$L^p$ bounds for Stein’s spherical maximal operators

Abstract

Let ${\mathfrak {M}}^\alpha$ be the spherical maximal operators of complex order $\alpha$ on ${{\mathbb {R}}^n}$. In this article we show that when $n\ge 2$, suppose

$$\begin{aligned} \Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^n})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^n})} \end{aligned}$$

holds for some $\alpha$ and $p\ge 2$, then we must have that $\textrm{Re}\,\alpha \ge \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$ In particular, when $n=2$, we prove that $\Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^2})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^2})}$ if $\textrm{Re}\ \! \alpha >\max \{1/p-1/2,\ -1/p\}$, and consequently the range of $\alpha$ is sharp in the sense that the estimate fails for $\textrm{Re}\ \alpha <\max \{1/p-1/2, -1/ p\}.$

1 Introduction

In 1976 Stein [19] introduced the spherical maximal means ${\mathfrak {M}}^\alpha f(x)= \sup _{t>0} |{\mathfrak {M}}^\alpha _tf (x) |$ of (complex) order $\alpha$, where

$$\begin{aligned} {\mathfrak {M}}^\alpha _t f (x) = {1\over \Gamma (\alpha ) } \int _{|y|\le 1} \left( 1-{|y|^2 }\right) ^{\alpha -1} f(x-ty)\,\text {d}y. \end{aligned}$$

(1.1)

These means are defined a priori only for $\textrm{Re}\ \alpha >0$, but the definition can be extended to all complex $\alpha$ by analytic continuation. In the case $\alpha =1$, ${\mathfrak {M}}^\alpha$ corresponds to the Hardy–Littlewood maximal operator and in the case $\alpha =0$, one recovers the spherical maximal means ${\mathfrak {M}} f(x)= \sup _{t>0} |{\mathfrak {M}}_tf (x) |$ in which

$$\begin{aligned} {\mathfrak {M}}_tf (x) = c_n \int _{{{\mathbb {S}}}^{n-1}} f(x-ty) \,\text {d}\sigma (y), \ \ \ (x, t)\in {{\mathbb {R}}^n}\times {\mathbb R^+}, \end{aligned}$$

(1.2)

where $c_n$ is a constant depending only on n, ${{\mathbb {S}}}^{n-1}$ denotes the standard unit sphere in ${{\mathbb {R}}^n}$ and $\text {d}\sigma$ is the induced Lebesgue measure on the unit sphere ${{\mathbb {S}}}^{n-1}.$ In [19, Theorem 2], Stein showed that

$$\begin{aligned} \Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^n})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^n})} \end{aligned}$$

(1.3)

in the following circumstances:

$$\begin{aligned} \textrm{Re}\, \alpha >1-n+ {n\over p} \ \ \ \ \textrm{when}\ 1<p\le 2; \end{aligned}$$

(1.4)

or

$$\begin{aligned} \ \ \textrm{Re}\, \alpha >{2-n\over p} \ \ \ \ \ \ \ \ \textrm{when}\ 2\le p\le \infty . \end{aligned}$$

(1.5)

The above maximal theorem tells us that when $\alpha =0$ and $n\ge 3$, the maximal operator ${\mathfrak {M}}$ is bounded on $L^p({{\mathbb {R}}^n})$ for the range of $p>n/(n-1)$. This range of p is sharp, as has been pointed out in [19, 21], no such result can hold for $p\le n/(n-1)$ if $n\ge 2$.

Some 10 years passed before Bourgain [1] finally proved that the maximal operator ${\mathfrak {M}}$ is bounded on $L^p{(\mathbb R^2)}$ for $p>2$. Bourgain’s theorem says that there exists $\epsilon (p)>0$ such that

$$\begin{aligned} \Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^2})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^2})}, \ \ \ \textrm{Re}\, \alpha > -\epsilon (p), \ \ \ 2< p< \infty . \end{aligned}$$

(1.6)

This result cannot hold even for $\alpha =0$ when $p=2$, see [19]. An alternative proof of Bourgain’s result was subsequently found by Mockenhaupt, Seeger and Sogge [11], who used a local smoothing estimate for the solutions of the wave operator. In 2017, Miao, Yang and Zheng [10] improved certain range of $\alpha$ for $L^p$-bounds for the operator ${\mathfrak {M}}^{\alpha }$ by using the Bourgain–Demeter decoupling theorem [2]. All these refinements can be stated altogether as follows: For $n\ge 2$ and $p\ge 2$, (1.3) holds whenever

$$\begin{aligned} \textrm{Re} \, \alpha >\max \left\{ { 1-n\over 4} +{3-n\over 2p}, \, {1-n\over p} \right\} . \end{aligned}$$

(1.7)

The above range $\alpha$ in (1.7) for $p>2$ is strictly wider than the range of $\alpha$ in (1.5). However, the range $\alpha$ in (1.7) is not optimal.

As mentioned above, the proof of the range of $\alpha$ in (1.7) relies on the progress concerning Sogge’s local smoothing conjecture, as originally formulated by Sogge [18]: For $n\ge 2$ and $p\ge 2n/(n-1)$, one has

$$\begin{aligned} \left\| u \right\| _{L^p({{\mathbb {R}}^n}\times [1,2])} \le C\left( \Vert f\Vert _{W^{\gamma , p}({{\mathbb {R}}^n})} + \Vert g\Vert _{W^{\gamma -1, p}({{\mathbb {R}}^n})} \right) , \ \ \ \ \ \ \ \textrm{if}\ \ \gamma >{n-1\over 2}-{n\over p},\nonumber \\ \end{aligned}$$

(1.8)

where

$$\begin{aligned} u(x,t)= \cos (t\sqrt{-\Delta }) f(x) +{\sin (t\sqrt{-\Delta }) \over \sqrt{-\Delta }} g(x) \end{aligned}$$

is the solution to the Cauchy problem for the wave equation in ${{\mathbb {R}}^{n}\times {{\mathbb {R}}}}:$

$$\begin{aligned} \left\{ \begin{array}{l} \left( (\partial /\partial t)^2-\Delta \right) u(x,t) =0, \\ u|_{t=0} = f, \\ (\partial /\partial t) u|_{t=0} =g. \end{array} \right. \end{aligned}$$

(1.9)

The local smoothing conjecture has been studied in numerous papers, see for instance [2, 4, 7, 8, 10, 11, 17, 24] and the references therein. When $n=2$, sharp results follow by the work of Guth, Wang and Zhang [7]. When $n\ge 3$, the conjecture holds for all $p\ge {2(n+1)/ (n-1)}$ by the Bourgain–Demeter decoupling theorem [2] and the method of [24].

The aim of this article is to prove the following result.

Theorem 1.1

Let $p\ge 2$.

• (i) Let $n\ge 2$. Suppose (1.3) holds for some $\alpha \in \mathbb {C}$. Then we must have

  $$\begin{aligned} \textrm{Re}\,\alpha \ge \max \left\{ {1\over p}-{n-1\over 2},\ -{n-1\over p}\right\} . \end{aligned}$$

• (ii) Let $n=2$. Then the estimate (1.3) holds if

  $$\begin{aligned} \textrm{Re}\,\alpha > \max \left\{ {1\over p}-{1\over 2},\ -{1\over p}\right\} , \end{aligned}$$

  and consequently the range of $\alpha$ is sharp in the sense that the estimate fails for $\textrm{Re}\ \alpha <\max \{1/p-1/2, -1/ p\}.$

Let $p\ge 2$ and $\alpha =(3-n)/2$. For an appropriate constant $c_n$, we have that $c_nt ({\mathfrak {M}}_t^\alpha g)(x)= u(x,t)$, where u is the solution to the wave equation (1.9) with $f=0$, see [20, 4.10, p.519]. As a consequence of (i) of Theorem 1.1, we have the following corollary.

Corollary 1.2

Let $n\ge 4$. Then

$$\begin{aligned} \left\| \sup _{t>0} \left| {u(x,t)\over t}\right| \right\| _{L^p({{\mathbb {R}}^n})} \le C_p\Vert g\Vert _{L^p({{\mathbb {R}}^n})} \end{aligned}$$

can not hold whenever $p> 2(n-1)/(n-3)$.

We would like to mention that for the range $\alpha$ in (1.5), it is commented in [20, 4.10, p.519] that the optimal results for $p>2$ and $n\ge 2$ “are still a mystery". Our Theorem 1.1 gives an affirmative answer in dimension $n=2$ to show sharpness of $\textrm{Re}\,\alpha > \max \left\{ {1/p}-{1/2},\ -{1/p}\right\}$ in the estimate (1.3) except the borderline.

The proof of (ii) of Theorem 1.1 can be shown by applying the work of Guth-Wang-Zhang [7] on local smoothing estimates along with the techniques previously used in [11] and [10]. The main contribution of this article is to show (i) of Theorem 1.1. From the asymptotic expansion of Fourier multiplier of the operator ${\mathfrak {M}}^{\alpha }_t$, it is seen that ${\mathfrak {M}}_t^\alpha$ are essentially the sum of half-wave operators $e^{it\sqrt{-\Delta }}$ and $e^{-it\sqrt{-\Delta }}$, and hence the complexity of the operator ${\mathfrak {M}}^{\alpha }_t$ comes from the interference between the operators $e^{it\sqrt{-\Delta }}$ and $e^{-it\sqrt{-\Delta }}$. To show the necessity of $L^p$-boundedness of ${\mathfrak {M}}^{\alpha }_t$, we make the following observations. For the case $p>2n/(n-1)$ we note that by the stationary phase argument, two waves $e^{ it\sqrt{-\Delta }}f$ and $e^{ -it\sqrt{-\Delta }}f$ concentrate on the opposite parts of sphere $\{x\in {\mathbb {R}^n}: |x|=t\}$, respectively, when ${{\hat{f}}}$ is supported on a small cone. For the case $2\le p\le 2n/(n-1)$, we let f be a wave packet of direction $\nu \in S^{n-1}$, then one can regard $e^{\pm it\sqrt{-\Delta }}f(x)$ as the translations $f(x\pm t\nu )$ of f(x), which concentrate on the opposite parts of sphere $\{x\in {\mathbb {R}^n}:\ |x|=t\}$. In Sect. 3, we construct two examples such that there is no interference between $e^{it\sqrt{-\Delta }}f$ and $e^{-it\sqrt{-\Delta }}f$ to obtain the desired range of $\alpha$ in (i) of Theorem 1.1.

The paper is organized as follows. In Sect. 2, we give some preliminary results including the properties of the Fourier multiplier associated to the spherical operators ${\mathfrak {M}}^\alpha _t$ by using asymptotic expansions of Bessel functions. The proof of (i) of Theorem 1.1 will be given in Sect. 3 by constructing two examples to obtain the necessarity of $L^p$-bounds for the maximal operator ${\mathfrak {M}}^\alpha$. In Sect. 4 we will give the proof of (ii) of Theorem 1.1.

2 Preliminary results

We begin with recalling the spherical function ${\mathfrak {M}}^\alpha _t f(x)=f*m_{\alpha ,t}(x)$ where $m_{\alpha ,t}(x)= t^{-n}m_{\alpha }(t^{-1}x)$ and

$$\begin{aligned} m_{\alpha }(x)=\Gamma (\alpha )^{-1}\big (1-|x|^{2}\big )_{+}^{\alpha -1}, \end{aligned}$$

where $\Gamma (\alpha )$ is the Gamma function and $(r)_+=\max \{0,r\}$ for $r\in {\mathbb {R}}$. Define the Fourier transform of f by ${\hat{f}}(\xi )=\int _{{\mathbb {R}^n}} e^{-2\pi ix \cdot \xi }f(x)\,\text {d}x.$ It follows by [22, p.171] that the Fourier transform of $m_\alpha$ is given by

$$\begin{aligned} \widehat{m_{\alpha }}(\xi )=\pi ^{-\alpha +1}|\xi |^{-n/2-\alpha +1} J_{n/2+\alpha -1}\big (2\pi |\xi |\big ). \end{aligned}$$

(2.1)

Here $J_\beta$ denotes the Bessel function of order $\beta$. For any complex number $\beta$, we can obtain the complete asymptotic expansion

$$\begin{aligned} J_\beta (r)\sim r^{-1/2}e^{ir} \sum _{j= 0}^{\infty }b_{j} r^{-j} + r^{-1/2}e^{-ir} \sum _{j= 0}^{\infty }d_j r^{-j},\, \ \ \ \ \ \ r\ge 1 \end{aligned}$$

(2.2)

for suitable coefficients $b_j$ and $d_j$ with $b_0,d_0\ne 0$. Note that when $\beta$ is a positive integer, (2.2) is given in [20, (15), p.338]. For general $\beta$, we refer it to [23, (1). 7.21, p.199].

Then there exists an error terms $E_{N,1}(r), E_{N,2}(r)$ and E(r) such that for any given $N\ge 1$ and $r\ge 1$,

$$\begin{aligned}&J_\beta (r) \nonumber \\&= r^{-1/2}e^{ir}\left( \sum _{j= 0}^{N-1}b_{j} r^{-j}+E_{N,1}(r)\right) + r^{-1/2}e^{-ir}\left( \sum _{j= 0}^{N-1}d_j r^{-j}+ E_{N,2}(r)\right) +E(r), \end{aligned}$$

(2.3)

where

$$\begin{aligned} \left| \left( \frac{d}{dr}\right) ^k E_{N,1}(r)\right| +\left| \left( \frac{d}{dr}\right) ^k E_{N,2}(r)\right| +\left| \left( \frac{d}{dr}\right) ^k E(r)\right| \le C_{k}r^{-N-k} \end{aligned}$$

(2.4)

for all $k\in \mathbb {Z}_+$. We rewrite (2.1) as

$$\begin{aligned} \widehat{m_\alpha }(\xi )&= \varphi (|\xi |)\widehat{m_\alpha }(\xi ) + (1-\varphi (|\xi |))\widehat{m_\alpha }(\xi ) \nonumber \\&= \left[ \varphi (|\xi |)\widehat{m_\alpha }(\xi )+{{\mathcal {E}}}(|\xi |) \right] \nonumber \\&\quad + \left[ e^{2\pi i|\xi |}{{\mathcal {E}}}_{N,1}(|\xi |)+e^{-2\pi i|\xi |}{{\mathcal {E}}}_{N,2}(|\xi |) \right] \nonumber \nonumber \\&\quad + |\xi |^{-(n-1)/2-\alpha }\left[ e^{2\pi i|\xi |}a_1(|\xi |)+e^{-2\pi i|\xi |}a_2(|\xi |) \right] , \end{aligned}$$

(2.5)

where

$$\begin{aligned}&{{\mathcal {E}}}(r) = (2\pi )^{1/2}c(\pi , \alpha )(1-\varphi (r))r^{-(n-2)/2-\alpha }E(2\pi r),\nonumber \\&{{\mathcal {E}}}_{N,\ell }(r) = c(\pi , \alpha ) E_{N,\ell }(2\pi r)(1-\varphi (r))r^{-(n-1)/2-\alpha },\ \ \ \ell =1,2, \nonumber \\&a_1(r) = c(\pi , \alpha ) \sum _{j= 0}^{N-1}b_j (2\pi r)^{-j}(1- \varphi (r)),\nonumber \\&a_2(r) = c(\pi , \alpha )\sum _{j= 0}^{N-1}d_j (2\pi r)^{-j}(1- \varphi (r)) \end{aligned}$$

(2.6)

with $c(\pi , \alpha )= 2^{-1/2}\pi ^{-\alpha +1/2}$. Here $\varphi \in C_0^{\infty }(\mathbb {R})$ is an even function, identically equals 1 on B(0, M) and supported on B(0, 2M), where $M=M(N)$ is large enough such that $|a_2(r)| \ge c_{low}>0$ for $|r|\ge M$. Then we can split the Fourier multiplier of the operator $\mathfrak {M}^\alpha _1$ into three parts as in (2.5) above. Firstly, we note that $\varphi (|\xi |)\widehat{m_\alpha }(\xi )$ is smooth and compactly supported and ${{\mathcal {E}}}(|\xi |)\in {{\mathscr {S}}}({\mathbb {R}^n})$. It is seen that $\sup _{t>0} |\widehat{m_\alpha }(tD) \varphi (t|D|)f|$ and $\sup _{t>0} |{{\mathcal {E}}}(t|D|)f|$ are bounded by the Hardy–Littlewood maximal function. Then for $p>1$,

$$\begin{aligned} \hspace{1cm} \left\| \sup _{t>0} |\widehat{m_\alpha }(tD) \varphi (t|D|))f|\right\| _{L^p(\mathbb {R}^n)}+\left\| \sup _{t>0} |{{\mathcal {E}}}(t|D|)f |\right\| _{L^p(\mathbb {R}^n)} \le C\Vert f\Vert _{L^p({\mathbb {R}^n})}.\nonumber \\ \end{aligned}$$

(2.7)

Secondly, we define

$$\begin{aligned} {{\mathscr {E}}}_{N}f(x,t)=\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi +t|\xi |)}{{\mathcal {E}}}_{N,1}\big (t|\xi |\big ) {\hat{f}}(\xi )\,\text {d}\xi +\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi -t|\xi |)}{{\mathcal {E}}}_{N,2}\big (t|\xi |\big ) {\hat{f}}(\xi )\,\text {d}\xi . \end{aligned}$$

Then we have the following lemma.

Lemma 2.1

Let $p\ge 2$. There exists a constant $C>0$ such that

$$\begin{aligned} \left\| \sup _{t\in [1,2]}|{{\mathscr {E}}}_{N}f(\cdot ,t)| \right\| _{L^p({\mathbb {R}^n})}\le C \Vert f\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

(2.8)

when

$$\begin{aligned} N>-{n-2\over p}-\textrm{Re}\,\alpha . \end{aligned}$$

The proof of Lemma 2.1 is based on the following elementary result (see [17, Lemma 2.4.2]).

Lemma 2.2

Let F be a smooth function defined on ${{\mathbb {R}}^n}\times [1,2]$. Then for $p>1$ and $1/p +1/p' =1,$

$$\begin{aligned} \left\| \sup _{1\le t\le 2} |F(\cdot ,t)|\right\| _{L^p({\mathbb {R}^n})}\le C_p\left( \Vert F(\cdot ,1)\Vert _{L^p({\mathbb {R}^n})} + \Vert F\Vert _{L^p({\mathbb {R}^n}\times [1,2])}^{1-1/p} \left\| \partial _t F\right\| _{L^p({\mathbb {R}^n}\times [1,2])}^{1/p}\right) . \end{aligned}$$

Proof of Lemma 2.1

We fix a function $\varphi$ as in (2.5). Let $\psi (r):=\varphi (r)-\varphi (2r)$ and $\psi _j(r):=\psi (2^{-j}r)$, for $j\ge 1$. So we have

$$\begin{aligned} 1\equiv \varphi (r) + \sum _{j\ge 1 }\psi _j(r), \quad r\ge 0. \end{aligned}$$

(2.9)

For $j\ge 1$, define

$$\begin{aligned} {{\mathscr {E}}}_{N,j}f(x,t)=\int _{{\mathbb {R}}^n} \left( e^{2\pi i(x\cdot \xi +t|\xi |)}{{\mathcal {E}}}_{N,1}\big (t|\xi |\big ) + e^{2\pi i(x\cdot \xi -t|\xi |)}{{\mathcal {E}}}_{N,2}\big (t|\xi |\big )\right) {\psi _j}(t|\xi |){\hat{f}}(\xi )d\xi . \end{aligned}$$

To prove (2.8), it suffices to show that there exists a constant $\delta >0$ such that for all $j\ge 1$,

$$\begin{aligned} \left\| \sup _{1\le t\le 2}|{{\mathscr {E}}}_{N,j}f(\cdot ,t)| \right\| _{L^p({\mathbb {R}^n})}\le C 2^{-\delta j}\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

(2.10)

Let us prove (2.10) by using Lemma 2.2. First, for each fixed $t\in [1,2]$, ${{\mathscr {E}}}_{N,j}f$ are the sum of two Fourier integral operators of order $-(n-1)/2-\textrm{Re}\, \alpha -N$ with phase $x\cdot \xi \pm t|\xi |$. By [20, Theorem 2, Chapter IX] and the fact that $e^{it\sqrt{-\Delta }}$ is local at scale t, we have

$$\begin{aligned} \sup _{1\le t\le 2}\left\| {\mathscr {E}}_{N,j}f(\cdot ,t)\right\| _{L^p({\mathbb {R}^n})}\le C 2^{-((n-1)/2+\textrm{Re}\,\alpha +N) j}2^{(n-1)(1/2-1/p)j}\Vert f\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

(2.11)

see also [16, Corollary 2.4]. Next, we write $\partial _{t}{{\mathscr {E}}}_{N,j}f(x,t)$ as the sum of following terms,

$$\begin{aligned}{} & {} \pm 2\pi it^{-1}\int e^{2\pi i(x\cdot \xi \pm t|\xi |)}t|\xi |{{\mathcal {E}}}_{N,1}\big (t|\xi |\big ) {\psi _j}(t|\xi |){\hat{f}}(\xi )d\xi ;\\{} & {} \pm 2\pi it^{-1}\int e^{2\pi i(x\cdot \xi \pm t|\xi |)}t|\xi |{{\mathcal {E}}}_{N,2}\big (t|\xi |\big ) {\psi _j}(t|\xi |){\hat{f}}(\xi )d\xi ;\\{} & {} t^{-1}\int e^{2\pi i(x\cdot \xi \pm t|\xi |)}t|\xi |({{\mathcal {E}}}_{N, 1}\psi _{j})'\big (t|\xi |\big ) {\hat{f}}(\xi )d\xi ;\\{} & {} t^{-1}\int e^{2\pi i(x\cdot \xi \pm t|\xi |)}t|\xi |({\mathcal E}_{N, 2}\psi _{j})'\big (t|\xi |\big ) {\hat{f}}(\xi )d\xi . \end{aligned}$$

By (2.4), we see that for each fixed $t\in [1,2]$, they are Fourier integral operators of order no more than $-(n-1)/2-\textrm{Re}\, \alpha -N+1$. By [20, Theorem 2, Chapter IX] again,

$$\begin{aligned} \sup _{1\le t\le 2}\left\| \partial _t{{\mathscr {E}}}_{N,j}f(\cdot ,t) \right\| _{L^p({\mathbb {R}^n})}\le C 2^{-((n-1)/2+\textrm{Re}\,\alpha +N-1) j}2^{(n-1)(1/2-1/p)j}\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

(2.12)

Lemma 2.2, together with (2.11) and (2.12), gives

$$\begin{aligned} \left\| \sup _{1\le t\le 2}|{{\mathscr {E}}}_{N,j}f(\cdot ,t)| \right\| _{L^p({\mathbb {R}^n})}\le C 2^{-((n-1)/2+\textrm{Re}\,\alpha +N-1/p) j}2^{(n-1)(1/2-1/p)j}\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

Choosing $N>-(n-2)/p-\textrm{Re}\,\alpha$ and letting $\delta = N+(n-2)/p+\textrm{Re}\,\alpha$, we obtain estimate (2.10). The proof of Lemma 2.1 is complete. $\square$

Finally, we define

$$\begin{aligned} {{\mathscr {A}}}_t f(x)= \int _{{\mathbb {R}^n}} \left( e^{2\pi i(x\cdot \xi + t|\xi |)}a_1(t|\xi |) + e^{2\pi i(x\cdot \xi - t|\xi |)}a_2(t|\xi |) \right) {\hat{f}}(\xi ) \,\text {d}\xi . \end{aligned}$$

(2.13)

From (2.5), (2.7) and Lemma 2.2, we see that the $L^p$-boundness of the operator $\mathfrak {M}^\alpha _t$ reduces to boundedness of the operator ${{\mathscr {A}}}_t$ on Sobolev spaces, which will be investigated in Sect. 3 below.

3 Proof of (i) of Theorem 1.1

To prove (i) of Theorem 1.1, we need to show the following proposition.

Proposition 3.1

Let $n\ge 2$ and $p\ge 2$. Suppose

$$\begin{aligned} \left\| {\mathfrak {M}}_1^\alpha f\right\| _{L^p({\mathbb {R}^n})}\le C\Vert f\Vert _{L^p({\mathbb {R}^n})} \end{aligned}$$

(3.1)

holds for some $\alpha \in \mathbb {C}$. Then, we have

$$\begin{aligned}\textrm{Re}\,\alpha \ge -{n-1\over p}. \end{aligned}$$

Let us prove Proposition 3.1. Fix $N>-{(n-2)/p}-\textrm{Re}\,\alpha$ as in Lemma 2.1. By (2.5), (2.7) and Lemma 2.1, we see that the proof of Proposition 3.1 reduces to the following lemma.

Lemma 3.2

Let $n\ge 2$ and $1<p<\infty$. Let ${{\mathscr {A}}}_1$ be an operator given in (2.13). Suppose

$$\begin{aligned} \Vert {{\mathscr {A}}}_1f \Vert _{L^p({\mathbb {R}^n})}\le C\Vert f\Vert _{W^{s,p}({\mathbb {R}^n})} \end{aligned}$$

(3.2)

holds for some $s\in \mathbb {R}$. Then, we have

$$\begin{aligned}s\ge (n-1)\left| \frac{1}{2}-\frac{1}{p}\right| .\end{aligned}$$

Proof

Let $\widehat{\gamma _\beta }(\xi ):=(1+|\xi |^{2})^{-\beta /2}$ with $\beta >(n-1)/2$. Recall that $\varphi$ is a function in (2.5). Let w belong to ${ S}^{0}$ (a symbol of order zero) satisfying $|w(r)|\ge c>0$ on $\mathbb {R}$ for some constant c. Moreover, w equals $\big (\sum _{j\ge 0}^{N-1}d_j r^{-j}\big )^{-1}$ on $\textrm{supp}{\hspace{.05cm}}(1-\varphi )$, and equals constant near zero. Assume that $\chi (\xi )\in C^\infty ({\mathbb {R}^n}\backslash \{0\})$ is homogeneous of order 0 and vanishes if $|{\xi \over |\xi |}-v_1|\ge 10^{-2}$, where $v_1:=(1,0,\ldots ,0)$. Define

$$\begin{aligned} {\hat{f}}_{\beta ,R}(\xi )=w(|\xi |)\varphi _R(|\xi |) \chi (\xi ){\widehat{\gamma }}_\beta (\xi ), \end{aligned}$$

where $\varphi _R(\cdot ):=\varphi (\cdot /R)$, and R is a large positive number. Since $w(|\xi |)\in S^{0}$ and $\chi$ is a Hörmander multiplier, w(|D|) and $\chi (D)$ are bounded on $L^{p}({\mathbb {R}^n})$. And $\varphi _R(|D|)$ is bounded on $L^{p}({\mathbb {R}^n})$ uniformly in R. So we have

$$\begin{aligned} \Vert f_{\beta ,R}\Vert _{W^{s,p}({\mathbb {R}^n})}=\Vert w(|D|) \varphi _R(|D|)\chi (D)\gamma _{\beta -s}\Vert _{L^{p}({\mathbb {R}^n})}\le C\Vert \gamma _{\beta -s}\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

(3.3)

where $C>0$ is a constant independent of R. On the other hand, it follows by [6, Proposition 1.2.5] that

$$\begin{aligned} |\gamma _{\beta -s}(x)|\le \left\{ \begin{array}{lll} C|x|^{-n+\beta -s} &{}\quad \textrm{if} &{} \, |x|\le 2,\\ Ce^{-|x|/2} &{}\quad \textrm{if} &{} \, |x|\ge 2 \end{array} \right. \end{aligned}$$

when $0<\beta -s<n$. From this, we see that $\Vert f_{\beta ,R}\Vert _{W^{s,p}({\mathbb {R}^n})}<\infty$ whenever $0<\beta -s<n$ and $(-n+\beta -s)p>-n$.

Now we turn to estimate $\Vert {{\mathscr {A}}}_1f_{\beta ,R} \Vert _{L^p({\mathbb {R}^n})}$. By using polar coordinate,

$$\begin{aligned}&\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi + |\xi |)}a_1(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \nonumber \\&\quad =\int _{0}^{\infty }\int _{S^{n-1}} e^{2\pi i(x\cdot r\theta + r)}a_{1}(r)w(r) \chi (\theta )(1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}\sigma (\theta )\text {d}r\nonumber \\&\quad =\int _{0}^{\infty } e^{2\pi ir}\widehat{\chi \text {d}\sigma }(-rx)a_1(r)w(r) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r. \end{aligned}$$

(3.4)

Note that $\chi (\xi )$ vanishes if $|{\xi \over |\xi |}-v_1|\ge 10^{-2}$. By the expansion in [20, p. 360], we can write that for $|x|\ge 1$ and $|{x\over |x|}-v_1|\le 10^{-2}$,

$$\begin{aligned} \widehat{\chi \text {d}\sigma }(-x)= e^{2\pi i|x|}h(-x)+e(-x), \end{aligned}$$

(3.5)

where e belongs to ${ S}^{-\infty }$ and $h\in { S}^{-(n-1)/2}$ can be splitted into two terms:

$$\begin{aligned} h(x)= c_0|x|^{-(n-1)/2}\chi (-x/|x|)+ {\tilde{e}}(x),\ \ {\tilde{e}}\in { S}^{-(n+1)/2} \end{aligned}$$

(3.6)

for all $|x|\ge 1$. Hence, if $|{x\over |x|}-v_1|\le 10^{-2}$, we then have

$$\begin{aligned}&\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi + |\xi |)}a_1(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \nonumber \\&\quad =\int _{0}^{\infty } e^{2\pi ir(|x|+1)}h(-rx)a_1(r)w(r) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r\nonumber \\&\qquad +\int _{0}^{\infty } e^{2\pi ir}e(-rx)a_1(r)w(r) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r. \end{aligned}$$

(3.7)

From (2.6), we have that $a_1=0$ near the origin. Since $\beta >(n-1)/2$, we see that if $|{x\over |x|}-v_1|\le 10^{-2}$ and $1/2\le |x|\le 2$,

$$\begin{aligned} \left| \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi + |\xi |)}a_1(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \right| \le C \end{aligned}$$

(3.8)

for some constant $C>0$ independent of R.

Next we calculate

$$\begin{aligned} \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - |\xi |)}a_2(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \end{aligned}$$

when $|{x\over |x|}-v_1|\le 10^{-2}$ and $1< |x| \le 1+\varepsilon$ ( $\varepsilon >0$ is a small constant that will be chosen later). As (3.4) and (3.7), we write

$$\begin{aligned}&\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - |\xi |)}a_2(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \\&\quad =C\int _{0}^{\infty }\int _{S^{n-1}} e^{2\pi i(x\cdot r\theta - r)}(1-\varphi (r)) \chi (\theta )(1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\, \text {d}\sigma (\theta )\text {d}r\\&\quad =C\int _{0}^{\infty } e^{-2\pi ir}\widehat{\chi \text {d}\sigma }(-rx)(1-\varphi (r)) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r\\&\quad =C\int _{0}^{\infty } e^{2\pi ir(|x|-1)}h(-rx)(1-\varphi (r)) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r\\&\qquad +C\int _{0}^{\infty } e^{-2\pi ir}e(-rx)(1-\varphi (r)) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r. \end{aligned}$$

The second term is bounded since $e\in { S}^{-\infty }$. Now we use (3.6) to write

$$\begin{aligned}&\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - |\xi |)}a_2(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \\&\qquad =C\int _{0}^{\infty } e^{2\pi ir(|x|-1)}\left[ c_0(r|x|)^{-\frac{n-1}{2}}\chi (x/|x|) + {\tilde{e}}(-rx) \right] \\&\qquad \quad (1-\varphi (r)) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r +O(1). \end{aligned}$$

To continue, we need the following result.

Lemma 3.3

Let g be a function satisfying $|g^{(k)}(r)|\le Cr^{m-k}, r\ge 1$ for some $m\in {{\mathbb {R}}}$ and for all $k\in \mathbb {Z}_+$. Then for all $\tau \ne 0$, we have

$$\begin{aligned} \left| \int _0^\infty e^{2\pi ir\tau } g(r)(1-\varphi (r)) \varphi _R(r)\,\text {d}r\right| \le C|\tau |^{-m-1} \end{aligned}$$

(3.9)

for some constant $C>0$ independent of R and $\tau$.

Proof

By (2.9), we write

$$\begin{aligned} \int _0^\infty e^{2\pi ir\tau } g(r)(1-\varphi (r)) \varphi _R(r) \,\text {d}r= \sum _{j\ge 1}\int _0^\infty e^{2\pi ir\tau } g(r)\psi _j(r) \varphi _R(r) \,\text {d}r. \end{aligned}$$

For each j and N, integration by parts shows

$$\begin{aligned}&\left| \int _0^\infty e^{2\pi ir\tau } g(r)\psi _j(r) \varphi _R(r)\,\text {d}r \right| \nonumber \\&\qquad = (2\pi )^{N}|\tau |^{-N}\left| \int _0^\infty e^{2\pi ir\tau }\left( \frac{\text {d}}{\text {d}r} \right) ^N\left( g(r)\psi _j(r) \varphi _R(r)\right) \,\text {d}r \right| \nonumber \\&\qquad \le C|\tau |^{-N}\int _{2^{j-1}\le r\le 2^{j+1}} r^{m-N} \text {d}r \nonumber \\&\qquad \le C|\tau |^{-N} 2^{j(m-N+1)}, \end{aligned}$$

(3.10)

where we applied the condition on g and for all $k\in \mathbb {Z}_+$

$$\begin{aligned} \left| {\text {d}^k\over \text {d}r^k} \big (\varphi _R(r)\big )\right| \le C_kr^{-k} \end{aligned}$$

for some constant $C_k>0$ independent of R and r.

Set $N=0$ for $2^j\le |\tau |^{-1}$, and $N>m+1$ otherwise. From this, it follows that

$$\begin{aligned}&\left| \int _0^\infty e^{2\pi ir\tau } g(r)(1-\varphi (r)) \varphi _R(r) \,\text {d}r\right| \\&\qquad \le C\sum _{2^j\le |\tau |^{-1}} 2^{j(m+1)}+ C\sum _{2^j\ge |\tau |^{-1}}|\tau |^{-N} 2^{j(m-N+1)}\\&\qquad \le C|\tau |^{-m-1}. \end{aligned}$$

This proves Lemma 3.3. $\square$

Back to the proof of Lemma 3.2. By Lemma 3.3,

$$\begin{aligned} \int _{0}^{\infty } e^{2\pi ir(|x|-1)} {\tilde{e}}(-rx) (1-\varphi (r)) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r =O\left( \big ||x|-1\big |^{\beta -(n-1)/2}\right) . \end{aligned}$$

Finally, for $|{x\over |x|}-v_1|\le 10^{-2}$ and $1< |x| \le 1+\varepsilon$, let us estimate

$$\begin{aligned} \int _{0}^{\infty } e^{2\pi ir(|x|-1)}(r|x|)^{-\frac{n-1}{2}}(1-\varphi (r)) (1+r^{2})^{-\beta /2}r^{n-1} \varphi _R(r)\,\text {d}r. \end{aligned}$$

Note that by Lemma 3.3 again,

$$\begin{aligned}{} & {} |x|^{-\frac{n-1}{2}}\int _{0}^{\infty } e^{2\pi ir(|x|-1)}(1-\varphi (r)) r^{\frac{n-1}{2}}((1+r^{2})^{-\beta /2}-r^{-\beta }) \varphi _R(r)\,\text {d}r\\{} & {} \quad =O\left( \big ||x|-1\big |^{\beta -(n-1)/2+1}\right) . \end{aligned}$$

For the term $|x|^{-\frac{n-1}{2}}\int _{0}^{\infty } e^{2\pi ir(|x|-1)}(1-\varphi (r)) r^{-\beta +\frac{n-1}{2}} \varphi _R(r)\,\text {d}r,$ we use scaling to obtain that if $-\beta +\frac{n-1}{2}>-1$,

$$\begin{aligned}&|x|^{-\frac{n-1}{2}}\int _{0}^{\infty } e^{2\pi ir(|x|-1)} (1-\varphi (r)) r^{-\beta +\frac{n-1}{2}} \varphi _R(r)\,\text {d}r\\&\quad =|x|^{-\frac{n-1}{2}}\int _{0}^{\infty } e^{2\pi ir(|x|-1)} r^{-\beta +\frac{n-1}{2}} \varphi _R(r)\,\text {d}r+O(1)\\&\quad =|x|^{-\frac{n-1}{2}}(|x|-1)^{\beta -\frac{n+1}{2}}\int _{0}^{\infty } e^{2\pi ir} r^{-\beta +\frac{n-1}{2}} \varphi \left( {r\over (|x|-1)R}\right) \,\text {d}r+O(1). \end{aligned}$$

Note that $1< |x| \le 1+\varepsilon$. When $\beta >\frac{n-1}{2}$ and $-\beta +\frac{n-1}{2}>-1$,

$$\begin{aligned} \lim _{R\rightarrow \infty } \int _{0}^{\infty } e^{2\pi ir} r^{-\beta +\frac{n-1}{2}} \varphi \left( {r\over (|x|-1)R}\right) \,\text {d}r = C_0, \end{aligned}$$

where $C_0$ is a non-zero constant. Hence, there exist $C>0$ and $\varepsilon _{1}\in (0,1/2)$ such that if $1< |x| \le 1+\varepsilon _1$,

$$\begin{aligned} \liminf _{R\rightarrow \infty } |x|^{-\frac{n-1}{2}}\left| \int _{0}^{\infty } e^{2\pi ir(|x|-1)}(1-\varphi (r)) r^{-\beta +\frac{n-1}{2}} \varphi _R(r)\,\text {d}r\right| \ge C \big ||x|-1\big |^{\beta -\frac{n+1}{2}}. \end{aligned}$$

Furthermore, we can find $0<\varepsilon \le \varepsilon _1$ such that for $1< |x| \le 1+\varepsilon$,

$$\begin{aligned}&\liminf _{R\rightarrow \infty }\left| \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - |\xi |)}a_2(\xi ) {\hat{f}}_{\beta ,R}(\xi ) \,\text {d}\xi \right| \\&\qquad \ge C \big ||x|-1\big |^{\beta -\frac{n+1}{2}}-O\left( \big ||x|-1\big |^{\beta -(n-1)/2+1}\right) \\&\qquad \ge {C\over 2}\big ||x|-1\big |^{\beta -\frac{n+1}{2}}. \end{aligned}$$

This, together with (3.8), tells us

$$\begin{aligned} \liminf _{R\rightarrow \infty } \Vert {{\mathscr {A}}}_1f_{\beta ,R} \Vert _{L^p(\Omega _\varepsilon )}\ge \Vert \liminf _{R\rightarrow \infty } |{\mathscr {A}}_1f_{\beta ,R}|\,\Vert _{L^p(\Omega _\varepsilon )}=\infty , \end{aligned}$$

(3.11)

if $\beta >\frac{n-1}{2}$, $-\beta +\frac{n-1}{2}>-1$, and $\big (\beta -\frac{n+1}{2}\big )p\le -1$. Here we applied Fatou’s lemma and $\Omega _\varepsilon := \{x\in {\mathbb {R}^n}: |{x\over |x|}-v_1|\le 10^{-2}$,$1< |x| \le 1+\varepsilon \}$.

Therefore, we have $\sup _{R>0}\Vert f_{\beta ,R}\Vert _{W^{s,p}({\mathbb {R}^n})}<\infty$ and $\liminf \limits _{R\rightarrow \infty } \Vert {{\mathscr {A}}}_1f_{\beta ,R} \Vert _{L^p({\mathbb {R}^n})} =\infty$ provided that

$$\begin{aligned} \left\{ \begin{array}{rcl} &{}0<\beta -s<n, \quad \\ &{}(-n+\beta -s)p>-n, \quad \\ &{}\beta>\frac{n-1}{2}, \quad \\ &{}-\beta +\frac{n-1}{2}>-1, \quad \\ &{}\big (\beta -\frac{n+1}{2}\big )p\le -1, \\ \end{array} \right. \end{aligned}$$

(3.12)

which is solvable when

$$\begin{aligned} -(n+1)/2<s< (n-1)(1/p-1/2). \end{aligned}$$

(3.13)

Hence, if (3.2) holds, then we must have $s\ge (n-1)(1/p-1/2)$ or $s\le -(n+1)/2$. However, once (3.2) holds for some $s_0\le -(n+1)/2$, it holds for all $s\ge s_0$, which is in contradiction with (3.13). So the only possible range of s where (3.2) holds is $s\ge (n-1)(1/p-1/2)$. By duality,

$$\begin{aligned} \Vert ({{\mathscr {A}}}_1)^{*} f\Vert _{L^{p'}({\mathbb {R}^n})}\le C\Vert f\Vert _{W^{s,p'}({\mathbb {R}^n})}. \end{aligned}$$

Because $({{\mathscr {A}}}_1)^{*}$ is essentially the same as ${\mathscr {A}}_1$, we must have $s\ge (n-1)(1/p'-1/2)=(n-1)(1/2-1/p)$ by the previous counterexample. This proves Lemma 3.2, and then the proof of Proposition 3.1 is complete. $\square$

Next, let us prove the following result.

Proposition 3.4

Let $n\ge 2$ and $p\ge 2$. Suppose

$$\begin{aligned} \left\| \sup _{1\le t\le 2}|\mathfrak {M}^\alpha _t f| \right\| _{L^p({\mathbb {R}^n})}\le C\Vert f\Vert _{L^p({\mathbb {R}^n})} \end{aligned}$$

(3.14)

holds for some $\alpha \in \mathbb {C}$. Then, we have

$$\begin{aligned} \textrm{Re}\,\alpha \ge \frac{1}{p}-\frac{n-1}{2}. \end{aligned}$$

Let us prove Proposition 3.4. Fix $N>-{(n-2)/p}-\textrm{Re}\,\alpha$ as in Lemma 2.1. By (2.7) and Lemma 2.1, the proof of Proposition 3.4 reduces to show the following lemma.

Lemma 3.5

Let $n\ge 2$ and $p>1$. Suppose

$$\begin{aligned} \left\| \sup _{1\le t\le 2}|{{\mathscr {A}}}_t f|\right\| _{L^p(\mathbb {R}^n)}\le C\Vert f\Vert _{W^{s,p}({\mathbb {R}^n})} \end{aligned}$$

(3.15)

holds for some $s\in \mathbb {R}$. Then, we have $s\ge 1/p$.

Proof

Let $\delta >0$ be a small number to be chosen later, and denote $\xi = (\xi _1, \xi ^\prime )\in {\mathbb {R}^n}$. For a given large $j\in {\mathbb N}$, we let ${\hat{f}}\ge 0$ be a smooth cut-off of the set

$$\begin{aligned} \left\{ (\xi _1,\xi ^\prime )\in {\mathbb {R}^n}:|\xi _1-2^j|\le \delta 2^{j-1}, |\xi ^\prime |\le \delta 2^{j/2}\right\} \end{aligned}$$

(3.16)

such that $\big |\partial _{\xi }^{\beta } {{\hat{f}}}(\xi )\big |\le C_{\delta ,\beta } 2^{-j|\beta '|/2}2^{-j|\beta _{1}|}$ for any $\beta =(\beta _{1},\beta ')\in \mathbb {Z}_{+}^{n}$. By a simple calculation, we see that

$$\begin{aligned} |\xi |-\xi _1\le C\delta ^2 \end{aligned}$$

(3.17)

in the support of ${\hat{f}}$. Let j be large enough such that $(1-\varphi (t|\xi |)){\hat{f}}(\xi )={\hat{f}}(\xi )$ for all $t\in [1,2]$, $\xi \in {\mathbb {R}^n}$ and

$$\begin{aligned} \inf _{\xi \in \textrm{supp}{\hspace{.05cm}}{{\hat{f}}}}|a_2(\xi )|\ge c_{low}>0. \end{aligned}$$

(3.18)

Note by [20, Chapter IX, Section 4] we have

$$\begin{aligned}\sup _{1\le t\le 2}\big |\partial _{\xi }^{\beta } \big (e^{2\pi it(|\xi |-\xi _1)}a_1(t|\xi |){{\hat{f}}}(\xi )\big )\big |\le C_{\delta ,\beta } 2^{-j|\beta '|/2}2^{-j|\beta _{1}|}.\end{aligned}$$

Then for $1\le t\le 2$ and $x_1>0$, we use integration by parts to bound that

$$\begin{aligned}&\left| \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi + t|\xi |)}a_1(t|\xi |) {\hat{f}}(\xi ) \,\text {d}\xi \right| \nonumber \\&\qquad = \left| \int _{{\mathbb {R}^n}} e^{2\pi i(x+t v_1)\cdot \xi }\left( e^{2\pi it(|\xi |-\xi _1)}a_1(t|\xi |) {\hat{f}}(\xi )\right) \,\text {d}\xi \right| \nonumber \\&\qquad \le C_{\delta } 2^{-jN}2^{j\frac{n+1}{2}}(x_1+t)^{-N}\le C_{\delta } 2^{-jN}2^{j\frac{n+1}{2}}, \end{aligned}$$

(3.19)

where $v_1=(1,0,\ldots ,0)$, $N\ge 1$ and the constant $C_\delta$ is independent of j and t.

As for $\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - t|\xi |)}a_2(t|\xi |) {\hat{f}}(\xi ) \,\text {d}\xi$ with $1\le t\le 2$, we split it into three terms

$$\begin{aligned}&\int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - t|\xi |)}a_2(t|\xi |) {\hat{f}}(\xi ) \,\text {d}\xi \nonumber \\&\qquad = \int _{{\mathbb {R}^n}} e^{2\pi i(x-t v_1)\cdot \xi }\big (e^{2\pi it(-|\xi |+\xi _1)}-1\big ) a_2(t|\xi |){\hat{f}}(\xi ) \,\text {d}\xi \nonumber \\&\quad \qquad + \int _{{\mathbb {R}^n}} (e^{2\pi i(x-t v_1)\cdot \xi }-1) a_2(t|\xi |){\hat{f}}(\xi )\,\text {d}\xi \nonumber \\&\quad \qquad +\int _{{\mathbb {R}^n}} a_2(t|\xi |){\hat{f}}(\xi ) \,\text {d}\xi . \end{aligned}$$

(3.20)

By (3.17), the first term of (3.20) is bounded by

$$\begin{aligned} C\int _{{\mathbb {R}^n}} \big |t(-|\xi |+\xi _1)\big | {\hat{f}}(\xi ) \,\text {d}\xi \le C\delta ^{2}\int _{{\mathbb {R}^n}} {\hat{f}}(\xi ) \,\text {d}\xi \le C\delta ^{n+2} 2^{j(n+1)/2}. \end{aligned}$$

If $|x_1-t|\le \delta 2^{-j}$ and $|x^\prime |\le 2^{-j/2}$, by the support condition (3.16) of ${\hat{f}}$, we have

$$\begin{aligned} \big |(x-tv_1)\cdot \xi \big |\le C\delta , \, \text {for all}\, \xi \in \textrm{supp}{\hspace{.05cm}}{\hat{f}}, \end{aligned}$$

which implies the second term of (3.20) is bounded by

$$\begin{aligned} C\int _{{\mathbb {R}^n}} \big |(x-tv_1)\cdot \xi \big | {\hat{f}}(\xi )\,\text {d}\xi \le C\delta \int _{{\mathbb {R}^n}} {\hat{f}}(\xi )\,\text {d}\xi \le C\delta ^{n+1} 2^{j(n+1)/2}. \end{aligned}$$

By (3.18), we have

$$\begin{aligned} \left| \int _{{\mathbb {R}^n}} a_2(t|\xi |){\hat{f}}(\xi ) \,\text {d}\xi \right| \ge \frac{c_{low}}{2}\int _{{\mathbb {R}^n}} {\hat{f}}(\xi ) \,\text {d}\xi \ge C_{L}\delta ^{n}2^{j\frac{n+1}{2}}. \end{aligned}$$

Then by (3.20) and the above estimates, if $\delta \le \min \{\frac{C_{L}}{2C_{U}},1\}$, we have

$$\begin{aligned}&\left| \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi - t|\xi |)}a_2(t|\xi |) {\hat{f}}(\xi ) \,\text {d}\xi \right| \nonumber \\&\qquad \ge \left| \int _{{\mathbb {R}^n}} a_2(t|\xi |){\hat{f}}(\xi ) \,\text {d}\xi \right| -C_{U}\delta ^{n+1}2^{j\frac{n+1}{2}}\ge \frac{C_{L}}{2}\delta ^{n}2^{j\frac{n+1}{2}} \end{aligned}$$

(3.21)

if $|x_1-t|\le \delta 2^{-j}$ and $|x^\prime |\le 2^{-j/2}$. It then follows from (3.19) and (3.21) that

$$\begin{aligned} \sup _{1\le t\le 2}|{{\mathscr {A}}}_t f|\ge \frac{C_{L}}{2}\delta ^{n}2^{j\frac{n+1}{2}} -C_{\delta } 2^{-jN}2^{j\frac{n+1}{2}}\ge \frac{C_{L}}{4}\delta ^{n}2^{j\frac{n+1}{2}}, \end{aligned}$$

(3.22)

when $1\le x_1\le 2$, $|x^\prime |\le 2^{-j/2}$ and $j\ge \frac{1}{N}\log _{2}(\frac{4C_{\delta }}{\delta ^{n}C_{L}}+1)$.

Assume (3.15) is true. Then from the definition of f and (3.22), we have

$$\begin{aligned} \frac{C_{L}}{4}\delta ^{n}2^{(n+1)j/2-(n-1)j/(2p)}&\le \left\| \sup _{1\le t\le 2}|{{\mathscr {A}}}_t f|\right\| _{L^p(\mathbb {R}^n)} \nonumber \\&\le C\Vert f\Vert _{W^{s,p}({\mathbb {R}^n})}\le C_\delta 2^{sj}2^{(n+1)j/2-(n+1)j/(2p)}. \end{aligned}$$

(3.23)

Let $j\rightarrow \infty$, then we obtain $s\ge 1/p$. This proves Lemma 3.5, and then the proof of Proposition 3.4 is complete. $\square$

We finally present the endgame in the

Proof of (i) of Theorem 1.1

This is a consequence of Proposition 3.1 and Proposition 3.4. $\square$

4 Proof of (ii) of Theorem 1.1

In this section, we give a criterion that allows us to derive $L^p$-boundedness for the maximal operator ${\mathfrak {M}}^{\alpha }$ on ${{\mathbb {R}}^n}, n\ge 2$. As a consequence, (ii) of Theorem 1.1 follows readily by applying the result of Guth, Wang and Zhang [7] on local smoothing estimate on ${\mathbb R^2}$. More precisely, we have the following result.

Proposition 4.1

Let $n\ge 2$ and $p>2$. If the local smoothing estimate

$$\begin{aligned} \left\| e^{it\sqrt{-\Delta }}f \right\| _{L^p({\mathbb {R}^n}\times [1,2])} \le C_{n,p} \Vert f\Vert _{W^{s, p}({\mathbb {R}^n})} \end{aligned}$$

(4.1)

holds for some $s\in \mathbb {R}$, then we have

$$\begin{aligned} \left\| \sup _{t>0}| {\mathfrak {M}}_t^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le C_{n,p,\alpha } \Vert f\Vert _{L^p({\mathbb {R}^n})} \end{aligned}$$

(4.2)

whenever $\textrm{Re}\, \alpha > \max \big \{ -{(n-1)/p}, \ s -{(n-1)/2} +{1/p}\big \}.$

The proof of Proposition 4.1 is inspired by [10]. Let $\varphi$ and $\{\psi _j\}_j$ be functions in (2.9). We write

$$\begin{aligned} \widehat{{\mathfrak {M}}_t^\alpha f}(\xi )&= \varphi (t|\xi |)\widehat{m_{\alpha }}(t\xi ){\hat{f}}(\xi )+\sum _{j\ge 1}\psi _j(t|\xi |)\widehat{m_{\alpha }}(t\xi ){\hat{f}}(\xi ) \nonumber \\&=:\widehat{{\mathfrak {M}}_{0,t}^\alpha f}(\xi )+\sum _{j\ge 1}\widehat{{\mathfrak {M}}_{j,t}^\alpha f}(\xi ). \end{aligned}$$

(4.3)

To prove Proposition 4.1, the first strategy is to show that if one modifies the definition so that for each operator ${\mathfrak {M}}_{j,t}^\alpha$, the supremum is taken over $1\le t\le 2$, then the resulting maximal function is bounded on $L^p({\mathbb R^n})$.

Lemma 4.2

Let $n\ge 2$ and $p>2$. Under the assumption (4.1) of Proposition 4.1, there exist $\delta >0$ and $C>0$, such that for all $j\ge 1$,

$$\begin{aligned} \left\| \sup _{t\in [1,2]}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le C2^{-\delta j} \Vert f\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

(4.4)

if $\textrm{Re}\, \alpha > \max \big \{ -{(n-1)/p}, s -{(n-1)/2} +{1/p}\big \}$.

Proof

By (2.5), (2.7) and (2.10), it suffices to show

$$\begin{aligned} \left\| \sup _{t\in [1,2]}| \mathscr {A}_{j,t}f |\right\| _{L^p({\mathbb {R}^n})}\le C2^{[ \max \{ (n-1)(1/2-1/p), \, s+{1/p}\}]j} \Vert f\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

(4.5)

where $\widehat{\mathscr {A}_{j,t}f}(\xi )=\psi _j(t|\xi |)\widehat{\mathscr {A}_{t}f}(\xi )$ and $\mathscr {A}_t f$ is defined in (2.13). By (2.6), we can write

$$\begin{aligned} {{\mathscr {A}}}_{j,t} f(x)= C\sum _{\ell =0}^{N-1}\int _{{\mathbb {R}^n}} \left( b_{\ell } e^{2\pi i(x\cdot \xi + t|\xi |)} + d_{\ell } e^{2\pi i(x\cdot \xi - t|\xi |)} \right) |t\xi |^{-\ell }\psi _j(t|\xi |) {\hat{f}}(\xi ) \,\text {d}\xi , \end{aligned}$$

which is a linear combination of

$$\begin{aligned} T_{\ell ,j}f(x,t):=\int _{{\mathbb {R}^n}}e^{2\pi i(x\cdot \xi \pm t|\xi |)}|t\xi |^{-\ell }\psi _j(t|\xi |){\hat{f}}(\xi )\,\text {d}\xi ,\ \ \ \ell =0,1,\ldots ,N-1. \end{aligned}$$

Hence, the proof of (4.5) reduces to showing that

$$\begin{aligned} \left\| \sup _{t\in [1,2]}| T_{0,j}f(\cdot ,t) |\right\| _{L^p({\mathbb {R}^n})}\le C2^{ [ \max \{ (n-1)(1/2-1/p), \, s+{1/p}\}]j} \Vert f\Vert _{L^p({\mathbb {R}^n})}, \ \ \ j\ge 1. \end{aligned}$$

(4.6)

Now we apply Lemma 2.2 to deal with (4.6). First, it follows from [20, Theorem 2, Chapter IX] that

$$\begin{aligned} \Vert T_{0,j}f(\cdot ,1) \Vert _{L^p({\mathbb {R}^n})}\le C 2^{(n-1)(1/2-1/p)j}\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

(4.7)

Next, we observe that for any $1\le t\le 2$ and $j\ge 1$, there holds

$$\begin{aligned} \big |\partial _\xi ^\beta \big (\psi _j(t|\xi |)\big )\big |\le C(1+|\xi |)^{-|\beta |}, \end{aligned}$$

where $\beta$ is any multi-index. So $\psi _j(t|\cdot |)\in S^0$ uniformly $1\le t\le 2$ and $j\ge 1$, hence

$$\begin{aligned}&\int _{{\mathbb {R}^n}}\left| \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi \pm t|\xi |)}\psi _j(t|\xi |){\hat{f}}(\xi )\,\text {d}\xi \right| ^p\,\text {d}x \nonumber \\&\quad \le C \int _{{\mathbb {R}^n}}\left| \int _{{\mathbb {R}^n}} e^{2\pi i(x\cdot \xi \pm t|\xi |)}{\tilde{\psi }}_j(\xi ){\hat{f}}(\xi )\,\text {d}\xi \right| ^p\,\text {d}x, \end{aligned}$$

(4.8)

where constant C is independent of t and j. Here ${\tilde{\psi }}_j$ equals to 1 if $|\xi |\in [2^{j-2}M,2^{j+1}M]$ and vanishes if $|\xi |\notin [2^{j-3}M,2^{j+2}M]$, so that ${\tilde{\psi }}_j$ equals to 1 on the support of $\psi _j(t|\cdot |)$ when $1\le t\le 2$. Then we apply our assumption (4.1) on local smoothing estimate to (4.8) to obtain

$$\begin{aligned} \Vert T_{0,j}f \Vert _{L^p({\mathbb {R}^n}\times [1,2])}\le C 2^{s j}\Vert f\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

and by the same token, the operator

$$\begin{aligned} \partial _t T_{0,j}(x,t) = \int _{{\mathbb {R}^n}}e^{2\pi i(x\cdot \xi \pm t|\xi |)}\big (\pm 2\pi i|\xi |\psi _j(t|\xi |)+|\xi |\psi _j^\prime (t|\xi |)\big ){\hat{f}}(\xi )\,\text {d}\xi . \end{aligned}$$

satisfies

$$\begin{aligned} \Vert \partial _tT_{0,j}f \Vert _{L^p({\mathbb {R}^n}\times [1,2])}\le C 2^{(s+1)j}\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

Thus, we use Lemma 2.2 to get

$$\begin{aligned} \left\| \sup _{t\in [1,2]}| T_{0,j}f(\cdot ,t) |\right\| _{L^p({\mathbb {R}^n})} \le C \big (2^{(n-1)(1/2-1/p)j}+2^{(s+1/p)j}\big )\Vert f\Vert _{L^p({\mathbb {R}^n})}, \end{aligned}$$

which implies estimate (4.6). $\square$

Finally, we can apply Lemma 4.2 to prove Proposition 4.1.

Proof of Proposition 4.1

By (4.3) and (2.7), (4.2) reduces to

$$\begin{aligned} \left\| \sup _{t>0}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le C2^{-\delta j} \Vert f\Vert _{L^p({\mathbb {R}^n})} \end{aligned}$$

(4.9)

for some $\delta >0$. Since $\ell ^p\subseteq \ell ^\infty$, we have

$$\begin{aligned} \left\| \sup _{t>0}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le \left( \sum _{k\in \mathbb {Z}} \left\| \sup _{t\in [2^k,2^{k+1}]}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}^p\right) ^{1/p}. \end{aligned}$$

(4.10)

However, it follows from Lemma 4.2 and a rescaling $t\rightarrow 2^{-k} t$ that

$$\begin{aligned} \left\| \sup _{t\in [2^k,2^{k+1}]}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le C2^{-\delta j}\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

(4.11)

Then for $2^k\le t\le 2^{k+1}$, there must be $|\xi |\in [2^{j-k-2}M,2^{j-k+1}M]$. This tells us that we can rewrite (4.11) as

$$\begin{aligned} \left\| \sup _{t\in [2^k,2^{k+1}]}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le C2^{-\delta j}\Vert P_{j-k}f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

This, together with (4.10), implies

$$\begin{aligned} \left\| \sup _{t>0}| {\mathfrak {M}}_{j,t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}&\le C2^{-\delta j}\left( \sum _{k\in \mathbb {Z}} \Vert P_{j-k}f\Vert _{L^p({\mathbb {R}^n})}^p\right) ^{1/p} \\&=C2^{-\delta j}\left\| \left( \sum _{k\in \mathbb {Z}}|P_{j-k}f|^p\right) ^{1/p}\right\| _{L^p({\mathbb {R}^n})}\\&\le C2^{-\delta j}\left\| \left( \sum _{k\in \mathbb {Z}}|P_{j-k}f|^2\right) ^{1/2}\right\| _{L^p({\mathbb {R}^n})} \end{aligned}$$

since $p>2$. By the Littlewood–Paley inequality [5],

$$\begin{aligned} \left\| \left( \sum _{k\in \mathbb {Z}}|P_{j-k}f|^2\right) ^{1/2}\right\| _{L^p({\mathbb {R}^n})}\le C\Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

This proves (4.9). Hence, the proof of Proposition 4.1 is complete. $\square$

Remark 4.3

(i) In the dimension $n\ge 3$ Gao et al. [3] obtained improved local smoothing estimates for the wave equation, that is, (4.1) holds with $s= (n-1)({1/2}-1/p)-\sigma$ for all $\sigma <2/p-1/2$ when

$$\begin{aligned} p> \left\{ \begin{array}{lll} {2(3n+5)\over 3n+1}, \ \ \ \textrm{for}\ &{} n \ \textrm{odd}; \\ {2(3n+6)\over 3n+2}, \ \ \ \textrm{for}\ &{}n \ \textrm{even}. \end{array} \right. \end{aligned}$$

Applying Proposition 4.1, we get that (1.3) holds if $\textrm{Re}\,\alpha > \alpha (p, n)$ where

$$\begin{aligned} \hspace{1cm} \alpha (p, n)= \left\{ \begin{array}{lll} \max \left\{ -{n-1\over p}, -\frac{3}{8}(n-1)+\frac{5-n}{4p},\frac{4(n-1)}{(3n+5)(n+3)}-\frac{n^{2}-5}{(n+3)p}\right\} , \ \ \ {} &{} \textrm{for}\ n \ \textrm{odd}; \\ \max \left\{ -{n-1\over p},-\frac{3n-2}{8}-\frac{n-6}{4p},-\frac{n-1}{n+4}-\frac{n^{2}+n-6}{(n+4)p}\right\} , \ \ \ &{} \textrm{for}\ n \ \textrm{even}. \end{array} \right. \qquad \end{aligned}$$

(4.12)

The above range $\alpha$ in (4.12) for $p>2$ is strictly wider than (1.7). However, the range p in (4.12) is not optimal. What happens when $n\ge 3$ (and $p>2$) remains open.

(ii) Under the assumption (4.1) of Proposition 4.1, it follows by (4.4) that for $n\ge 2$ and $p>2$,

$$\begin{aligned} \left\| \sup _{t\in [1,2]}| {\mathfrak {M}}_{t}^{\alpha }f |\right\| _{L^p({\mathbb {R}^n})}\le C \Vert f\Vert _{L^p({\mathbb {R}^n})} \end{aligned}$$

provided that $\textrm{Re}\, \alpha > \max \big \{ -{(n-1)/p}, s -{(n-1)/2} +{1/p}\big \}$. It is interesting to describe the full range of (p, q) such that

$$\begin{aligned} \left\| \sup _{t\in [1,2]}| {\mathfrak {M}}_{t}^{\alpha }f |\right\| _{L^q({\mathbb {R}^n})}\le C \Vert f\Vert _{L^p({\mathbb {R}^n})}. \end{aligned}$$

For $\alpha =0$, we refer it to [9, 13,14,15] and the references therein.