{3})}a_{k,l,j,2}(t\xi)\tilde{\chi}\biggl{(}\frac{\xi_{1}}{\xi_{2}}\biggl{)}\tilde{\chi}\biggl{(}\frac{\xi_{2}}{\xi_{3}}\biggl{)}$
		$\displaystyle\quad\quad\times\chi_{0}\biggl{(}\frac{2^{k-dl}\|\xi_{1}\|+\|\xi_{2}\|}{\|\xi_{3}\|}\biggl{)}\beta(2^{-j}\|\delta_{2^{k-dl}}\xi\|)\hat{f}(\xi)d\xi,$

$\Psi_{1}(\xi_{1}/\xi_{2})$ is a small perturbation of $c_{d}(-\xi_{1}/\xi_{2})^{d/(d-1)}$ , and $a_{k,l,j,2}$ is a symbol of order zero.

By replacing $2^{-Ml}$ with $2^{k-dl}$ , a similar proof implies that the results in Theorem 2.1 remain valid for Fourier integral operators $F_{k,l,j}^{1,1}$ and $F_{k,l,j}^{1,2}$ .

By the estimate (E1) in Theorem 2.1, for $1\leq j\leq dl-k$ , we have

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3})}\lesssim 1,

(5.8)

and for $j>dl-k$ ,

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3})}\lesssim 2^{-\frac{j}{2}}2^{\frac{dl-k}{2}}.

(5.9)

The estimates (E5) and (E4) in Theorem 2.1 imply that for each $j\geq 1$ , there hold

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{1}(\mathbb{R}^{3})\rightarrow L^{1}(\mathbb{R}^{3})}\lesssim 2^{j},

(5.10)

and

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{\infty}(\mathbb{R}^{3})\rightarrow L^{\infty}(\mathbb{R}^{3})}\lesssim 1.

(5.11)

The interpolation argument implies that inequality (5.2) holds for $p>\max\{3/2,2d/(d+1)\}$ and $i=1$ .

By the estimate (E2) in Theorem 2.1, for each $j\geq 1$ , there holds

\biggl{\|}\sup_{t\in[1,2]}|\widetilde{A_{t,k,l,j}^{1}|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})\rightarrow L^{6}(\mathbb{R}^{3})}\lesssim 2^{\frac{dl-k}{2}}.

(5.12)

Interpolation with the $L^{2}$ -estimates (5.8) and (5.9) implies that when $\frac{1}{2}-\frac{1}{dm+d+1}<\frac{1}{q}\leq\frac{1}{2}$ , inequality (5.3) for $i=1$ holds. This completes the proof of Lemma 5.2 for $i=1$ .

Now let’s turn to consider the maximal operator $\sup_{t\in[1,2]}|A_{t}^{2}|$ , there holds

\biggl{\|}\sup_{t\in[1,2]}|A_{t}^{2}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}\leq\sum_{k\geq 1}\sum_{1\leq l\leq\frac{k-N}{d}}2^{[(d+1)l+mk](\frac{1}{p}-\frac{1}{q})}2^{-l-k}\biggl{\|}\sup_{t\in[1,2]}|A_{t,k,l}^{2}|\biggl{\|}_{L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})}.

(5.13)

Here,

A_{t,k,l}^{2}f(y):=\int_{\mathbb{R}^{2}}f(y_{1}-tx_{1},y_{2}-t(2^{dl-k}x_{2}+x_{1}^{d}\phi(2^{-l}x_{1})),y_{3}-tx_{2}^{m})\eta_{k,l}^{2}(x_{1},x_{2})dx_{1}dx_{2},

and

\displaystyle\eta_{k,l}^{2}(x_{1},x_{2})

\displaystyle=\tilde{\rho}(x_{1})\tilde{\rho}(x_{2})\chi_{0}\biggl{(}2^{N}2^{dl-k}\frac{x_{2}}{x_{1}^{d}\phi(2^{-l}x_{1})}\biggl{)}\eta(2^{-l}x_{1},2^{-k}x_{2}+2^{-dl}x_{1}^{d}\phi(2^{-l}x_{1})).

By an argument similar to that of $\sup_{t\in[1,2]}|A_{t}^{1}|$ , we can get $L^{p}(\mathbb{R}^{3})$ -boundedness of $\sup_{t\in[1,2]}|A_{t}^{2}|$ provided that $p>3/2$ , and $L^{2}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ -boundedness of $\sup_{t\in[1,2]}|A_{t}^{2}|$ for $\frac{1}{2}-\frac{d}{md+d+1}<\frac{1}{q}\leq\frac{1}{2}$ .

Then we completed the proof of the positive results in Theorem 1.6.

Finally, we will show that $\sup_{t>0}|A_{t}|$ cannot be $L^{p}(\mathbb{R}^{3})$ -bounded for $p<2d/(d+1)$ when $d\geq 3$ . By a change of coordinates, it is equivalent to consider the maximal operator $\sup_{t>0}|\widetilde{A}_{t}|$ , where $\widetilde{A_{t}}$ is the averaging operator along the hypersurface $\{(x_{1},x_{2}+x_{1}^{d}\phi(x_{1}),x_{2}^{m}):(x_{1},x_{2})\in U\}$ , $U$ is a small neighborhood of the origin. Let $\tilde{U}=(0,\delta)\times(\epsilon/2,\epsilon)\subset U$ , $0<\delta\ll\epsilon\ll 1$ . Let

T=\{t(x_{1},x_{2},x_{2}^{m}):(x_{1},x_{2})\in\tilde{U},t\in[1,2]\}.

It is obvious that $|T|\sim_{\epsilon}\delta$ , where the implied constants depend on $\epsilon$ . We define

D=[-10\delta,10\delta]\times[-10\delta^{d},10\delta^{d}]\times[-10m\delta^{d},10m\delta^{d}],

and $f=\chi_{D}$ which is the characteristic function over $D$ . Then

\|f\|_{L^{p}(\mathbb{R}^{3})}\sim\delta^{\frac{2d+1}{p}}.

(5.14)

Notice that for each $y\in T$ , there exists a $s_{y}\in[1,2]$ and $(z_{1},z_{2})\in\tilde{U}$ (which depend on $y$ ) such that $y=s_{y}(z_{1},z_{2},z_{2}^{m})$ . Then there holds

	$\displaystyle\sup_{t>0}\|\widetilde{A_{t}}f(y)\|$	$\displaystyle\geq\int_{\tilde{U}}\chi_{D}(s_{y}(z_{1}-x_{1},z_{2}-x_{2}-x_{1}^{d}\phi(x_{1}),z_{2}^{m}-x_{2}^{m}))dx_{1}dx_{2}$
		$\displaystyle\geq\int_{\tilde{U}(z_{1},z_{2})}\chi_{D}(s_{y}(z_{1}-x_{1},z_{2}-x_{2}-x_{1}^{d}\phi(x_{1}),z_{2}^{m}-x_{2}^{m}))dx_{1}dx_{2},$		(5.15)

where $\tilde{U}(z_{1},z_{2})=\{(x_{1},x_{2})\in\tilde{U}:|x_{1}-z_{1}|\leq\delta/2,|x_{2}-z_{2}|\leq\delta^{d}/2\}$ . It is clear that $|\tilde{U}(z_{1},z_{2})|\gtrsim\delta^{d+1}$ . Moreover, for each $(x_{1},x_{2})\in\tilde{U}(z_{1},z_{2})$ , there holds

|s_{y}(z_{1}-x_{1})|\leq 10\delta,\quad|s_{y}(z_{2}^{m}-x_{2}^{m})|\leq 10m\delta^{d},

|s_{y}(z_{2}-x_{2}-x_{1}^{d}\phi(x_{1}))|\leq 10\delta^{d},

which implies that $s_{y}(z_{1}-x_{1},z_{2}-x_{2}-x_{1}^{d}\phi(x_{1}),z_{2}^{m}-x_{2}^{m})\in D$ . Hence for each $y\in T$ , there holds

\sup_{t>0}|\widetilde{A_{t}}f(y)|\geq|\tilde{U}(z_{1},z_{2})|\gtrsim\delta^{d+1}.

Furthermore,

\biggl{\|}\sup_{t>0}|\widetilde{A_{t}}f|\biggl{\|}_{L^{p}(\mathbb{R}^{3})}\geq\biggl{\|}\sup_{t>0}|\widetilde{A_{t}}f|\biggl{\|}_{L^{p}(T)}\gtrsim_{\epsilon}\delta^{d+1+\frac{1}{p}}.

(5.16)

If $\sup_{t>0}|\widetilde{A_{t}}|$ is $L^{p}(\mathbb{R}^{3})$ bounded, then the inequalities (5.14) and (5.16) imply that

\delta^{d+1+\frac{1}{p}}\lesssim_{\epsilon}\delta^{\frac{2d+1}{p}}.

Then it follows that $\frac{1}{p}\leq\frac{d+1}{2d}$ since $\delta$ can be sufficiently small.

When $d=2$ , we can get the necessity of $p>3/2$ by modifying the above arguments, i.e., just choose $\tilde{U}=(0,\epsilon^{\prime})\times(\epsilon,2\epsilon)$ , $T=\{t(x_{1},x_{2},(x_{2}-x_{1}^{2}\phi(x_{1}))^{m}):(x_{1},x_{2})\in\tilde{U},t\in[1,2]\}$ , $\tilde{U}(z_{1},z_{2})=\{(x_{1},x_{2})\in\tilde{U}:|z_{i}-x_{i}|<\frac{\delta}{2},i=1,2\}$ , and $f$ as the characteristic function over $B(0,\delta)$ with $0<\delta\ll\epsilon^{\prime}\ll\epsilon\ll 1$ .

We will prove Theorem 1.7 in the rest of this section.

Proof of Theorem 1.7: Our construction of $\gamma(x_{1})$ was inspired by Example 6.1.1 in [16]. Let $\chi$ be a smooth cutoff function, $0\leq\chi\leq 1$ , $\chi=1$ on $[-1,1]$ and supp $\chi\subset[-2,2]$ . For each $k\in\mathbb{N}$ , set the dyadic interval $I_{k}=[2^{-k-1},2^{-k}]$ with center $c(I_{k})=3/4\cdot 2^{-k}$ . Put

\gamma(x_{1})=\sum_{k\geq 1}2^{-2^{k}}\chi\biggl{(}\frac{x_{1}-c(I_{k})}{2^{-k-10}}\biggl{)}.

It is clear that $\gamma(x_{1})$ is a smooth function and $\gamma^{(j)}(0)=0$ for every integer $j\geq 0$ . We notice that for each $k\in\mathbb{N}$ , $\gamma(x_{1})=2^{-2^{k}}$ on $\tilde{I_{k}}=[-2^{-k-10}+c(I_{k}),2^{-k-10}+c(I_{k})]$ .

For arbitrary small neighborhood $U$ of the origin, we choose $k$ sufficiently large such that $\tilde{I_{k}}\times(0,\delta)\subset U$ , where $\delta\ll 2^{-2^{k}}$ which is allowed to be sufficiently small. Let the vectors

e_{1}=(1,0,0),\quad\quad e_{2}=(0,1,m(-2^{-2^{k}})^{m-1}),\quad\quad e_{3}=(0,-m(-2^{-2^{k}})^{m-1},1),

and the rectangle

D=\{y\in\mathbb{R}^{3}:|y\cdot e_{1}|\leq 10\cdot 2^{-k};|y\cdot e_{2}|\leq 10\cdot\delta;|y\cdot e_{3}|\leq 10m^{2}\cdot 2^{-(m-2)2^{k}}\delta^{2}\}.

For each $t\in[1,2]$ , set

	$\displaystyle D_{t}$	$\displaystyle=\{y\in\mathbb{R}^{3}:\|[y-t(0,0,(-2^{-2^{k}})^{m})]\cdot e_{1}\|\leq 2^{-k};\|[y-t(0,0,(-2^{-2^{k}})^{m})]\cdot e_{2}\|\leq\delta;$
		$\displaystyle\quad\|[y-t(0,0,(-2^{-2^{k}})^{m})]\cdot e_{3}\|\leq 2^{-(m-2)2^{k}}\delta^{2}\}.$

Notice that if $t,t^{\prime}\in[1,2]$ and $|t-t^{\prime}|\geq 10\cdot 2^{2\cdot 2^{k}}\delta^{2}$ , then $D_{t}\cap D_{t^{\prime}}=\emptyset$ .

Let $f=\chi_{D}$ be the characteristic function over the set $D$ . Then

\|f\|_{L^{p}(\mathbb{R}^{3})}\sim_{k}\delta^{3/p},

(5.17)

where the implied constant depends on $k$ (notice that $k$ is fixed). Moreover, for fixed $t\in[1,2]$ and every $y\in D_{t}$ , $(x_{1},x_{2})\in\tilde{I_{k}}\times(0,\delta)$ , there holds

|[y-t(x_{1},x_{2},(x_{2}-\gamma(x_{1}))^{m})]\cdot e_{1}|\leq|[y-t(0,0,(-2^{-2^{k}})^{m})]\cdot e_{1}|+2\cdot 2^{-k}\leq 3\cdot 2^{-k},

|[y-t(x_{1},x_{2},(x_{2}-\gamma(x_{1}))^{m})]\cdot e_{2}|\leq|[y-t(0,0,(-2^{-2^{k}})^{m})]\cdot e_{2}|+2\delta+2m2^{-(m-1)2^{k}}\delta\leq 3\cdot\delta,

|[y-t(x_{1},x_{2},(x_{2}-\gamma(x_{1}))^{m})]\cdot e_{3}|\leq|[y-t(0,0,(-2^{-2^{k}})^{m})]\cdot e_{3}|+2m(m-1)2^{-(m-2)2^{k}}\delta^{2}\leq 2m^{2}\cdot 2^{-(m-2)2^{k}}\delta^{2}.

Hence,

A_{t}f(y)\geq\int_{\tilde{I_{k}}\times(0,\delta)}\chi_{D}(y_{1}-tx_{1},y_{2}-tx_{2},y_{3}-t(x_{2}-\gamma(x_{1}))^{m})dx_{1}dx_{2}\gtrsim_{k}\delta.

We choose a sequence $\{t_{i}\}_{i\geq 1}\subset[1,2]$ such that $t_{i+1}-t_{i}=10\cdot 2^{2\cdot 2^{k}}\delta^{2}$ , there holds

\displaystyle\biggl{\|}\sup_{t\in[1,2]}|A_{t}f(y)|\biggl{\|}_{L^{p}(\cup_{i}D_{t_{i}})}\gtrsim_{k}|\cup_{i}D_{t_{i}}|^{1/p}\delta\gtrsim_{k}\delta^{1+1/p}.

(5.18)

Finally, Theorem 1.7 follows from inequality (5.17) and (5.18) since $\delta$ can be sufficiently small.

6 Appendix

This appendix consists of two parts. Firstly, we provide some counterexamples related to Theorem 1.2 and Theorem 1.3, and discuss the necessary conditions for the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of the corresponding local maximal operators. Secondly, for the completeness of the paper, we explain how to prove the Hölder continuity property starting from the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded estimates of the local maximal operators in this paper. Once we obtain the Hölder continuity property of the local maximal operator, we can immediately obtain the weighted estimate for the corresponding global maximal operator by combining it with Theorem 1.20 and Corollary 1.21 in [12].

6.1 Some counterexamples and necessary conditions

Theorem 6.1.

Let $\nu_{1}>\nu_{2}\geq 2$ , $c\in\mathbb{R}$ , $U\subset\mathbb{R}^{2}$ be a small neighborhood of the origin $(0,0)$ , $S=\{(x_{1},x_{2},x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}+c):(x_{1},x_{2})\in U\}$ . Then the local maximal operator associated with $S$ cannot be $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded if $(\frac{1}{p},\frac{1}{q})\notin\overline{\Delta_{0}}$ . Moreover, when $c=0$ , $\frac{\nu_{1}+1}{p}-\frac{\nu_{1}+1}{q}-1\leq 0$ is necessary for the local maximal operator to be $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded; when $c\neq 0$ , $\frac{\nu_{1}+1}{p}-\frac{1}{q}-1\leq 0$ is a necessary condition for the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of the local maximal operator.

Proof. Let

A_{t}f(y)=\int_{U}f(y-t(x_{1},x_{2},x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}+c))dx_{1}dx_{2}.

We first show that $\sup_{t\in[1,2]}|A_{t}|$ cannot be $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded unless $\frac{1}{p}\leq\frac{2}{3}$ , $\frac{1}{q}\leq\frac{1}{p}\leq\frac{3}{q}$ and $\frac{1}{q}\geq\frac{2}{p}-1$ . By selecting the function $f$ as the characteristic function on balls $B(0,\delta^{-1})$ and the set $T$ as the $\delta^{-1}$ neighborhood of $S$ , where $\delta>0$ can be sufficiently small, if $\sup_{t\in[1,2]}|A_{t}|$ is $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded, then we can obtain $|T|^{\frac{1}{q}}\lesssim|B(0,\delta^{-1})|^{\frac{1}{p}}$ , it follows the necessity of $\frac{1}{q}\leq\frac{1}{p}$ . In order to establish the necessary condition $\frac{1}{p}\leq\frac{3}{q}$ , we choose the function $f$ as the characteristic function on the set $T$ , where the set $T$ is defined as the $\delta$ neighborhood of $S$ . It is important to note that for any $y$ belonging to the ball $B(0,\delta)$ and $(x_{1},x_{2})\in U$ , $y+(x_{1},x_{2},x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}+c)$ is an element of $T$ . Hence the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ implies the validity of $|B(0,\delta)|^{\frac{1}{q}}\lesssim|T|^{\frac{1}{p}}$ , which means that the condition $\frac{1}{p}\leq\frac{3}{q}$ is necessary.

To establish the necessary condition $\frac{1}{p}\leq\frac{2}{3}$ , we define the function $f$ as the characteristic function on the ball $B(0,\delta)$ , and the set $T=\{t(x_{1},x_{2},x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}+c):t\in[1,2],(x_{1},x_{2})\in(\frac{\epsilon}{2},\epsilon)\times(\frac{\epsilon}{2},\epsilon)\}$ , $\delta\ll\epsilon\ll 1$ . Then $|T|\sim\epsilon^{2}$ , and for each $y\in T$ , there exists a $t(y)\in[1,2]$ and $(z_{1},z_{2})\in(\frac{\epsilon}{2},\epsilon)\times(\frac{\epsilon}{2},\epsilon)$ (which depend on $y$ ) such that $y=t(y)(z_{1},z_{2},z_{1}^{\nu_{1}}z_{2}^{\nu_{2}}+c)\in T$ , there holds

\sup_{t\in[1,2[}|A_{t}f(y)|\geq|A_{t(y)}f(y)|\geq|U(z_{1},z_{2})|,

where the set $U(z_{1},z_{2})$ is defined by $U(z_{1},z_{2})=\{(x_{1},x_{2}):|x_{i}-z_{i}|<\delta,i=1,2\}$ with measure $\delta^{2}$ . Then it follows from the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ that $|T|^{\frac{1}{q}}|U(z_{1},z_{2})|\lesssim|B(0,\delta)|^{\frac{1}{p}}$ , which requires $\frac{1}{p}\leq\frac{2}{3}$ since $\delta$ can be sufficiently small.

Next we will prove that the condition $\frac{1}{q}\leq\frac{2}{p}-1$ is necessary. We choose $f$ as the characteristic function on the set $D=\{z\in\mathbb{R}^{3}:|z\cdot e_{1}|\leq 10\delta,|z\cdot e_{2}|\leq 10\delta,|z\cdot e_{3}|\leq 10\delta^{2}\}$ , where $e_{1}=(1,0,\nu_{1}\epsilon^{\nu_{1}+\nu_{2}-1})$ , $e_{2}=(0,1,\nu_{2}\epsilon^{\nu_{1}+\nu_{2}-1})$ , $e_{3}=(-\nu_{1}\epsilon^{\nu_{1}+\nu_{2}-1},-\nu_{2}\epsilon^{\nu_{1}+\nu_{2}-1},1)$ , $\delta\ll\epsilon\ll 1$ . Let $R_{t}=\{y\in\mathbb{R}^{3}:|(y-t(\epsilon,\epsilon,\epsilon^{\nu_{1}+\nu_{2}}+c))\cdot e_{1}|\leq\delta,|(y-t(\epsilon,\epsilon,\epsilon^{\nu_{1}+\nu_{2}}+c))\cdot e_{2}|\leq\delta,|(y-t(\epsilon,\epsilon,\epsilon^{\nu_{1}+\nu_{2}}+c))\cdot e_{3}|\leq\delta^{2}\}$ , $t\in[1,2]$ . Then for each $(x_{1},x_{2})\in U(\epsilon):=\{(x_{1},x_{2}):|x_{i}-\epsilon|\leq\delta,i=1,2\}$ and $y\in R_{t}$ , $t\in[1,2]$ , we have $y-t(x_{1},x_{2},x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}+c)\in D$ . Hence the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ yields $|\cup_{t\in[1,2]}R_{t}|^{\frac{1}{q}}|U(\epsilon)|\lesssim|D|^{\frac{1}{p}}$ . This implies that $1+\frac{1}{q}\geq\frac{2}{p}$ since $|\cup_{t\in[1,2]}R_{t}|\sim_{\epsilon}\delta^{2}$ .

Finally, in order to complete the proof of the theorem, we choose $f$ as the characteristic function on the set $D=(-10\delta,10\delta)\times(-\epsilon,\epsilon)\times(-10\delta^{\nu_{1}},10\delta^{\nu_{1}})$ , and the set $U=(0,\delta)\times(0,\epsilon)$ , $\delta\ll\epsilon\ll 1$ . When $c=0$ , for each $(x_{1},x_{2})\in U$ , and $y\in R:=(0,\delta)\times(0,\epsilon)\times(0,\delta^{\nu_{1}})$ , we have $y-(x_{1},x_{2},x_{1}^{\nu_{1}})\in D$ . Then it follows from the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ that $|R|^{\frac{1}{q}}|U|\lesssim|D|^{\frac{1}{p}}$ . As a result, we have obtained the necessity of $\frac{\nu_{1}+1}{p}-\frac{\nu_{1}+1}{q}-1\leq 0$ . When $c\neq 0$ , we choose the set $R_{t}=(0,\delta)\times(0,\epsilon)\times(tc,\delta^{\nu_{1}}+tc)$ , $t\in[1,2]$ . Then for each $y\in R_{t}$ , $t\in[1,2]$ and $(x_{1},x_{2})\in U$ , we have $y-t(x_{1},x_{2},x_{1}^{\nu_{1}}+c)\in D$ . Then the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ implies $|\cup_{t\in[1,2]}R_{t}|^{\frac{1}{q}}|U|\lesssim|D|^{\frac{1}{p}}$ , which dedicates the necessity of $\frac{\nu_{1}+1}{p}-\frac{1}{q}-1\leq 0$ since $|\cup_{t\in[1,2]}R_{t}|\sim_{\epsilon}\delta$ . $\Box$

Theorem 6.2.

Let $M\geq 6$ , $\epsilon\ll 1$ , $S=\{(x_{1},x_{2},x_{1}^{M}+x_{2}^{M}+c):(x_{1},x_{2})\in U\}$ . Here, $U$ is a small neighborhood of the origin $(0,0)$ . The local maximal operator along $S$ cannot be $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded unless $\frac{1}{q}\leq\frac{1}{p}\leq\frac{3}{q}$ . Furthermore, when $c\neq 0$ , we have an additional necessary condition that $(\frac{1}{p},\frac{1}{q})$ satisfies $\frac{M/2+1}{p}-\frac{1}{q}-1\leq 0$ ; when $c=0$ , $\frac{1}{p}\leq\frac{2M}{M+1}$ and $\frac{M+2}{p}-\frac{M+2}{q}-2\leq 0$ are also necessary for the local maximal operator along $S$ to be $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded.

Proof. The argument on the necessity of condition $\frac{1}{q}\leq\frac{1}{p}\leq\frac{3}{q}$ in Theorem 6.1 applies to the maximal operator in Theorem 6.2, we prefer to omit its proof here. Let

A_{t}f(y)=\int_{U}f(y-t(x_{1},x_{2},x_{1}^{M}+x_{2}^{M}+c))dx_{1}dx_{2}.

We first consider the case $c\neq 0$ . Let $f$ be the characteristic function on the set $D:=(-10\delta,10\delta)\times(-10\delta,10\delta)\times(-10\delta^{M},10\delta^{M})$ . For each $y\in R_{t}:=(-\delta,\delta)\times(-\delta,\delta)\times(-\delta^{M}+tc,\delta^{M}+tc)$ , $(x_{1},x_{2})\in\tilde{U}=(-\delta,\delta)\times(-\delta,\delta)$ and $t\in[1,2]$ , we have $y-t(x_{1},x_{2},x_{1}^{M}+x_{2}^{M}+c)$ belongs to $D$ . Hence the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ implies that $|\cup_{t\in[1,2]}R_{t}|^{\frac{1}{q}}|\tilde{U}|\lesssim|D|^{\frac{1}{p}}$ , which requires $\frac{M/2+1}{p}-\frac{1}{q}-1\leq 0$ .

Secondly, when $c=0$ , we will prove the necessity of conditions $\frac{M+2}{p}-\frac{M+2}{q}-2\leq 0$ and $\frac{1}{p}\leq\frac{M+1}{2M}$ . Given any $(x_{1},x_{2})\in(0,\delta)\times(0,\delta)$ , $y\in(0,\delta)\times(0,\delta)\times(0,\delta^{M})$ and $t\in[1,2]$ , we have $y-t(x_{1},x_{2},x_{1}^{M}+x_{2}^{M})\in D=(-10\delta,10\delta)\times(-10\delta,10\delta)\times(-10\delta^{M},10\delta^{M})$ . Therefore, we choose $f$ as the characteristic function on the set D, then the $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ implies the necessity of the condition $\frac{M+2}{p}-\frac{M+2}{q}-2\leq 0$ . In order to illustrate the necessity of $\frac{1}{p}\leq\frac{M+1}{2M}$ , we choose $f$ as the characteristic function on the set $D:=(-10\delta,10\delta)\times(-10\delta^{M},10\delta^{M})\times(-10M\delta^{M},10M\delta^{M})$ . For any given $y=t(y)(z_{1},z_{2},z_{2}^{M})\in T:=\{t(z_{1},z_{2},z_{2}^{M}):(z_{1},z_{2})\in(0,\delta)\times(\frac{\epsilon}{2},\epsilon),t\in[1,2]\}$ , $\delta\ll\epsilon\ll 1$ , and $(x_{1},x_{2})\in U(z_{1},z_{2}):=(z_{1}-\delta,z_{1}+\delta)\times(z_{2}-\delta^{M},z_{2}+\delta^{M})$ , we have $y-t(y)(x_{1},x_{2},x_{1}^{M}+x_{2}^{M})\in D$ . Then the $L^{p}(\mathbb{R}^{3})\rightarrow L^{p}(\mathbb{R}^{3})$ boundedness of $\sup_{t\in[1,2]}|A_{t}|$ yields $|T|^{\frac{1}{p}}|U(z_{1},z_{2})|\lesssim|D|^{\frac{1}{p}}$ . This implies the necessity of $\frac{1}{p}\leq\frac{M+1}{2M}$ . $\Box$

6.2 Hölder continuity property

This subsection is a brief overview of how we can prove the Hölder continuity property of the local maximal operator considered in this article, that is to prove that $\sup_{t\in[1,2]}|A_{t}|$ satisfies

\biggl{\|}\sup_{t\in[1,2]}\biggl{|}A_{t}f(y+z)-A_{t}f(y)\biggl{|}\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim|z|^{\epsilon}\|f\|_{L^{p}(\mathbb{R}^{3})}

(6.1)

for some real number $\epsilon>0$ and any $z\in\mathbb{R}^{3}$ , $|z|\leq 1$ , whenever $(\frac{1}{p},\frac{1}{q})\neq(0,0)$ and belongs to the region $\Delta$ where $\sup_{t\in[1,2]}|A_{t}|$ is $L^{p}(\mathbb{R}^{3})\rightarrow L^{q}(\mathbb{R}^{3})$ bounded. Since the hypersurface studied in this paper are of finite type, the Fourier transform of their surface measures decays to a certain extent. Therefore, we can utilize the argument in Section 1.1.3 of reference [12] to deduce the problem to obtaining the inequality

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}A_{t+h}f(y)-A_{t}f(y)\biggl{|}\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\delta^{\epsilon_{1}}\|f\|_{L^{p}(\mathbb{R}^{3})}

(6.2)

for every $\delta\in(0,1)$ and some $\epsilon_{1}>0$ . Noting that for each $(\frac{1}{p},\frac{1}{q})\in\Delta$ we have

\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}A_{t+h}f(y)-A_{t}f(y)\biggl{|}\biggl{\|}_{L^{q}(\mathbb{R}^{3})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{3})},

then we can get inequality (6.2) by interpolating with the results in the following theorem.

Theorem 6.3.

(1) If $(\frac{1}{p},\frac{1}{q})=(\frac{1}{2},\frac{1}{2})$ , then the estimate (6.2) holds true for $A_{t}$ that is considered in Theorem 1.2 (1), Theorem 1.3 (1) and Theorem 1.6, respectively.
(2) Let $A_{t}$ be the averaging operator described in Theorem 1.2 (2) or Theorem 1.3 (2). Then the estimate (6.2) holds true with $(\frac{1}{p},\frac{1}{q})=(\frac{1}{8h_{\Phi}},\frac{1}{8h_{\Phi}})$ .

Proof. We firstly prove the corresponding results for the averaging operator $A_{t}$ considered in Theorem 1.2 (1) and Theorem 1.3 (1). After dyadic decomposition and isometric transformation, we have

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}A_{t+h}f(y)-A_{t}f(y)\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim\sum_{k\geq 1}2^{-2k}\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A_{t+h,k}}f(y)-\widetilde{A_{t,k}}f(y)\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})},

(6.3)

where $\widetilde{A_{t,k}}$ is defined by equality (4.8), then the proof of this part can be finished if we can show that

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A_{t+h,k}}f(y)-\widetilde{A_{t,k}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\delta^{\epsilon_{1}}\|f\|_{L^{2}(\mathbb{R}^{3})}

(6.4)

for some $\epsilon_{1}>0$ . We present the principal part of our proof, which is to demonstrate that the operator $\widetilde{A_{t,k}^{\lambda_{h_{0}}}}$ in Lemma 4.2 and the operator $\widetilde{A_{t,k}^{\tau}}$ in Lemma 4.4 satisfy (6.4). In the context of operator $\widetilde{A_{t,k}^{\lambda_{h_{0}}}}$ , for any $p>1$ , we decompose $\widetilde{A_{t,k}^{\lambda_{h_{0}}}}$ such that

		$\displaystyle\biggl{\\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{\|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k}}f(y)\biggl{\|}\biggl{\\|}_{L^{p}(\mathbb{R}^{3})}$
		$\displaystyle\leq\sum_{l\geq\log\frac{1}{\epsilon}}2^{-l}\biggl{\\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{\|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}f(y)\biggl{\|}\biggl{\\|}_{L^{p}(\mathbb{R}^{3})},$		(6.5)

where $\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}$ denotes the averaging operator along the hypersurface $S^{h_{0}}_{k,l}$ defined by (4.5) (with $c=0$ ) whose Gaussian curvature is non-vanishing. We further introduce the operator $\widetilde{A_{t,k,l,j}^{\lambda_{h_{0}}}}$ such that the Fourier transform of $\widetilde{A_{t,k,l,j}^{\lambda_{h_{0}}}}f$ is supported in $\{\xi\in\mathbb{R}^{3}:|\xi|\sim 2^{j}\}$ , $j\geq 1$ , then Sobolev embedding and Plancherel’s theorem imply that

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l,j}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\delta^{\frac{1}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}

(6.6)

and

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l,j}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{-\frac{j}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.7)

Interpolation between (6.6) and (6.7) implies that for $\epsilon_{1}\in(0,1)$ , there holds

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l,j}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\delta^{\frac{\epsilon_{1}}{2}}2^{-\frac{(1-\epsilon_{1})j}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.8)

Then the estimate (6.4) for $\widetilde{A_{t,k}^{\lambda_{h_{0}}}}$ follows from (6.8) and (6.2).

Regarding the operator $\widetilde{A_{t,k}^{\tau}}$ , we introduce the averaging operator $\widetilde{A_{t,k,j}^{\tau}}$ along the hypersurface $\tilde{S_{k}}$ defined by (4.15) with $c=0$ , and the Fourier transform of $\widetilde{A_{t,k,j}^{\tau}}f$ is supported in $\{\xi\in\mathbb{R}^{3}:|\xi|\sim 2^{j}\}$ , $j\geq 1$ . Sobolev embedding and the maximal estimate (2.23) yield

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k,j}}f(y)-\widetilde{A^{\tau}_{t,k,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\delta^{\frac{1}{2}}2^{(\frac{1}{2}-\frac{1}{M})j}\|f\|_{L^{2}(\mathbb{R}^{3})}

(6.9)

and

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k,j}}f(y)-\widetilde{A^{\tau}_{t,k,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{-\frac{j}{M}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.10)

Interpolating between (6.9) and (6.10), we obtain that for $\epsilon_{1}\in(0,\frac{2}{M})$ , there holds

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k,j}}f(y)-\widetilde{A^{\tau}_{t,k,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\delta^{\frac{\epsilon_{1}}{2}}2^{(\frac{\epsilon_{1}}{2}-\frac{1}{M})j}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.11)

Then we have obtained the estimate (6.4) for $\widetilde{A_{t,k}^{\tau}}$ .

Next we will show the results for $A_{t}$ considered in Theorem 1.2 and Theorem 1.3 with $c\neq 0$ . Let $\widetilde{A_{t,k}}$ be defined by (4.8). Then the desired results follow if we can prove that

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A_{t+h,k}}f(y)-\widetilde{A_{t,k}}f(y)\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim 2^{\frac{2mk}{p}}\delta^{\frac{\epsilon_{1}}{p}}\|f\|_{L^{p}(\mathbb{R}^{3})}

(6.12)

for $p>2h_{\Phi}$ . We demonstrate how to prove this result for operators $\widetilde{A_{t,k}^{\lambda_{h_{0}}}}$ and $\widetilde{A_{t,k}^{\tau}}$ . For the operator $\widetilde{A_{t,k}^{\lambda_{h_{0}}}}$ , we define $\widetilde{A_{t,k,l}^{\lambda_{h_{0}}}}$ and $\widetilde{A_{t,k,l,j}^{\lambda_{h_{0}}}}$ as in the case when $c=0$ . It follows from Sobolev embedding and Plancherel’s theorem that

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l,j}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{mk+n_{h_{0}}l}\delta^{\frac{1}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}

(6.13)

and

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l,j}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{\frac{mk+n_{h_{0}}l}{2}}2^{-\frac{j}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.14)

Inequalities (6.13) and (6.14) imply that for $\epsilon_{1}\in(0,1)$ , there holds

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l,j}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{mk+n_{h_{0}}l}\delta^{\frac{\epsilon_{1}}{2}}2^{-\frac{(1-\epsilon_{1})j}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})},

(6.15)

this yields

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{mk+n_{h_{0}}l}\delta^{\frac{\epsilon_{1}}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.16)

Moreover, we have the following trivial estimate,

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}f(y)\biggl{|}\biggl{\|}_{L^{\infty}(\mathbb{R}^{3})}\lesssim\|f\|_{L^{\infty}(\mathbb{R}^{3})}.

(6.17)

Interpolation between (6.16) and (6.17) yields

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\lambda_{h_{0}}}_{t+h,k,l}}f(y)-\widetilde{A^{\lambda_{h_{0}}}_{t,k,l}}f(y)\biggl{|}\biggl{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim 2^{\frac{2mk+2n_{h_{0}}l}{p}}\delta^{\frac{\epsilon_{1}}{p}}\|f\|_{L^{p}(\mathbb{R}^{3})}.

(6.18)

Inequalities (6.18) and (6.2) imply that (6.12) holds for $\widetilde{A_{t,k}^{\tau}}$ with $p>2h_{\Phi}$ .

As for the operator $\widetilde{A_{t,k}^{\tau}}$ , we define the operators $\widetilde{A_{t,k,j}^{\tau}}$ in the same way as in the case when $c=0$ . Sobolev embedding and the maximal estimate (2.23) yield

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k,j}}f(y)-\widetilde{A^{\tau}_{t,k,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{mk}\delta^{\frac{1}{2}}2^{(\frac{1}{2}-\frac{1}{M})j}\|f\|_{L^{2}(\mathbb{R}^{3})}

(6.19)

and

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k,j}}f(y)-\widetilde{A^{\tau}_{t,k,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{\frac{mk}{2}}2^{-\frac{j}{M}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.20)

Interpolating between (6.19) and (6.20), we obtain that for $\epsilon_{1}\in(0,\frac{2}{M})$ , there holds

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k,j}}f(y)-\widetilde{A^{\tau}_{t,k,j}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{mk}\delta^{\frac{\epsilon_{1}}{2}}2^{(\frac{\epsilon_{1}}{2}-\frac{1}{M})j}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.21)

It follows that

\displaystyle\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k}}f(y)-\widetilde{A^{\tau}_{t,k}}f(y)\biggl{|}\biggl{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim 2^{mk}\delta^{\frac{\epsilon_{1}}{2}}\|f\|_{L^{2}(\mathbb{R}^{3})}.

(6.22)

Then the estimate (6.12) for $\widetilde{A_{t,k}^{\tau}}$ can be obtained by interpolating between (6.22) and the trivial estimate .

\biggl{\|}\sup_{t\in[1,2],h\in[0,\delta]}\biggl{|}\widetilde{A^{\tau}_{t+h,k}}f(y)-\widetilde{A^{\tau}_{t,k}}f(y)\biggl{|}\biggl{\|}_{L^{\infty}(\mathbb{R}^{3})}\lesssim\|f\|_{L^{\infty}(\mathbb{R}^{3})}.

$\Box$

Finally, we tend to omit the proof of the corresponding conclusion for the operator $A_{t}$ considered in Theorem 1.6, as the idea of its proof is very similar to that of the operators involved in Theorem 1.2 (1) and 1.3 (1).

References

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[2] B. Buschenhenke, I. A. Ikromov and D. Müller, Estimates for maximal functions associated to hypersurfaces in $\mathbbm{R}^{3}$ with height $h<2$ : part II: A geometric conjecture and its proof for generic 2-surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., (2023). https://doi.org/10.2422/2036-2145.202301-016.
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	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{q}(\mathbb{R}^{3})}$	$\displaystyle\leq\max\{1,c^{\frac{1}{q}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{(M-1)l(\frac{1}{q}-\frac{1}{p})-l}2^{\frac{j}{q}-\frac{j}{2}}\biggl{\\|}\tilde{\rho}(t)F_{j,l,1}(y,t)\biggl{\\|}_{L^{q}(\mathbb{R}^{3}\times[\frac{1}{2},4])}$
		$\displaystyle\quad\quad+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{(M-1)l(\frac{1}{q}-\frac{1}{p})-l}2^{\frac{j}{q}-j+\frac{Ml}{2}}\biggl{\\|}\tilde{\rho}(t)F_{j,l,2}(y,t)\biggl{\\|}_{L^{q}(\mathbb{R}^{3}\times[\frac{1}{2},4])}\biggl{)}.$		(2.22)

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{M+2}{2}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{2}{M+2}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{2}{M+2}j-\frac{j}{2}}2^{\frac{8}{M+2}(\frac{3j}{8}+\frac{Ml}{4})}2^{\frac{M-6}{2(M+2)}j}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{2}{M+2}j-j+\frac{Ml}{2}}2^{\frac{4}{M+2}j}2^{\frac{Ml}{M+2}}2^{\frac{(M-6)j}{M+2}-\frac{(M-6)Ml}{2(M+2)}}\biggl{)}\\|f\\|_{L^{\frac{M+2}{4}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{2}{M+2}}\}\\|f\\|_{L^{\frac{M+2}{4}}(\mathbb{R}^{3})}.$

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{5M+2}{M+2}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{M+2}{5M+2}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{(M+2)(M-1)}{5M+2}l}2^{\frac{M+2}{5M+2}j-\frac{j}{2}}2^{\frac{3M+6}{5M+2}(\frac{j}{2}+\frac{Ml}{3})}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l\leq\frac{j}{M}}2^{-l}2^{-\frac{(M+2)(M-1)}{5M+2}l}2^{\frac{M+2}{5M+2}j-j+\frac{Ml}{2}}2^{\frac{2M+4}{5M+2}j}2^{\frac{(\frac{M}{2}+1)Ml}{5M+2}}\biggl{)}\\|f\\|_{L^{\frac{5M+2}{2M+4}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{M+2}{5M+2}}\}\\|f\\|_{L^{\frac{5M+2}{2M+4}}(\mathbb{R}^{3})}.$

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{M+2}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{1}{M+2}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{j}{M+2}-\frac{j}{2}}2^{\frac{6}{M+2}(\frac{j}{2}+\frac{Ml}{3})}2^{\frac{M-4}{2(M+2)}j}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{-\frac{2(M-1)}{M+2}l}2^{\frac{j}{M+2}-j+\frac{Ml}{2}}2^{\frac{5}{M+2}j}2^{\frac{(M-4)j}{M+2}-\frac{(M-4)Ml}{2(M+2)}}\biggl{)}\\|f\\|_{L^{\frac{M+2}{3}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{1}{M+2}}\}j\\|f\\|_{L^{\frac{M+2}{3}}(\mathbb{R}^{3})}.$

	$\displaystyle\biggl{\\|}\sup_{t\in[1,2]}\|\widetilde{A_{t,j}}f\|\biggl{\\|}_{L^{\frac{8m+4}{M+2}}(\mathbb{R}^{3})}$
	$\displaystyle\leq\max\{1,c^{\frac{M+2}{8M+4}}\}\biggl{(}\sum_{l\geq\frac{j}{M}}2^{-l}2^{-\frac{(2M+4)(M-1)}{8M+4}l}2^{\frac{M+2}{8M+4}j-\frac{j}{2}}2^{\frac{5M+10}{8M+4}(\frac{3j}{5}+\frac{2Ml}{5})}$
	$\displaystyle+\sum_{\log{\frac{1}{\epsilon_{0}}}\leq l<\frac{j}{M}}2^{-l}2^{-\frac{(2M+4)(M-1)}{8M+4}l}2^{\frac{M+2}{8M+4}j-j+\frac{Ml}{2}}2^{\frac{7M+2}{8M+4}j}2^{\frac{Ml(4-M)}{4M+2}}\biggl{)}\\|f\\|_{L^{\frac{8M+4}{3M+6}}(\mathbb{R}^{3})}$
	$\displaystyle\lesssim\max\{1,c^{\frac{M+2}{8M+4}}\}j\\|f\\|_{L^{\frac{8M+4}{3M+6}}(\mathbb{R}^{3})}.$

Boundedness for maximal operators over hypersurfaces in ℝ3\mathbb{R}^{3} ††thanks: This work is supported by the National Key R&D Program of China (No.2023YFA1010800); the Natural Science Foundation of China (No.12271435, No.12301113).

1 Introduction

1.1 Maximal operators associated with homogeneous hypersurfaces

Proposition 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

Theorem 1.5.

1.2 Other maximal operators over hypersurfaces

Theorem 1.6.

Theorem 1.7.

2 Proof of Theorem 1.4

Theorem 2.1.

3 Proof of Theorem 2.1

Theorem 3.1.

Lemma 3.2.

4 Proofs of Theorem 1.2 and Theorem 1.3

Lemma 4.1.

Lemma 4.2.

Remark 4.3.

Lemma 4.4.

Remark 4.5.

5 Proofs of Theorem 1.6 and Theorem 1.7

Lemma 5.1.

Lemma 5.2.

6 Appendix

6.1 Some counterexamples and necessary conditions

Theorem 6.1.

Theorem 6.2.

6.2 Hölder continuity property

Theorem 6.3.

References

Boundedness for maximal operators
over hypersurfaces in $\mathbb{R}^{3}$ ^†^†thanks: This work is supported by the National Key R&D Program of China (No.2023YFA1010800); the Natural Science Foundation of China (No.12271435, No.12301113).