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Linear adjoint restriction estimates for paraboloid

  • Changxing Miao
  • Junyong ZhangEmail author
  • Jiqiang Zheng
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Abstract

We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127–1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091–2102]. As an application, we show a local smoothing estimate for a solution of the linear Schrödinger equation under the assumption that the initial datum has additional angular regularity.

Keywords

Linear adjoint restriction estimate Local restriction estimate Bessel function Spherical harmonics Local smoothing 

Mathematics Subject Classification

42B37 42B10 42B25 35Q55 

1 Introduction

Let S be a non-empty smooth compact subset of the paraboloid,
$$\begin{aligned} \big \{~(\tau ,\xi )\displaystyle \int {\mathbb {R}}\times {\mathbb {R}}^n: \tau =|\xi |^2~\big \}, \end{aligned}$$
where \(n\ge 1\). We denote by \(d\sigma \) the pull-back of the n-dimensional Lebesgue measure \(d\xi \) under the projection map \((\tau ,\xi )\mapsto \xi \). Let f be a Schwartz function and define the inverse space-time Fourier transform of the measure \(fd\sigma \)
$$\begin{aligned} (fd\sigma )^{\vee }(t,x)&=\int \limits _{S} f(\tau , \xi )e^{2\pi i(x\cdot \xi +t\tau )}d\sigma (\xi ) \\&=\int \limits _{{\mathbb {R}}^n} f(|\xi |^2, \xi )e^{2\pi i(x\cdot \xi +t|\xi |^2)}d\xi .\nonumber \end{aligned}$$
(1.1)
The classical linear adjoint restriction estimate for the paraboloid reads
$$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\le C_{p,q,n,S}\Vert f\Vert _{L^p(S;d\sigma )}, \end{aligned}$$
(1.2)
where \(1\le p,q\le \infty \). The famous restriction problem is to find the optimal range of p and q such that the estimate (1.2) holds. It is known that the condition
$$\begin{aligned} q>\frac{2(n+1)}{n} \quad \text {and}\quad \frac{n+2}{q}\le \frac{n}{p'}, \end{aligned}$$
(1.3)
is necessary for (1.2), see [24, 29]. Here \(p'\) denotes the conjugate exponent of p. The adjoint restriction estimate conjecture on paraboloid reads as follows.

Conjecture 1.1

The inequality (1.2) holds true if and only if inequalities (1.3) are valid.

There is a large amount of literature on this problem. For \(n=1\), Conjecture 1.1 was proved by Fefferman-Stein [11] for the non-endpoint case and by Zygmund [36] for the endpoint case. Conjecture 1.1 in high dimension case becomes much more difficult. For \(n\ge 2\), Tomas [33] showed (1.2) for \(q>{2(n+2)}/n\), and Stein [25] fixed the limit case \(q={2(n+2)}/n\). Bourgain [1] further proved estimate (1.2) for \(q>2(n+2)/n-\epsilon _n\) with some \(\epsilon _n>0\); in particular, \(\epsilon _n=\frac{2}{15}\) when \(n=2\). Further improvements were made by Moyua-Vargas-Vega [16] and Wolff [34]. Tao [31] used the bilinear argument to show that estimate (1.2) holds true for \(q>{2(n+3)}/{(n+1)}\) with \(n\ge 2\). This result was improved by Bourgain-Guth [2] when \(n\ge 4\). This conjecture is so difficult that it remains open up to now. For more details, we refer the reader to [2, 29, 30, 31, 32, 34].

On the other hand, the restriction conjecture becomes simpler (but not trivial) when a test function has some angular regularity. For example, Conjecture 1.1 is proved by Shao [22] when test functions are cylindrically symmetric and are supported on a dyadic subset of the paraboloid in the form of
$$\begin{aligned} \Big \{(\tau ,\xi )\in {\mathbb {R}}\times {\mathbb {R}}^n:\; M\le |\xi |\le 2M,\;\; \tau =|\xi |^2,\;\; M\in 2^{{\mathbb {Z}}}\Big \}. \end{aligned}$$
Indeed, many famous conjectures in harmonic analysis (such as Fourier restriction estimates, Bochner-Riesz estimate etc.) have easier counterparts when the corresponding operators act on radial functions. Let \({\mathbb {S}}^{n-1}\) denote the unit sphere in \({\mathbb {R}}^n\) and \(L^q_{\text {sph}} := L^{q}_\theta ({\mathbb {S}}^{n-1})\), the intermediate situation is to replace the \(L^q({\mathbb {R}}^n)\) by \(L^q_{r^{n-1}dr}L^2_{\text {sph}}\) in (1.2). This intermediate case has been settled for adjoint restriction estimates for a cone by the authors of [17]. More precisely, if S is a non-empty smooth compact subset of the cone:
$$\begin{aligned} S=\big \{(\tau ,\xi )\in {\mathbb {R}}\times {\mathbb {R}}^n:\;\; \tau =|\xi |\big \}, \end{aligned}$$
then for \(q>{2n}/(n-1)\) and \((n+1)/q\le (n-1)/p'\) we have
$$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^q_t({\mathbb {R}};L^q_{r^{n-1}dr}L^{2}_{\text {sph}})}\le C_{p,q,n,S}\Vert f\Vert _{L^p(S;d\sigma )}. \end{aligned}$$
(1.4)
The \(L^2_{\text {sph}}\)-norm allows us to use spherical harmonic expanding, so the problem is converted to \(L^q(\ell ^2)\)-bounds for sequences of operators \(\{H_{k}\}\) where each \(H_k\) is an operator acting on radial functions. The pioneering paper using such intermediate space is the Mockenhaupt Diploma in which he proved weighted \(L^p\) inequalities and then sharp \(L^p_{\mathrm {rad}}(L^2_\mathrm {sph})\rightarrow L^p_{\mathrm {rad}}(L^2_\mathrm {sph})\) estimates for the disc multiplier operator, see either Mockenhaupt [14] or Córdoba [5]. Sharp endpoint bounds for the disk multiplier were obtained by Carbery-Romera-Soria [4]. Müller-Seeger [15] established some sharp mixed spacetime \(L^p_{\mathrm {rad}}(L^2_\mathrm {sph})\) estimates in order to study a local smoothing of solutions for the linear wave equation. Córdoba-Latorre [9] revisited some classical conjecture including restriction estimate in harmonic analysis in this kind of mixed space-time. Gigante-Soria [12] studied a related mixed norm problem for Schrödinger maximal operators. Concerning the sphere restriction conjecture, Carli-Grafakos [7] also treated the same problem for spherically-symmetric functions and Cho-Guo-Lee [8] showed a restriction estimate for \(q>2(n+1)/n\) and \(s\ge (n+2)/q-n/2\)
$$\begin{aligned} \left\| \int \limits _{{\mathbb {S}}^{n}} e^{2\pi ix\cdot \xi } f(\xi )d\sigma (\xi )\right\| _{L^q({\mathbb {R}}^{n+1})}\le C\Vert f\Vert _{H^s({\mathbb {S}}^{n})}, \quad x\in {\mathbb {R}}^{n+1}, \end{aligned}$$
(1.5)
where \(d\sigma \) is the induced Lebesgue measure on \({\mathbb {S}}^{n}\) and \(H^s({\mathbb {S}}^{n})\) denote the \(L^2\)-Sobolev space of order s on the sphere. An advantage of the proof consists in a fact that inequality (1.5) is based on \(L^2\)-spaces. The advantage of using the \(L^2\)-based Hilbert space also allows us to use effective the \(TT^*\) arguments to obtain Strichartz estimate with a wider range of admissible indexes by compensating with extra regularity in angular direction; see Sterbenz [21] for wave equation, Cho-Lee [9] for general dispersive equations and the authors [18] for wave equation with an inverse-square potential. Concerning other results in this direction, Cho-Hwang-Kwon-Lee [10] studied profile decompositions of fractional Schrödinger equations under the angular regularity assumption.

In this paper, we prove that estimate (1.2) holds for all pq in (1.3) by compensating with some loss of angular derivatives. Our strategy is to use a spherical harmonic expanding as well as localized restriction estimates. In contrast to the radial case, e.g. [7, 22], the main difficulty comes from the asymptotic behavior of the Bessel function \(J_{\nu }(r)\) when \(\nu \gg 1\). It is worth to point out that the method of treating cone restriction [17] is not valid since it can not be used to exploit the curvature property of paraboloid multiplier \(e^{it|\xi |^2}\). We note that the bilinear argument used in [22], which is in spirit of Carleson-Sjölin argument or equivalently the \(TT^*\) argument, can be used to deal with the oscillation of the paraboloid multiplier. To use this argument, one needs to write the Bessel function \(J_\nu (r)\sim c_\nu r^{-1/2}e^{ir}\) when \(r\gg 1\). This expression works well for small \(\nu \) (corresponding to the radial case) but it seems complicate to write the Bessel function in that form when \(\nu \gg 1\). Indeed, as in [37], one can do this when \(\nu ^2\ll r\), but it will cause more loss of derivative for the case \(\nu \lesssim r\lesssim \nu ^2\), since it is difficult to capture simultaneously the oscillation and decay behavior of \(J_{\nu }(r)\). Our new idea here is to establish a \(L^4_{t,x}\)-localized restriction estimate by directly analyzing the kernel associated with the Bessel function. The key ingredient is to explore the decay and oscillation property of \(J_\nu (r)\) for \(r\gg \nu \), and resonant property of paraboloid multiplier. We also have to overcome low decay shortage of \(J_{\nu }(r)\) (when \(\nu \sim r\gg 1\)) by compensating a loss of angular regularity.

Before stating the main theorem, we introduce some notation. Incorporating the angular regularity, we set the infinitesimal generators of the rotations on Euclidean space:
$$\begin{aligned} \Omega _{j,k}:=x_j\partial _k-x_k\partial _j \end{aligned}$$
and define for \(s\in {\mathbb {R}}\)
$$\begin{aligned} \Delta _\theta :=\sum \limits _{j<k}\Omega _{j,k}^2,\quad |\Omega |^s=(-\Delta _{\theta })^{\frac{s}{2}}. \end{aligned}$$
Hence \(\Delta _{\theta }\) is the Laplace-Beltrami operator on \({\mathbb {S}}^{n-1}\). Define the Sobolev norm \(\Vert \cdot \Vert _{H^{s,p}_{\text {sph}}({\mathbb {R}}^n)}\) by setting
$$\begin{aligned} \Vert g\Vert ^p_{H^{s,p}_{\text {sph}}({\mathbb {R}}^n)}=\int \limits _0^\infty \int \limits _{{\mathbb {S}}^{n-1}}|(1-\Delta _{\theta })^{s/2}g(r\theta )|^p d\theta ~r^{n-1}dr. \end{aligned}$$
(1.6)
Given a constant A, we briefly write \(A+\epsilon \) as \(A_+\) or \(A-\epsilon \) as \(A_-\) for \(0<\epsilon \ll 1\).

Our main result is the following one.

Theorem 1.1

Let \(n\ge 2\). The following estimates hold for all Schwartz functions f
  • if \(q_0=(2(n+1)/n)_+\) and \((n+2)/q_0=n/p_0'\), then
    $$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^{q_0}_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\le C_{p,q_0,n,S}\Vert f(|\xi |^2,\xi )\Vert _{H^{\sigma _0,p_0}_{\mathrm {sph}}({\mathbb {R}}^n_{\xi })}, \end{aligned}$$
    (1.7)
    where \(\sigma _0=(n-2)\big (\frac{1}{2}-\frac{1}{q_0}\big )+\frac{2}{q_0}\);
  • if \(1\le q,p\le \infty \) satisfy (1.3), then
    $$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\le C_{p,q,n,S}\Vert (1+|\Omega |)^{s} f\Vert _{L^p(S;d\sigma )}, \end{aligned}$$
    (1.8)
where \(s=s(q,n)=\sigma _0\alpha \) and \(0\le \alpha \le 1\) satisfying \(1/q=\alpha /q_0+(1-\alpha )/q_1\). Here \(q_1=q(n)_+\) with \(q(n)=2+12/(4n+1-k)\) if \(n+1\equiv k (\text {mod}~3), k=-1,0,1\) as in Bourgain-Guth [2, Theorem 1].

Remark 1.1

Estimate (1.8) is an interpolation consequence of (1.7) and \(L^p\)-estimates in Bourgain-Guth [2]. Inequality (1.8) leads to the linear adjoint restriction estimate when \(q\in (2(n+1)/n, q(n)]\) with some loss of angular derivatives.

Remark 1.2

Since the sphere \({\mathbb {S}}^n=\{(\tau ,\xi ): |\tau |^2+|\xi |^2=1\}\) is closely related to the paraboloid in sense of Taylor expansion \(\sqrt{1-\rho ^2}=1-\frac{1}{2} \rho ^2+O(\rho ^4)\) near \(\rho =0\), it seems to be possible to show some modified version of (1.5) with \(H^{s,p}({\mathbb {S}}^n)\)-norm on right hand side.

As an application of the modified restriction estimate, we show a result on the local smoothing estimate for the Schödinger equation for initial data with additional conditions angular regularity by Rogers’s argument in [20]. Our result here extend [20, Theorem 1] from \(q>2(n+3)/(n+1)\) to \(q>2(n+1)/n\) under the assumption that initial data has additional angular regularity.

More precisely, we have the following local smoothing result.

Corollary 1.1

Let \(n\ge 2\), \(q>2(n+1)/n\) and s be as in Theorem 1.1. Then
$$\begin{aligned} \Vert e^{it\Delta }u_0\Vert _{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)}\le C\big \Vert (1+|\Omega |)^su_0\big \Vert _{W^{\alpha ,q}({\mathbb {R}}^n)}, \end{aligned}$$
(1.9)
where \(\alpha >2n(1/2-1/q)-2/q\) and \(W^{\alpha ,q}({\mathbb {R}}^n)\) is the Sobolev space.

This paper is organized as follows: In Sect. 2, we introduce notation and present some basic facts about spherical harmonics and Bessel functions. Furthermore, we use the stationary phase argument to prove some properties of Bessel functions. Section 3 is devoted to the proof of Theorem 1.1. In Sect. 4, we prove the key Proposition 3.1. We prove Corollary 1.1 in the final section.

2 Preliminaries

2.1 Notation

We use \(A\lesssim B\) to denote the statement that \(A\le CB\) for some large constant C which may vary from line to line and depend on various parameters, and similarly employ \(A\sim B\) to denote the statement that \(A\lesssim B\lesssim A\). We also use \(A\ll B\) to denote the statement \(A\le C^{-1} B\). If a constant C depends on a special parameter other than the above, we shall write it explicitly by subscripts. For instance, \(C_\epsilon \) should be understood as a positive constant not only depending on pqn and S, but also on \(\epsilon \). Throughout this paper, pairs of conjugate indices are written as \(p, p'\), where \(\frac{1}{p}+\frac{1}{p'}=1\) with \(1\le p\le \infty \). Let \(R>0\) be a dyadic number, we define the dyadic annulus in \({\mathbb {R}}^n\) by
$$\begin{aligned} A_{R}:=\big \{~x\in {\mathbb {R}}^n: \;\; R/2\le |x|\le R~\big \},\quad S_{R}:=[R/2, R]. \end{aligned}$$
For each \(M\in 2^{{\mathbb {Z}}}\), we define \({{\mathbb {L}}}_M\) to be the class of Schwartz functions supported on a dyadic subset of the paraboloid in the form of
$$\begin{aligned} \big \{(\tau ,\xi )\in {\mathbb {R}}\times {\mathbb {R}}^n: M\le |\xi |\le 2M, \tau =|\xi |^2\big \}. \end{aligned}$$
(2.1)

2.2 Spherical harmonics expansions and Bessel function

We recall an expansion formula with respect to the spherical harmonics. Let
$$\begin{aligned} \xi =\rho \omega \quad \text {and}\quad x=r\theta \quad \text {with}\quad \omega ,\theta \in {\mathbb {S}}^{n-1}. \end{aligned}$$
(2.2)
For every \(g\in L^2({\mathbb {R}}^n)\), we have the expansion formula
$$\begin{aligned} g( \xi )=\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)}a_{k,\ell }(\rho )Y_{k,\ell }(\omega ), \end{aligned}$$
where
$$\begin{aligned} \big \{Y_{k,1},\ldots , Y_{k,d(k)}\big \} \end{aligned}$$
is the orthogonal basis of the spherical harmonics space of degree k on \({\mathbb {S}}^{n-1}\). This space is recorded by \({\mathcal {H}}^{k}\) and it has the dimension
$$\begin{aligned} d(k)=\frac{2k+n-2}{k}C^{k-1}_{n+k-3}\simeq \langle k\rangle ^{n-2}. \end{aligned}$$
It is clear that we have the orthogonal decomposition of \(L^2({\mathbb {S}}^{n-1})\)
$$\begin{aligned} L^2({\mathbb {S}}^{n-1})=\bigoplus _{k=0}^\infty {\mathcal {H}}^{k}. \end{aligned}$$
It follows that
$$\begin{aligned} \Vert g(\xi )\Vert _{L^2_\omega }=\Vert a_{k,\ell }(\rho )\Vert _{\ell ^2_{k,\ell }}. \end{aligned}$$
(2.3)
Using the spherical harmonic expansion, as well as [19, 28], we define the action of \((1-\Delta _\omega )^{s/2}\) on g as follows
$$\begin{aligned} (1-\Delta _\omega )^{s/2} g=\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)}(1+k(k+n-2))^{s/2}a_{k,\ell }(\rho )Y_{k,\ell }(\omega ). \end{aligned}$$
(2.4)
Given \(s,s'\ge 0\) and \(p,q\ge 1\), define
$$\begin{aligned} \Vert g\Vert _{H^{s,q}_{\rho }H^{s',p}_{\omega }}:=\big \Vert (1-\Delta )^{\frac{s}{2}}\big ((1-\Delta _\omega )^{\frac{s'}{2}}g\big )\big \Vert _{L^{q}_{\mu (\rho )}({\mathbb {R}}^+;L^{p}_{\omega }({\mathbb {S}}^{n-1}))}, \end{aligned}$$
where \(\mu (\rho )=\rho ^{n-1}d\rho \).
For our purpose, we need the inverse Fourier transform of \(a_{k,\ell }(\rho )Y_{k,\ell }(\omega )\). We recall the Bochner-Hecke formula, see [13] and [26, Theorem 3.10]
$$\begin{aligned} {\check{g}}(r\theta )=\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)}2\pi i^{k}Y_{k,\ell }(\theta )r^{-\frac{n-2}{2}}\int \limits _0^\infty J_{\nu (k)}(2\pi r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}d\rho . \end{aligned}$$
(2.5)
Here \(\nu (k)=k+\frac{n-2}{2}\) and the Bessel function \(J_{\nu }(r)\) of order \(\nu \) is defined by
$$\begin{aligned} J_{\nu }(r)=\frac{(r/2)^\nu }{\Gamma (\nu +\frac{1}{2})\Gamma (1/2)}\int \limits _{-1}^{1}e^{isr}(1-s^2)^{(2\nu -1)/2}\mathrm {d }s, \end{aligned}$$
where \(\nu >-1/2\) and \(r>0\). It is easy to verify that there exists a constant C independent of \(\nu \) such that
$$\begin{aligned} |J_\nu (r)|\le \frac{Cr^\nu }{2^\nu \Gamma (\nu +\frac{1}{2})\Gamma (1/2)}\Big (1+\frac{1}{\nu +1/2}\Big ). \end{aligned}$$
(2.6)
To investigate a behavior of asymptotic bound on \(\nu \) and r, we recall the Schläfli integral representation [35] of the Bessel function: for \(r\in {\mathbb {R}}^+\) and \(\nu >-\frac{1}{2}\)
$$\begin{aligned} J_\nu (r)= & {} \frac{1}{2\pi }\int \limits _{-\pi }^\pi e^{ir\sin \theta -i\nu \theta }d\theta -\frac{\sin (\nu \pi )}{\pi }\int \limits _0^\infty e^{-(r\sinh s+\nu s)}ds\nonumber \\=: & {} {\tilde{J}}_\nu (r)-E_\nu (r). \end{aligned}$$
(2.7)
Clearly, \(E_\nu (r)=0\) when \(\nu \in {\mathbb {Z}}^+\). An easy computation shows that
$$\begin{aligned} |E_\nu (r)|=\Big |\frac{\sin (\nu \pi )}{\pi }\int \limits _0^\infty e^{-(r\sinh s+\nu s)}ds\Big |\le C (r+\nu )^{-1}. \end{aligned}$$
(2.8)
There is a number of references for the asymptotic behavior of a Bessel function, see e.g. [9, 23, 25, 35]. We recall some properties of a Bessel function for a convenience.

Lemma 2.1

(Asymptotics of Bessel functions) Let \(\nu \gg 1\) and let \(J_\nu (r)\) be the Bessel function of order \(\nu \) defined as above. Then there exists a large constant C and small constant c independent of \(\nu \) and r such that:
  • When \(r\le \frac{\nu }{2}\), we have
    $$\begin{aligned} |J_\nu (r)|\le C e^{-c(\nu +r)}; \end{aligned}$$
    (2.9)
  • When \(\frac{\nu }{2}\le r\le 2\nu \), we have
    $$\begin{aligned} |J_\nu (r)|\le C \nu ^{-\frac{1}{3}}(\nu ^{-\frac{1}{3}}|r-\nu |+1)^{-\frac{1}{4}}; \end{aligned}$$
    (2.10)
  • When \(r\ge 2\nu \), we have
    $$\begin{aligned} J_\nu (r)=r^{-\frac{1}{2}}\sum \limits _{\pm }a_\pm (\nu ,r) e^{\pm ir}+E(\nu ,r), \end{aligned}$$
    (2.11)
where \(|a_\pm (\nu ,r)|\le C\) and \(|E(\nu ,r)|\le Cr^{-1}\).

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1 by using some localized linear estimates whose proof are postpone to the next section. Since inequality (1.7) is a special case of (1.8), we aim to prove (1.8). Since (1.8) is a direct consequence of the Stein-Tomas inequality [25] for the case \(p\le 2\), it suffices to prove (1.8) for the case \(p\ge 2\). More precisely, we will only establish the estimate for \(q>{2(n+1)}/{n}\), \((n+2)/q={n}/{p'}\) with \(p\ge 2\)
$$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\le C_{p,q,n,S}\Vert (1+|\Omega |\big )^{s}f\Vert _{L^p(S;d\sigma )}. \end{aligned}$$
(3.1)
Recall the notation \({\mathbb {L}}_M\) and \(A_R\) in the Sect. 2.1. We decompose f into a sum of dyadic supported functions
$$\begin{aligned} f=\sum \limits _{M}f_M, \end{aligned}$$
where \(f_M=f\chi _{\{(\tau ,\xi ):\tau =|\xi |^2, M\le |\xi |\le 2M\}}\in {{\mathbb {L}}}_{M}\). It follows that
$$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}= & {} \bigg \Vert \sum \limits _{M}(f_Md\sigma )^{\vee }\bigg \Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\nonumber \\= & {} \bigg (\sum \limits _{R}\Big \Vert \sum \limits _{M}(f_Md\sigma )^{\vee }\Big \Vert ^q_{L^q_{t,x}({\mathbb {R}}\times A_R)}\bigg )^{\frac{1}{q}}\nonumber \\\lesssim & {} \bigg (\sum \limits _{R}\Big (\sum \limits _{M}\left\| (f_Md\sigma )^{\vee }\right\| _{L^q_{t,x}({\mathbb {R}}\times A_R)}\Big )^q\bigg )^{\frac{1}{q}}. \end{aligned}$$
(3.2)
To prove (3.1), we need localized linear restriction estimates.

Proposition 3.1

Assume \(f\in {{\mathbb {L}}}_1\) and \(R>0\) is a dyadic number. Then the following linear restriction estimates hold true.
  • Let \(q=2\), then
    $$\begin{aligned} \Vert (fd\sigma )^\vee \Vert _{L^2_{t,x}({\mathbb {R}}\times A_R)}\lesssim \min \left\{ R^\frac{1}{2}, R^{\frac{n}{2}}\right\} \Vert f\Vert _{L^2(S;d\sigma )}. \end{aligned}$$
    (3.3)
  • Let \(q=3p'\) with \(2\le p\le 4\) and \(\sigma =(n-2)(\frac{1}{2}-\frac{1}{q})+\frac{2}{q}\), \(0<\epsilon \ll 1\), then
    $$\begin{aligned} \Vert (fd\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\lesssim \min \left\{ R^{(n-1)(\frac{1}{q}-\frac{1}{2})+\epsilon }, R^{\frac{n}{q}}\right\} \left\| \big (1+|\Omega |\big )^{\sigma }f\right\| _{L^p(S;d\sigma )}. \end{aligned}$$
    (3.4)
We postpone the proof of Proposition 3.1 to the next section, and we complete the proof of Theorem 1.1 by this proposition. By a scaling argument, we conclude from (3.3) that
$$\begin{aligned} \Vert (f_Md\sigma )^\vee \Vert _{L^2_{t,x}({\mathbb {R}}\times A_R)}\lesssim \min \left\{ (RM)^\frac{1}{2}, (RM)^{\frac{n}{2}}\right\} M^{n-\frac{n+2}{2}-\frac{n}{2}}\Vert f_M\Vert _{L^2(S;d\sigma )}. \end{aligned}$$
For any (qp) satisfying
$$\begin{aligned} q>{2(n+1)}/{n},\;\; (n+2)/q={n}/{p'} \;\;\;\text {with}\;\; \;p\ge 2, \end{aligned}$$
let \(\alpha =2-\frac{3}{q}-\frac{1}{p}\), then we choose \({{\bar{q}}}=3{{\bar{p}}}'\) such that
$$\begin{aligned} \frac{1}{q}=\frac{1-\alpha }{2}+\frac{\alpha }{{\bar{q}}},\qquad \frac{1}{p}=\frac{1-\alpha }{2}+\frac{\alpha }{{\bar{p}}}. \end{aligned}$$
From (3.4), we have that for \({\bar{q}}=3{\bar{p}}'\) with \(2\le {\bar{p}}\le 4\) and \({\bar{\sigma }}=(n-2)(\frac{1}{2}-\frac{1}{{\bar{q}}})+\frac{2}{{\bar{q}}}\)
$$\begin{aligned}&\Vert (f_Md\sigma )^\vee \Vert _{L^{{\bar{q}}}_{t,x}({\mathbb {R}}\times A_R)}\\&\quad \lesssim \min \left\{ (RM)^{(n-1)(\frac{1}{{\bar{q}}}-\frac{1}{2})+{\bar{\epsilon }}}, (RM)^{\frac{n}{{\bar{q}}}}\right\} M^{n-\frac{n+2}{{\bar{q}}}-\frac{n}{{\bar{p}}}}\left\| (1+|\Omega |\big )^{{\bar{\sigma }}}f_M\right\| _{L^{{\bar{p}}}(S;d\sigma )}, \end{aligned}$$
where \(0<{\bar{\epsilon }}\ll 1\). Therefore we obtain by an interpolation theorem
$$\begin{aligned}&\Vert (f_Md\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\nonumber \\&\quad \lesssim \min \{(RM)^{\frac{n}{q}}, (RM)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]+\epsilon }\}\left\| (1+|\Omega |\big )^{\sigma }f_M\right\| _{L^{p}(S;d\sigma )}. \end{aligned}$$
(3.5)
Here \(0<\epsilon :={\bar{\epsilon }}\alpha \ll 1\). According to (3.2), we obtain
$$\begin{aligned}&\Vert (fd\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)} \\&\quad \lesssim \left( \sum \limits _{R}\left( \sum \limits _{M}\min \left\{ (RM)^{\frac{n}{q}}, (RM)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]+\epsilon }\right\} \Vert (1+|\Omega |\big )^{\sigma }f_M\Vert _{L^{p}(S;d\sigma )} \right) ^q\right) ^{\frac{1}{q}}. \end{aligned}$$
Since \(q>{2(n+1)}/n\), \(\epsilon \ll 1\), and RM are both dyadic number, we have
$$\begin{aligned}&\sup _{R>0}\bigg (\sum \limits _M \min \Big \{(RM)^{\frac{n}{q}}, (RM)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]+\epsilon }\Big \}\bigg )<\infty ,\\&\sup _{M>0}\bigg (\sum \limits _R \min \Big \{(RM)^{\frac{n}{q}}, (RM)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]+\epsilon }\Big \}\bigg )<\infty . \end{aligned}$$
Note that for \(q>{2(n+1)}/n>p\ge 2\), we have by the Schur lemma and embedding inequality
$$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\lesssim & {} \bigg (\sum \limits _{M}\Vert (1+|\Omega |\big )^{\sigma }f_M\Vert ^p_{L^{p}(S;d\sigma )}\bigg )^{\frac{1}{p}}\\= & {} \left\| (1+|\Omega |\big )^{\sigma }f\right\| _{L^{p}(S;d\sigma )}. \end{aligned}$$
Choosing \(q=q_0=\left( 2(n+1)/n \right) _+\) and \((n+2)/q_0={n}/{p_0'}\), we have
$$\begin{aligned} \Vert (fd\sigma )^{\vee }\Vert _{L^{q_0}_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)} \lesssim \left\| (1+|\Omega |\big )^{\sigma _0}f\right\| _{L^{p_0}(S;d\sigma )}. \end{aligned}$$
This implies (1.7). Interpolating this inequality with the restriction estimate by Bourgain-Guth [2, Theorem 1], we prove (3.1). Hence, the proof of estimate (1.8) is completed.

4 Localized restriction estimate

In this section we prove Proposition 3.1. We start our proof by recalling
$$\begin{aligned} (f(\tau ,\xi )d\sigma )^\vee (t,x) =\int \limits _{{\mathbb {R}}^{n}} g(\xi )e^{2\pi i(x\cdot \xi +t|\xi |^2)}d\xi , \end{aligned}$$
(4.1)
where \(g(\xi )=f(|\xi |^2, \xi )\in {\mathcal {S}}({\mathbb {R}}^n)\) with \(\text {supp}~g\subset \{\xi :|\xi |\in [1,2]\}\). We apply the spherical harmonic expansion to g to obtain
$$\begin{aligned} g( \xi )=\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)}a_{k,\ell }(\rho )Y_{k,\ell }(\omega ). \end{aligned}$$
Recalling \(\nu (k)=k+(n-2)/2\), we have by (2.5)
$$\begin{aligned} (fd\sigma )^\vee (t,x) =2\pi r^{-\frac{n-2}{2}}\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{-2\pi it\rho ^2} J_{\nu (k)}(2\pi r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )d\rho .\nonumber \\ \end{aligned}$$
(4.2)
Here we insert a harmless smooth bump function \(\varphi \) supported on the interval (1 / 2, 4) into the above integral, since \(a_{k,\ell }(\rho )\) is supported on [1, 2]. Now we estimate the quantity \(\Vert (fd\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\). To this end, we first prove the following lemma.

Lemma 4.1

Let \(\mu (r)=r^{n-1}dr\) and \(\omega (k)\) be a weight specified below. For \(q\ge 2\), we have
$$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big |\int \limits _0^\infty e^{it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{ \frac{n-2}{2}}\rho d\rho \big |^2\Big )^{\frac{1}{2}} \bigg \Vert _{L^q_t({\mathbb {R}};L^q_{\mu (r)}(S_R))}\nonumber \\&\quad \lesssim \bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big \Vert J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{ \frac{n-2}{2}+\frac{1}{q'}}\big \Vert ^2_{L^{q'}_\rho }\Big )^{\frac{1}{2}} \bigg \Vert _{L^q_{\mu (r)}(S_R)}.\nonumber \\ \end{aligned}$$
(4.3)

Proof

Since \(q\ge 2\), the Minkowski inequality and the Fubini theorem show that the left hand side of (4.3) is bounded by
$$\begin{aligned} \bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\Big \Vert \int \limits _0^\infty e^{it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{\frac{n-2}{2}}\rho ~d\rho \Big \Vert ^2_{L^q_t({\mathbb {R}})}\Big )^{\frac{1}{2}}\bigg \Vert _{L^q_{\mu (r)}(S_R)}. \end{aligned}$$
We rewrite this by making the variable change \(\rho ^2\rightsquigarrow \rho \)
$$\begin{aligned} \bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\Big \Vert \int \limits _0^\infty e^{it\rho } J_{\nu (k)}( r\sqrt{\rho })a_{k,\ell }(\sqrt{\rho })\varphi (\sqrt{\rho })\rho ^{\frac{n-2}{4}}~d\rho \Big \Vert ^2_{L^q_t({\mathbb {R}})}\Big )^{\frac{1}{2}}\bigg \Vert _{L^q_{\mu (r)}(S_R)}.\nonumber \\ \end{aligned}$$
(4.4)
We use the Hausdorff-Young inequality with respect to t and we change variables back to obtain
$$\begin{aligned} \text {LHS of }~(4.3)\lesssim \Big \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k) \big \Vert J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{(n-2)/2+1/q'}\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{1}{2}} \Big \Vert _{L^q_{\mu (r)}(S_R)}. \end{aligned}$$
\(\square \)

Now we prove that the inequalities (3.3) and (3.4) with \( R\lesssim 1\). For doing this, we need

Lemma 4.2

Let \(q\ge 2\) and \(R\lesssim 1\), we have the following estimate
$$\begin{aligned} \Vert (f~d\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\lesssim R^{\frac{n}{q}} \bigg (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big \Vert a_{k,\ell }(\rho )\varphi (\rho )\big \Vert _{L^{q'}_{\rho }}^2\bigg )^{\frac{1}{2}}, \end{aligned}$$
(4.5)
where \(\omega (k)=(1+k)^{2(n-1)(1/2-1/q)}\).
We postpone the proof of this lemma for a moment. Note that for \(q'\le 2\le p\), we use (4.5), (2.4), the Minkowski inequality and the Hölder inequality to obtain
$$\begin{aligned} \Vert (f~d\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)} \lesssim&R^{\frac{n}{q}} \bigg \Vert \Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big |a_{k,\ell }(\rho )\big |^2\Big )^{\frac{1}{2}}\varphi (\rho )\bigg \Vert _{L^{q'}_{\rho }}\\ \lesssim&R^{\frac{n}{q}} \left\| g\right\| _{L^{q'}_{\rho }H_\omega ^{m}({\mathbb {S}}^{n-1})}\lesssim R^{\frac{n}{q}} \left\| g\right\| _{L^{p}_{\rho }H_\omega ^{m,p}({\mathbb {S}}^{n-1})}, \end{aligned}$$
where \(m=(n-1)(\frac{1}{2}-\frac{1}{q})\). In particular, for \(q=2\) and \(4\le q\le 6\), this proves (3.3) and (3.4) when \(R\lesssim 1\). Hence it suffices to consider the case \(R\gg 1\) once we prove Lemma 4.2.

Proof of Lemma 4.2

By scaling argument in variables tx and (4.2), we obtain
$$\begin{aligned}&\Vert (f~d\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\nonumber \\&\quad \lesssim \bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}. \end{aligned}$$
(4.6)
By Sobolev’s embedding, (2.3) and (2.4), we have
$$\begin{aligned}&\Vert (f~d\sigma )^\vee \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\\&\quad \lesssim \bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\Big |\int \limits _0^\infty e^{it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{ \frac{n-2}{2}}\rho ~d\rho \Big |^2\Big )^{\frac{1}{2}} \bigg \Vert _{L^q_t({\mathbb {R}};L^q_{\mu (r)}(S_R))}. \end{aligned}$$
By Lemma 4.1, it is enough to show
$$\begin{aligned}&\Big \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k) \big \Vert J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{(n-2)/2+1/q'}\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{1}{2}} \Big \Vert _{L^q_{\mu (r)}(S_R)}\\&\quad \lesssim R^{\frac{n}{q}} \bigg (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big \Vert a_{k,\ell }(\rho )\varphi (\rho )\big \Vert _{L^{q'}_{\rho }}^2\bigg )^{\frac{1}{2}}. \end{aligned}$$
Writing briefly \(\nu =\nu (k)\), and noting that \(R<r<2R\) and \(1<\rho <2\), we have by (2.6)
$$\begin{aligned}&\Big \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big \Vert J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{(n-2)/2+1/q'}\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{1}{2}} \Big \Vert _{L^q_{\mu (r)}([R,2R])} \\&\quad \lesssim \bigg (\int \limits _{R}^{2R} r^{-\frac{(n-2)q}{2}}\Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)} \omega (k)\Big |\frac{(4 r)^{\nu }}{2^{\nu }\Gamma (\nu +\frac{1}{2})\Gamma (\frac{1}{2})}\Big |^2\big \Vert a_{k,\ell }(\rho )\rho ^\nu \varphi (\rho )\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{q}{2}} r^{n-1}~dr\bigg )^{\frac{1}{q}}\\&\quad \lesssim R^{\frac{n}{q}}\bigg (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)} \omega (k)\Big [\frac{(2 R)^{\nu -\frac{n-2}{2}}}{\Gamma (\nu +\frac{1}{2})}\Big ]^2\big \Vert a_{k,\ell }(\rho )\rho ^\nu \varphi (\rho )\big \Vert _{L^{q'}_\rho }^2\bigg )^{\frac{1}{2}}\\&\quad \lesssim R^{\frac{n}{q}}\bigg (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)} \omega (k)\big \Vert a_{k,\ell }(\rho )\varphi (\rho )\big \Vert _{L^{q'}_\rho }^2\bigg )^{\frac{1}{2}} . \end{aligned}$$
In the last inequality, we use the Stirling formula \(\Gamma \left( \nu +1\right) \sim \sqrt{\nu }(\nu /e)^\nu \) and the fact that \(R\lesssim 1\) and \(\nu \ge (n-2)/2\). \(\square \)

Now we are in a position to prove Proposition 3.1 when \(R\gg 1\). We first prove (3.3) by making use of (4.1). Since \(\text {supp}~g\subset \{\xi :|\xi |\in [1,2]\}\), we may assume \(|\xi _n|\sim 1\). Then we freeze one spatial variable, say \(x_n\), with \(|x_n|\lesssim R\) and free other spatial variables \(x'=(x_1,\ldots , x_{n-1})\). After making the change of variables \(\eta _j=\xi _j,~ \eta _n=|\xi |^2\) with \(j=1,\ldots n-1\), we use the Plancherel theorem on the spacetime Fourier transform in \((t,x')\) to obtain (3.3).

When \(R\gg 1\), inequality (3.4) is a consequence of the interpolation theorem and the following proposition.

Proposition 4.1

Assume \(f\in {\mathbb {L}}_1\) and \(R\gg 1\) is a dyadic number. For every small constant \(0<\epsilon \ll 1\), we have the following inequalities
  • For \(q=4\), we have
    $$\begin{aligned} \Vert (f~d\sigma )^\vee \Vert _{L^4_{t,x}({\mathbb {R}}\times A_R)}\lesssim R^{-\frac{n-1}{4}+\epsilon } \Vert (1+|\Omega |\big )^{\frac{n}{4}}f\Vert _{L^4(S;~d\sigma )}. \end{aligned}$$
    (4.7)
  • For \(q=6\), we have
    $$\begin{aligned} \Vert (f~d\sigma )^\vee \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)}\lesssim R^{-\frac{n-1}{3}+\epsilon } \Vert \big (1+|\Omega |\big )^{\frac{n-1}{3}} f\Vert _{L^2(S;~d\sigma )}. \end{aligned}$$
    (4.8)

Remark 4.1

It seems to be possible to remove the \(\epsilon \)-loss in (4.8), but we do not purchase this option here because we do not need it in this paper.

To prove this proposition, we firstly show

Lemma 4.3

Assume \(f\in {{\mathbb {L}}}_1\) and \(R\gg 1\). We have the following estimate
$$\begin{aligned} \Vert (f~d\sigma )^\vee \Vert _{L^4_{t,x}({\mathbb {R}}\times A_R)}\lesssim R^{-\frac{n-1}{4}+\epsilon } \Vert g\Vert _{L_\rho ^4 H_\omega ^{\frac{n}{4},4}({\mathbb {S}}^{n-1})}, \end{aligned}$$
(4.9)
where \(0<\epsilon \ll 1\), and \(g(\xi )=f(|\xi |^2,\xi )\).

Proof

By the scaling argument and (4.2), it suffices to estimate the quantity
$$\begin{aligned} \bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^4_{t,x}({\mathbb {R}}\times A_R)}. \end{aligned}$$
(4.10)
In the following, we consider the three cases. For the first two cases, we establish the estimates for general \(q\ge 4\) so that we can use them directly for \(q=6\) later.
  • Case 1: \(k\in \Omega _1:=\{k:R\ll \nu (k)\}\). Let \(\omega (k)=(1+k)^{2(n-1)(1/2-1/q)}\) again. We have by a similar argument as in the proof of Lemma 4.2:
    $$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\\&\quad \lesssim \bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k\in \Omega _1} \sum \limits _{\ell =1}^{d(k)}\omega (k)\Big |\int \limits _0^\infty e^{it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{ \frac{n-2}{2}}\rho ~d\rho \Big |^2\Big )^{\frac{1}{2}} \bigg \Vert _{L^q_t({\mathbb {R}};L^q_{\mu (r)}(S_R))}\\&\quad \lesssim \bigg \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)}\omega (k) \big \Vert J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{(n-2)/2+1/q'}\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{1}{2}} \bigg \Vert _{L^q_{\mu (r)}(S_R)}. \end{aligned}$$
    Recall that for \(R\gg 1\) and \(k\in \Omega _1\), we have \(|J_{\nu (k)}(r)|\lesssim e^{-c(r+\nu )}\) by (2.9). Using \(R<r<2R\) and \(1<\rho <2\), we obtain
    $$\begin{aligned}&\Big \Vert r^{-\frac{n-2}{2}}\Big (\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)}\omega (k)\big \Vert J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\varphi (\rho )\rho ^{(n-2)/2+1/q'}\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{1}{2}} \Big \Vert _{L^q_{\mu (r)}([R,2R])} \\&\quad \lesssim \bigg (\int \limits _{R}^{2R} r^{-\frac{(n-2)q}{2}}\Big (\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} \omega (k)e^{-(r+\nu )}\big \Vert a_{k,\ell }(\rho )\rho ^\nu \varphi (\rho )\big \Vert _{L^{q'}_\rho }^2\Big )^{\frac{q}{2}} r^{n-1}~dr\bigg )^{\frac{1}{q}}\\&\quad \lesssim e^{-cR}\bigg (\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} \omega (k)e^{-\nu (k)} \big \Vert a_{k,\ell }(\rho )\rho ^\nu \varphi (\rho )\big \Vert _{L^{q'}_\rho }^2\bigg )^{\frac{1}{2}}\\&\quad \lesssim e^{-cR}\bigg (\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} \omega (k)\big \Vert a_{k,\ell }(\rho )\varphi (\rho )\big \Vert _{L^{q'}_\rho }^2\bigg )^{\frac{1}{2}} . \end{aligned}$$
    By Minkowski’s inequality and Hölder’s inequality, we obtain
    $$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)} \nonumber \\&\quad \lesssim e^{-cR} \bigg \Vert \Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}\omega (k)\big |a_{k,\ell }(\rho )\big |^2\Big )^{\frac{1}{2}}\varphi (\rho )\bigg \Vert _{L^{p}_{\rho }}. \end{aligned}$$
    (4.11)
    Applying this with \(q=4=p\), we have
    $$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^4_{t,x}({\mathbb {R}}\times A_R)} \\&\quad \lesssim e^{-cR} \bigg \Vert \Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2}\big |a_{k,\ell }(\rho )\big |^2\Big )^{\frac{1}{2}} \varphi (\rho )\bigg \Vert _{L^{4}_{\rho }} \\&\quad \lesssim R^{-\frac{n-1}{4}+\epsilon }\Vert g\Vert _{L_\rho ^4 H_\omega ^{(n-1)/4,4}({\mathbb {S}}^{n-1})}. \end{aligned}$$
  • Case 2: \(k\in \Omega _2:=\{k: \nu (k)\sim R \}\). Recalling \(g(\xi )=f(|\xi |^2, \xi )\), and using the Sobolev embedding, the Strichartz estimate and the fact \(\text {supp}~g\subset \{\xi \in {\mathbb {R}}^n:|\xi |\in [1,2]\}\), we have for \(q\ge 4\) and \(\frac{2}{q}=n(\frac{1}{2}-\frac{1}{r})\)
    $$\begin{aligned} \Vert (f~d\sigma )^{\vee }\Vert _{L^q_{t,x}({\mathbb {R}}\times {\mathbb {R}}^n)}\lesssim \Vert (f~d\sigma )^{\vee }\Vert _{L^q({\mathbb {R}}; H^m_{r}({\mathbb {R}}^n))}\lesssim \Vert {\hat{g}}\Vert _{H^{m}({\mathbb {R}}^n)}\lesssim \Vert g\Vert _{L^2({\mathbb {R}}^n)}\nonumber \\ \end{aligned}$$
    (4.12)
    where \(m=\frac{(q-2)n-4}{2q}\ge 0\) since \(n\ge 2\). If \(g=\bigoplus _{k\in \Omega _2} {\mathcal {H}}^{k}\), then
    $$\begin{aligned} \Vert g\Vert ^2_{L_\omega ^2({\mathbb {S}}^{n-1})}=&\sum \limits _{k\in \Omega _2}\sum \limits _{\ell =1}^{{d}(k)}|a_{k,\ell }|^2\nonumber \\ \lesssim&R^{-2(n-1)(1/2-1/q)} \sum \limits _{k\in \Omega _2}\sum \limits _{\ell =1}^{{d}(k)}(1+k)^{2(n-1)(1/2-1/q)}|a_{k,\ell }|^2\nonumber \\ \lesssim&R^{-2(n-1)(1/2-1/q)}\Vert g\Vert ^2_{H_\omega ^{(n-1)(\frac{1}{2}-\frac{1}{q}),2}({\mathbb {S}}^{n-1})}. \end{aligned}$$
    (4.13)
    Since \(\text {supp} g\subset \{\xi \in {\mathbb {R}}^n: |\xi |\in [1,2]\}\) and \(p\ge 2\), we have by Hölder’s inequality and (4.12)
    $$\begin{aligned}&\Big \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _2}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big \Vert _{L^q_{t,x}({\mathbb {R}}\times A_R)}\nonumber \\&\quad \lesssim R^{-(n-1)(1/2-1/q)}\Vert g\Vert _{L_\rho ^p H_\omega ^{(n-1)(\frac{1}{2}-\frac{1}{q}),p}({\mathbb {S}}^{n-1})}. \end{aligned}$$
    (4.14)
    In particular, when \(q=p=4\), inequality (4.14) implies that
    $$\begin{aligned}&\Big \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _2}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \nonumber \\&\quad \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big \Vert _{L^4_{t,x}({\mathbb {R}}\times A_R)}\nonumber \\&\qquad \lesssim R^{-(n-1)/4}\Vert g\Vert _{L_\rho ^4 H_\omega ^{(n-1)/4,4}({\mathbb {S}}^{n-1})}. \end{aligned}$$
    (4.15)
  • Case 3: \(k\in \Omega _3:=\{k: \nu (k)\ll R\}\). We need the following lemma about the oscillation and decay property of a Bessel function. This lemma was proved by Barcelo-Cordoba [3].

Lemma 4.4

(Oscillation and asymptotic property, [3]). Let \(\nu >1/2\) and \(r>\nu +\nu ^{1/3}\). There exists a constant number C independent of r and \(\nu \) such that
$$\begin{aligned} J_{\nu }(r)=\sqrt{\frac{2}{\pi }}\frac{\cos \theta (r)}{(r^2-\nu ^2)^{1/4}}+h_\nu (r), \end{aligned}$$
(4.16)
where \(\theta (r)=(r^2-\nu ^2)^{1/2}-\nu \arccos \frac{\nu }{r}-\frac{\pi }{4}\) and
$$\begin{aligned} |h_{\nu }(r)|\le C\left( \left( \frac{\nu ^2}{(r^2-\nu ^2)^{7/4}}+\frac{1}{r}\right) 1_{[\nu +\nu ^{1/3},2\nu ]}(r)+\frac{1}{r}1_{[2\nu ,\infty )}(r)\right) . \end{aligned}$$
(4.17)
Note that \(\nu (k)=k+(n-2)/2\) and \(k\in \Omega _3\), we can write
$$\begin{aligned} J_\nu (r)=I_{\nu }(r)+{\bar{I}}_{\nu }(r)+h_\nu (r), \quad \;\text {where}\;\; |h_\nu (r)|\lesssim r^{-1}, \end{aligned}$$
and
$$\begin{aligned} I_{\nu }(r)=\frac{\sqrt{2/\pi }e^{i\theta (r)}}{\left( r^2-\nu ^2\right) ^{1/4}}. \end{aligned}$$
A simple computation yields to
$$\begin{aligned} \left\{ \begin{array}{ll} &{}\theta '(r)=(r^2-\nu ^2)^{1/2}r^{-1},\\ &{}\theta ''(r)=(r^2-\nu ^2)^{-1/2}-(r^2-\nu ^2)^{1/2}r^{-2}=(r^2-\nu ^2)^{-1/2}\nu ^2r^{-2},\\ &{}\theta '''(r)=\frac{\nu ^2}{r}(r^2-\nu ^2)^{-3/2}\nu ^2r^{-2}\left( -3+\frac{2\nu ^2}{r^2}\right) . \end{array}\right. \end{aligned}$$
(4.18)
Using Sobolev embedding on sphere and Minkowski’s inequality, we estimate
$$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^4_{t,x}({\mathbb {R}}\times A_R)}\\&\quad \lesssim R^{-\frac{n-2}{2}}\bigg \Vert \Big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \Big |\int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big |^2\Big )^{1/2} \bigg \Vert _{L^4_t({\mathbb {R}};L^4_{\mu (r)}(S_R))}\\&\quad \lesssim R^{-\frac{n-3}{4}}\bigg \Vert \bigg (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \Big |\int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big |^2\Big )^{1/2} \bigg \Vert _{L^4_t({\mathbb {R}};L^4_{r}(S_R))}. \end{aligned}$$
Since \(J_\nu (r)=I_{\nu }(r)+{\bar{I}}_{\nu }(r)+h_\nu (r)\), it suffices to estimate two terms
$$\begin{aligned}&\bigg (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \Big \Vert \int \limits _0^\infty e^{- it\rho ^2} h_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big \Vert _{L^4_t({\mathbb {R}};L^4_{r}(S_R))}^{2}\bigg )^{1/2}\nonumber \\&\quad \lesssim R^{-3/4}\Vert g\Vert _{L_\rho ^4 H_\omega ^{\frac{n-1}{4},4}({\mathbb {S}}^{n-1})} \end{aligned}$$
(4.19)
and
$$\begin{aligned}&\bigg \Vert \bigg (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \Big |\int \limits _0^\infty e^{- it\rho ^2} I_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big |^2\Big )^{1/2} \bigg \Vert _{L^4_t({\mathbb {R}};L^4_{r}(S_R))}\nonumber \\&\quad \lesssim R^{-1/2+\epsilon }\Vert g\Vert _{L_\rho ^4 H_\omega ^{\frac{n}{4},4}({\mathbb {S}}^{n-1})}. \end{aligned}$$
(4.20)
For the first purpose, we consider the operator
$$\begin{aligned} T_{\nu }(a)(t,r)=\chi \Big (\frac{r}{R}\Big )\int \limits _0^\infty e^{- it\rho ^2} h_{\nu }( r\rho )a(\rho )\rho ^{\frac{n}{2}}\varphi (\rho ) d\rho \end{aligned}$$
where \(|h_\nu (r)|\le C/r\). By a similar argument as in the proof of Lemma 4.1, it is easy to see
$$\begin{aligned} \Vert T_{\nu }(a)(t,r)\Vert _{L^q_{t,r}}\le R^{-1/q'}\Vert a\varphi \Vert _{L^{q'}_\rho }. \end{aligned}$$
(4.21)
Hence we have
$$\begin{aligned}&\Big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \Big \Vert \int \limits _0^\infty e^{- it\rho ^2} h_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \Big \Vert _{L^4_t({\mathbb {R}};L^4_{r}(S_R))}^{2}\Big )^{1/2}\\&\quad \lesssim R^{-3/4}\Big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \Big \Vert a_{k,\ell }(\rho )\varphi (\rho )\Big \Vert ^2_{L^{4/3}}\Big )^{1/2}\\&\quad \lesssim R^{-3/4}\Big \Vert \Big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \big |a_{k,\ell }(\rho )\big |^2\Big )^{1/2}\varphi \Big \Vert _{L^{4/3}}\\&\quad \lesssim R^{-3/4}\Vert g\Vert _{L_\rho ^4 H_\omega ^{\frac{n-1}{4},4}({\mathbb {S}}^{n-1})} \end{aligned}$$
which implies (4.19).
Next we prove (4.20). To this end, let \(\beta (\rho )=\rho ^{\frac{n}{2}}\varphi (\rho )\), we see that
$$\begin{aligned}&\left\| \left( \sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \left| \int \limits _0^\infty e^{- it\rho ^2} I_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )\mathrm {d}\rho \right| ^{2}\right) ^{1/2}\right\| ^4_{L^4_t({\mathbb {R}};L^4_{r}(S_R))}\nonumber \\&\quad =\Big \Vert \sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \int \limits _{{\mathbb {R}}^2}e^{- it(\rho _1^2-\rho _2^2)} I_{\nu (k)}( r\rho _1)\overline{I_{\nu (k)}( r\rho _2)}\nonumber \\&\qquad \times a_{k,\ell }(\rho _1)\overline{a_{k,\ell }(\rho _2)}\beta (\rho _1)\beta (\rho _2)\mathrm {d}\rho _1\mathrm {d}\rho _2\Big \Vert ^2_{L^2_t({\mathbb {R}};L^2_{r}(S_R))}\nonumber \\&\quad \le \Big (\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2} \Big \Vert \int \limits _{{\mathbb {R}}^2}e^{- it(\rho _1^2-\rho _2^2)} I_{\nu (k)}( r\rho _1)\overline{I_{\nu (k)}( r\rho _2)}\nonumber \\&\qquad \times \sum \limits _{\ell =1}^{d(k)} a_{k,\ell }(\rho _1)\overline{a_{k,\ell }(\rho _2)}\beta (\rho _1)\beta (\rho _2)\mathrm {d}\rho _1\mathrm {d}\rho _2\Big \Vert _{L^2_t({\mathbb {R}};L^2_{r}(S_R))}\Big )^2\nonumber \\&\quad =\Big (\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2} \Big ( \int \limits _{{\mathbb {R}}^4} \sum \limits _{\ell =1}^{d(k)} a_{k,\ell }(\rho _1) \overline{a_{k,\ell }(\rho _2)}\sum \limits _{\ell '=1}^{d(k)}\overline{a_{k,\ell '}(\rho _3)}a_{k,\ell '}(\rho _4)\beta (\rho _1)\beta (\rho _2)\beta (\rho _3)\beta (\rho _4)\nonumber \\&\qquad \int \limits _{\mathbb {R}}e^{- it(\rho _1^2-\rho _2^2+\rho _3^2-\rho _4^2)} dt K(R,\nu ;\rho _1,\rho _2,\rho _3,\rho _4) d\rho _1d\rho _2d\rho _3d\rho _4\Big )^{1/2}\Big )^2 \end{aligned}$$
(4.22)
where the kernel
$$\begin{aligned}&K(R,\nu ;\rho _1,\rho _2,\rho _3,\rho _4)\nonumber \\&\quad =\int \limits _0^\infty \frac{\chi (\frac{r}{R}) e^{i\left( \theta (\rho _1r)-\theta (\rho _2r)+\theta (\rho _3r)-\theta (\rho _4r)\right) }}{\left( (r\rho _1)^2-\nu ^2\right) ^{1/4} \left( (r\rho _2)^2-\nu ^2\right) ^{1/4}\left( (r\rho _3)^2-\nu ^2\right) ^{1/4}\left( (r\rho _4)^2-\nu ^2\right) ^{1/4}}dr.\nonumber \\ \end{aligned}$$
(4.23)
Now we analyze the kernel K. Let
$$\begin{aligned} \phi (r;\rho _1,\rho _2,\rho _3,\rho _4)=\theta (\rho _1r)-\theta (\rho _2r)+\theta (\rho _3r)-\theta (\rho _4r). \end{aligned}$$
Hence if \(\rho _1^2-\rho _2^2=\rho _4^2-\rho _3^2\), we have by (4.18)
$$\begin{aligned} \phi '_r= & {} (\rho _1^2-\rho _2^2)r\Big (\frac{1}{\sqrt{(r\rho _1)^2-\nu ^2}+\sqrt{(r\rho _2)^2-\nu ^2}}-\frac{1}{\sqrt{(r\rho _3)^2-\nu ^2} +\sqrt{(r\rho _4)^2-\nu ^2}}\Big )\\= & {} \frac{(\rho _1^2-\rho _2^2)(\rho _3^2-\rho _2^2)r^3}{\Big (\sqrt{(r\rho _1)^2-\nu ^2}+\sqrt{(r\rho _2)^2-\nu ^2}\Big )\Big (\sqrt{(r\rho _3)^2-\nu ^2} +\sqrt{(r\rho _4)^2-\nu ^2}\Big )}\\&\times \qquad \bigg (\frac{1}{\sqrt{(r\rho _3)^2-\nu ^2}+\sqrt{(r\rho _2)^2-\nu ^2}}+\frac{1}{\sqrt{(r\rho _1)^2-\nu ^2}+\sqrt{(r\rho _4)^2-\nu ^2}}\bigg ). \end{aligned}$$
Since \(k\in \Omega _3\), one has \(r\gg \nu (k)\). Therefore we have
$$\begin{aligned} |\phi '_r|&\ge |\rho _1^2-\rho _2^2|\cdot |\rho _3^2-\rho _2^2|. \end{aligned}$$
Applying integration by parts with respect to r to (4.23), we have for any \(N\ge 0\)
$$\begin{aligned} K(R,\nu ;\rho _1,\rho _2,\rho _3,\rho _4)\lesssim R^{-1}\left( 1+R|\rho _1^2-\rho _2^2|\cdot |\rho _3^2-\rho _2^2|\right) ^{-N}, \end{aligned}$$
(4.24)
when \(\rho _1^2-\rho _2^2=\rho _4^2-\rho _3^2\). Let \(b_{k,\ell }(\rho )=2a_{k,\ell }(\sqrt{\rho })\beta ({\sqrt{\rho })}/\sqrt{\rho }\), from (4.22) and (4.24), it suffices to estimate
$$\begin{aligned}&\Big (\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2} \Big (\int \limits _{{\mathbb {R}}^4} \delta (\rho _1-\rho _2+\rho _3-\rho _4) K(R,\nu (k);\sqrt{\rho _1},\sqrt{\rho _2},\sqrt{\rho _3},\sqrt{\rho _4})\\&\qquad \times \sum \limits _{\ell =1}^{d(k)} b_{k,\ell }(\rho _1) \overline{b_{k,\ell }(\rho _2)}\sum \limits _{\ell '=1}^{d(k)}\overline{b_{k,\ell '}(\rho _3)}b_{k,\ell '}(\rho _4) d\rho _1d\rho _2d\rho _3d\rho _4 \Big )^{1/2}\Big )^2\\&\quad =\Big (\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2} \Big (\int \limits _{{\mathbb {R}}^3} K(R,\nu (k);\sqrt{\rho _1},\sqrt{\rho _2},\sqrt{\rho _3},\sqrt{\rho _1-\rho _2+\rho _3}) \\&\qquad \times \sum \limits _{\ell =1}^{d(k)} b_{k,\ell }(\rho _1) \overline{b_{k,\ell }(\rho _2)}\sum \limits _{\ell '=1}^{d(k)}\overline{b_{k,\ell '}(\rho _3)}b_{k,\ell '}(\rho _1-\rho _2+\rho _3) d\rho _1d\rho _2d\rho _3\Big )^{1/2}\Big )^2\\&\quad \le R^{-1}\Big (\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2} \Big (\int \limits _{{\mathbb {R}}^3} (1+R|\rho _1-\rho _2||\rho _3-\rho _2|)^{-N} \\&\qquad \times \sum \limits _{\ell =1}^{d(k)} \big | b_{k,\ell }(\rho _1) \overline{b_{k,\ell }(\rho _2)}\big |\sum \limits _{\ell '=1}^{d(k)}\big |\overline{b_{k,\ell '}(\rho _3)}b_{k,\ell '}(\rho _1-\rho _2+\rho _3)\big | d\rho _1d\rho _2d\rho _3\Big )^{1/2}\Big )^2\\&\quad \lesssim R^{-1}\Big (\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2}\Big ( \int \limits _{{\mathbb {R}}^3} (1+R|\rho _1-\rho _2||\rho _3-\rho _2|)^{-N}\\&\qquad \times b_k(\rho _1)b(\rho _2)b_k(\rho _3)b_k(\rho _1-\rho _2+\rho _3)d\rho _1d\rho _2d\rho _3\Big )^{1/2}\Big )^2 \end{aligned}$$
where \(b_k(\rho )=\big (\sum \limits _{\ell =1}^{d(k)} |b_{k,\ell }(\rho )|^2\big )^{1/2}\). Then we aim to estimate
$$\begin{aligned} \int \limits _{{\mathbb {R}}^3}\frac{b(\rho _1)b(\rho _2)b(\rho _3)b(\rho _1-\rho _2+\rho _3)}{(1+R|\rho _1-\rho _2||\rho _3-\rho _2|)^{N}}d\rho _1d\rho _2d\rho _3\lesssim R^{-1+\epsilon }\Vert b\Vert _{L^4}^4. \end{aligned}$$
(4.25)
Indeed once we have proved (4.25), we show
$$\begin{aligned}&\left\| \left( \sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{(n-1)/2} \left| \int \limits _0^\infty e^{- it\rho ^2} I_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )\mathrm {d}\rho \right| ^{2}\right) ^{1/2}\right\| ^2_{L^4_t({\mathbb {R}};L^4_{r}(S_R))}\nonumber \\&\quad \lesssim R^{-1+\epsilon }\sum \limits _{k\in \Omega _3}(1+k)^{(n-1)/2+\frac{1}{2}+\epsilon }(1+k)^{-\frac{1}{2}-\epsilon }\left\| b_k\right\| ^2_{L^4}\nonumber \\&\quad \lesssim R^{-1+2\epsilon }\Big (\sum \limits _{k\in \Omega _3}(1+k)^{n}\Big \Vert (\sum \limits _{\ell =1}^{d(k)} | b_{k,\ell }(\rho )|^2)^{1/2} \Big \Vert ^4_{L^4}\Big )^{1/2} \nonumber \\&\quad \lesssim R^{-1+2\epsilon }\Big \Vert \big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)} (1+k)^{\frac{n}{2}}| a_{k,\ell }(\rho )|^2\big )^{1/2} \Big \Vert ^2_{L^4} \end{aligned}$$
which implies (4.20). Therefore, it remains to prove
$$\begin{aligned} \int \limits _{{\mathbb {R}}^3}\frac{b(\rho _1)b(\rho _2)b(\rho _3)b(\rho _1-\rho _2+\rho _3)}{\left( 1+R|\rho _1-\rho _2|\cdot |\rho _3-\rho _2|\right) ^N}d\rho _1 d\rho _2 d\rho _3 \lesssim R^{-1+\epsilon }\Vert b\Vert ^4_{L^4}. \end{aligned}$$
(4.26)
For \(R=2^{k_0}\gg 1\), we decompose the integral into
$$\begin{aligned}&\int \limits _{{\mathbb {R}}^3}\frac{b(\rho _1)b(\rho _2)b(\rho _3)b(\rho _1-\rho _2+\rho _3)}{\left( 1+R|\rho _1-\rho _2||\rho _3-\rho _2|\right) ^N}d\rho _1 d\rho _2 d\rho _3\nonumber \\&\quad =\bigg (\sum \limits _{\{(i,j)\in {\mathbb {N}}^2; i+j\ge k_0\}}+\sum \limits _{\{(i,j)\in {\mathbb {N}}^2; i+j\lesssim k_0\}}R^{-N}2^{N(i+j)}\bigg )\nonumber \\&\qquad \int b(\rho _2) d\rho _2 \int \limits _{|\rho _1-\rho _2|\sim 2^{-i}}b(\rho _1) d\rho _1 \int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}b(\rho _3)b(\rho _1-\rho _2+\rho _3) d\rho _3.\nonumber \\ \end{aligned}$$
(4.27)
To estimate it, we need the following lemma.

Lemma 4.5

We have the following estimate for the integral
$$\begin{aligned} \int b(\rho _2) d\rho _2 \int \limits _{|\rho _1-\rho _2|\sim 2^{-i}}b(\rho _1) d\rho _1 \int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}b(\rho _3)b(\rho _1-\rho _2+\rho _3) d\rho _3\lesssim 2^{-(i+j)}\Vert b\Vert _{L^4}^4.\nonumber \\ \end{aligned}$$
(4.28)

Proof

We first have by Hölder’s inequality
$$\begin{aligned}&\int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}b(\rho _3)b(\rho _1-\rho _2+\rho _3) d\rho _3 \nonumber \\&\quad \lesssim \left( \int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}|b(\rho _3)|^2 d\rho _3 \int \limits _{|\rho _3-\rho _2|\sim {2^{-j}}}|b(\rho _1-\rho _2+\rho _3)|^2 d\rho _3\right) ^{1/2} \nonumber \\&\quad \lesssim \left( \int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}|b(\rho _3)|^2 d\rho _3 \int \limits _{|\rho |\sim {2^{-j}}}|b(\rho _1+\rho )|^2 d\rho \right) ^{1/2}\nonumber \\&\quad \lesssim \left( \int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}|b(\rho _3)|^2 d\rho _3 \int \limits _{|\rho -\rho _1|\sim {2^{-j}}}|b(\rho )|^2 d\rho \right) ^{1/2}. \end{aligned}$$
(4.29)
Let I be the left hand side of (4.28). We estimate I by (4.29) and Hölder’s inequality
$$\begin{aligned}&\int b(\rho _2) \int \limits _{|\rho _1-\rho _2|\sim 2^{-i}}\Big (\int \limits _{|\rho _1-\rho |\sim 2^{-j}}|b(\rho )|^2 d\rho \Big )^{1/2}b(\rho _1) d\rho _1 \Big (\int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}|b(\rho _3)|^2 d\rho _3\Big )^{1/2} d\rho _2\\&\quad \lesssim \Vert b\Vert _{L^4} \bigg \Vert \int \limits _{|\rho _1-\rho _2| \sim 2^{-i}}\Big (\int \limits _{|\rho _1-\rho |\sim 2^{-j}}|b(\rho )|^2 d\rho \Big )^{1/2}|b(\rho _1)| d\rho _1 \bigg \Vert _{L^2_{\rho _2}} \bigg \Vert \Big (\int \limits _{|\rho _3-\rho _2|\sim 2^{-j}}|b(\rho _3)|^2 d\rho _3\Big )^{1/2}\bigg \Vert _{L^4} \\&\quad \lesssim \Vert b\Vert _{L^4}\Big \Vert \chi _i*\big ((\chi _j*|b|^2)^\frac{1}{2}|b|\big )\Big \Vert _{L^2}\big \Vert \chi _j*|b|^2\big \Vert ^{1/2}_{L^2}, \end{aligned}$$
where \(\chi _j=\chi _j(\rho )=\chi (2^j\rho )\) and \(\chi \in C_c^\infty ([\frac{1}{4},4])\). It is easy to see by the Young inequality
$$\begin{aligned} \left\| \chi _j*|b|^2\right\| ^{1/2}_{L^2}\lesssim \Vert \chi _j\Vert ^{1/2}_{L^1}\left\| b\right\| _{L^4}\lesssim 2^{-j/2}\left\| b\right\| _{L^4}, \end{aligned}$$
and
$$\begin{aligned} \left\| \chi _i*\left( (\chi _j*|b|^2)^\frac{1}{2}|b|\right) \right\| _{L^2} \lesssim&\Vert \chi _i\Vert _{L^1}\left\| (\chi _j*|b|^2)^\frac{1}{2}|b|\right\| _{L^2}\\ \lesssim&\Vert \chi _i\Vert _{L^1}\big \Vert \chi _j*|b|^2\big \Vert _{L^2}^\frac{1}{2}\Vert b\Vert _{L^4}\\ \lesssim&2^{-i}2^{-j/2}\left\| b\right\| ^2_{L^4}. \end{aligned}$$
Collecting the above estimates, we obtain
$$\begin{aligned} I\lesssim 2^{-(i+j)}\Vert b\Vert ^4_{L^4}. \end{aligned}$$
This completes the proof of Lemma 4.5. \(\square \)
Now we return to prove (4.26). Applying Lemma 4.5 to (4.27), we have
$$\begin{aligned}&\int \limits _{{\mathbb {R}}^3}\frac{b(\rho _1)b(\rho _2)b(\rho _3)b(\rho _1-\rho _2+\rho _3)}{\left( 1+R|\rho _1-\rho _2||\rho _3-\rho _2|\right) ^N}d\rho _1 d\rho _2 d\rho _3\nonumber \\&\quad \lesssim \bigg (\sum \limits _{\{(i,j)\in {\mathbb {N}}^2; i+j\ge k_0\}}2^{-(i+j)}+R^{-N}\sum \limits _{\{(i,j)\in {\mathbb {N}}^2; i+j\lesssim k_0\}}2^{(N-1)(i+j)}\bigg ) \Vert b\Vert ^4_{L^4}\nonumber \\&\quad \lesssim R^{-1+\epsilon }\Vert b\Vert ^4_{L^4}. \end{aligned}$$
(4.30)
Hence we prove (4.26), and so, we finish the proof of (4.7). \(\square \)

We next prove (4.8) in Proposition 4.1. We need to prove the following lemma.

Lemma 4.6

Let \(R\gg 1\) and \(f\in {\mathbb {L}}_1\), we have the following estimate for every \(0<\epsilon \ll 1\)
$$\begin{aligned} \Vert (f~d\sigma )^\vee \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)}\lesssim R^{-\frac{n-1}{3}+\epsilon }\Vert g\Vert _{L^2_{\rho }H_\omega ^{\frac{n-1}{3},2}({\mathbb {S}}^{n-1}) }, \end{aligned}$$
(4.31)
where \(g(\xi )=f(|\xi |^2,\xi )\).

Proof

It suffices to estimate, by a scaling argument, the following quantity
$$\begin{aligned} \bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k=0}^{\infty }\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)}. \end{aligned}$$
(4.32)
We divide the above integral into three cases.
\(\bullet \) Case 1: \(k\in \Omega _1:=\{k:R\ll \nu (k)\}\). Using (4.11) with \(q=6\), we prove
$$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _1}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)} \\&\quad \lesssim e^{-cR} \bigg \Vert \Big (\sum \limits _{k=0}^\infty \sum \limits _{\ell =1}^{d(k)}(1+k)^{2(n-1)/3}\left| a_{k,\ell }(\rho )\right| ^2\Big )^{\frac{1}{2}} \varphi (\rho )\bigg \Vert _{L^{2}_{\rho }}\lesssim e^{-cR} \Vert g\Vert _{L_\rho ^2 H_\omega ^{\frac{n-1}{3},2}({\mathbb {S}}^{n-1})}. \end{aligned}$$
\(\bullet \) Case 2: \(k\in \Omega _2:=\{k: \nu (k)\sim R \}\). Applying (4.14) with \(q=6\) and \(p=2\), we show
$$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _2}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)}\nonumber \\&\quad \lesssim R^{-(n-1)/3}\Vert g\Vert _{L_\rho ^2 H_\omega ^{\frac{n-1}{3},2}({\mathbb {S}}^{n-1})}. \end{aligned}$$
(4.33)
\(\bullet \) Case 3: \(k\in \Omega _3:=\{k: \nu (k)\ll R\}\). We introduce the operator
$$\begin{aligned} T_{\nu }(a)(t,r)=\chi \Big (\frac{r}{R}\Big )\int \limits _0^\infty e^{- it\rho ^2} h_{\nu }( r\rho )a(\rho )\rho ^{\frac{n}{2}}\varphi (\rho ) d\rho \end{aligned}$$
where \(|h_\nu (r)|\le C/r\) and the operator
$$\begin{aligned} H_{\nu }(a)(t,r)=\chi \Big (\frac{r}{R}\Big )\int \limits _0^\infty e^{- it\rho ^2} I_{\nu }( r\rho )a(\rho )\rho ^{\frac{n}{2}}\varphi (\rho ) d\rho , \end{aligned}$$
where \(\nu =\nu (k)=k+(n-2)/2\). Since
$$\begin{aligned} J_\nu (r)=I_{\nu }(r)+{\bar{I}}_{\nu }(r)+h_\nu (r), \end{aligned}$$
our aim here is to estimate
$$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)}\\&\quad \lesssim R^{-\frac{n-1}{3}+\frac{1}{2}}\bigg (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{2(n-1)/3} \Big ( \left\| T_{\nu (k)}(a_{k,\ell })(t,r)\right\| _{L^6_t({\mathbb {R}};L^6_{r}(S_R))}^{2}\\&\qquad \qquad +\left\| H_{\nu (k)}(a_{k,\ell })(t,r) \right\| _{L^6_t({\mathbb {R}};L^6_{r}(S_R))}^{2}\Big )\bigg )^{1/2}. \end{aligned}$$
By making use of (4.21) with \(q=6\), we have
$$\begin{aligned} \Vert T_{\nu }(a)(t,r)\Vert _{L^6_{t,r}}\le R^{-5/6}\Vert a\varphi \Vert _{L^{6/5}}. \end{aligned}$$
This implies that
$$\begin{aligned}&\bigg (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{2(n-1)/3} \left\| T_{\nu (k)}(a_{k,\ell })(t,r)\right\| _{L^6_t({\mathbb {R}};L^6_{r}(S_R))}^{2}\bigg )^{1/2}\nonumber \\&\quad \lesssim R^{-5/6}\bigg \Vert \Big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{2(n-1)/3} \left| a_{k,\ell }(\rho )\right| ^2\Big )^{1/2}\varphi \bigg \Vert _{L^{6/5}}\nonumber \\&\quad \lesssim R^{-5/6}\Vert g\Vert _{L_\rho ^2 H_\omega ^{\frac{n-1}{3},2}({\mathbb {S}}^{n-1})}. \end{aligned}$$
(4.34)
On the other hand, by (2.11), one has \(|I_\nu (r)|\lesssim r^{-1/2}\) when \(k\in \Omega _3\). Consider the operator
$$\begin{aligned} H_{\nu }(a)(t,r)=\chi \Big (\frac{r}{R}\Big )\int \limits _0^\infty e^{- it\rho ^2} I_{\nu }( r\rho )a(\rho )\rho ^{\frac{n}{2}}\varphi (\rho ) d\rho , \end{aligned}$$
where \(\nu =\nu (k)=k+(n-2)/2\) with \(k\in \Omega _3\).
On the one hand, it is easy to see
$$\begin{aligned} \left\| H_{\nu }(a)(t,r)\right\| _{L^\infty _{t,r}({\mathbb {R}}\times {\mathbb {R}}^n)}\lesssim R^{-1/2}\Vert a \varphi \Vert _{L^1}. \end{aligned}$$
On the other hand, we have the claim that for any \(\epsilon >0\)
$$\begin{aligned} \left\| H_{\nu }(a)(t,r)\right\| _{L^4_{t,r}({\mathbb {R}}\times {\mathbb {R}})}\lesssim R^{-1/2+\epsilon }\Vert a \varphi \Vert _{L^4_\rho }. \end{aligned}$$
(4.35)
We postpone the proof of this claim to the end of this section. Hence, by the interpolation of the above two estimates, for any \(\epsilon >0\), we obtain that
$$\begin{aligned} \left\| H_{\nu }(a)(t,r)\right\| _{L^6_{t,r}({\mathbb {R}}\times {\mathbb {R}}^n)}\lesssim R^{-1/2+\epsilon }\Vert a \varphi \Vert _{L^2}. \end{aligned}$$
This shows
$$\begin{aligned}&\bigg (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{2(n-1)/3} \left\| H_{\nu (k)}(a_{k,\ell })(t,r)\right\| _{L^6_t({\mathbb {R}};L^6_{r}(S_R))}^{2}\bigg )^{1/2}\nonumber \\&\quad \lesssim R^{-1/2+\epsilon }\Big (\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)}(1+k)^{2(n-1)/3} \left\| a_{k,\ell }(\rho )\varphi (\rho )\right\| ^2_{L^{2}}\Big )^{1/2}\nonumber \\&\quad \lesssim R^{-1/2+\epsilon }\Vert g\Vert _{L^2_{\rho }H_\omega ^{\frac{n-1}{3},2}({\mathbb {S}}^{n-1}) }. \end{aligned}$$
(4.36)
Collecting (4.34) and (4.36) yields
$$\begin{aligned}&\bigg \Vert r^{-\frac{n-2}{2}}\sum \limits _{k\in \Omega _3}\sum \limits _{\ell =1}^{d(k)} i^{k}Y_{k,\ell }(\theta ) \int \limits _0^\infty e^{- it\rho ^2} J_{\nu (k)}( r\rho )a_{k,\ell }(\rho )\rho ^{\frac{n}{2}}\varphi (\rho )~d\rho \bigg \Vert _{L^6_{t,x}({\mathbb {R}}\times A_R)}\\&\quad \lesssim R^{-\frac{n-1}{3}+\epsilon }\Vert g\Vert _{L^2_{\rho }H_\omega ^{\frac{n-1}{3},2}({\mathbb {S}}^{n-1}) }. \end{aligned}$$
This implies (4.31), which completes the proof of Lemma 4.6. \(\square \)

The proof of claim (4.35)

The same argument in the proof the (4.20) shows the claim (4.35). Recall the kernel (4.23), it is enough to estimate the integral
$$\begin{aligned}&\left\| H_{\nu }(a)(t,r)\right\| ^4_{L^4_{t,r}({\mathbb {R}}\times {\mathbb {R}}^n)}\\&\quad =\int \limits _{{\mathbb {R}}^4} \int \limits _{\mathbb {R}}e^{- it(\rho _1^2-\rho _2^2+\rho _3^2-\rho _4^2)} K(R,\nu ;\rho _1,\rho _2,\rho _3,\rho _4) a(\rho _1) a(\rho _2)a(\rho _3)a(\rho _4)\\&\qquad \qquad \beta (\rho _1)\beta (\rho _2)\beta (\rho _3)\beta (\rho _4) dt d\rho _1d\rho _2d\rho _3d\rho _4, \end{aligned}$$
where \(\beta (\rho )=\rho ^{\frac{n}{2}}\varphi (\rho )\). For \(b(\rho )=2a(\sqrt{\rho })\beta ({\sqrt{\rho })}/\sqrt{\rho }\), therefore we obtain
$$\begin{aligned}&\left\| H_{\nu }(a)(t,r)\right\| ^4_{L^4_{t,r}({\mathbb {R}}\times {\mathbb {R}}^n)}\\&\quad =\int \limits _{{\mathbb {R}}^4} \delta (\rho _1-\rho _2+\rho _3-\rho _4) K(R,\nu ; \sqrt{\rho _1},\sqrt{\rho _2},\sqrt{\rho _3},\sqrt{\rho _4}) b(\rho _1) b(\rho _2)b(\rho _3)b(\rho _4)d\rho _1d\rho _2d\rho _3d\rho _4\\&\quad =\int \limits _{{\mathbb {R}}^3} K(R,\nu ; \sqrt{\rho _1},\sqrt{\rho _2},\sqrt{\rho _3},\sqrt{\rho _1-\rho _2+\rho _3}) b(\rho _1) b(\rho _2)b(\rho _3)b(\rho _1-\rho _2+\rho _3)d\rho _1d\rho _2d\rho _3\\&\quad \lesssim R^{-2+\epsilon }\Vert b\Vert ^4_{L^4}\lesssim R^{-2+\epsilon }\Vert a\varphi \Vert ^4_{L^4} . \end{aligned}$$
where we use the kernel estimate (4.24) and (4.26) in the first inequality. \(\square \)

5 Local smoothing estimate

K. M. Rogers [20] developed an argument showing that a restriction estimate implies a local smoothing estimate under some suitable conditions. For the sake of convenience, we closely follow this argument to prove Corollary 1.1. In fact, by making use of the standard Littlewood-Paley argument, it can be reduced to prove the claim
$$\begin{aligned} \Vert e^{it\Delta }(1-\Delta _\theta )^{-s/2}u_0\Vert _{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)} \lesssim N^{(2n(1/2-1/q)-2/q)_+}\left\| u_0\right\| _{L^q_{x}},\quad \forall ~ N\gg 1 \end{aligned}$$
(5.1)
where
$$\begin{aligned}\mathrm {supp}~ {\mathcal {F}}({(1-\Delta _\theta )^{-s/2}u_0})\subset \{\xi :|\xi |\le N\}. \end{aligned}$$
Here we denote by \({\mathcal {F}}\) the Fourier transform. We also use the notation \({\hat{h}}\) to express the Fourier transform of h. Let \(h=(1-\Delta _\theta )^{-s/2}u_0\). Denote by \(P_N\) the Littlewood-Paley projector, i.e.
$$\begin{aligned} P_Nh={\mathcal {F}}^{-1}\Big (\chi \Big (\frac{|\xi |}{N}\Big ){\hat{h}}\Big ),\quad \; \chi \in {\mathbb {C}}_c^\infty ([1/2,1]). \end{aligned}$$
By the Littlewood-Paley theory and the claim (5.1), one has for \(\alpha >2n(1/2-1/q)-2/q\)
$$\begin{aligned} \Vert e^{it\Delta }h\Vert ^2_{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)}\lesssim&\Vert e^{it\Delta }P_{\lesssim 1}h\Vert ^2_{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)}+ \sum \limits _{N\gg 1}\left\| e^{it\Delta }P_Nh\right\| ^2_{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)}\\ \lesssim&\Vert u_0\Vert ^2_{L_x^q({\mathbb {R}}^n)}+\sum \limits _{N\gg 1}N^{2[2n(1/2-1/q)-2/q]+}\big \Vert P_Nu_0\big \Vert ^2_{L^q_{x}}\\ \lesssim&\Vert u_0\Vert ^2_{L_x^q({\mathbb {R}}^n)}+ \bigg \Vert \Big (\sum \limits _{N\gg 1}N^{q\alpha } \left| P_Nu_0\right| ^q\Big )^{1/q}\bigg \Vert ^2 _{L^q_{x}}\\ \lesssim&\Vert u_0\Vert ^2_{L_x^q({\mathbb {R}}^n)}+\bigg \Vert \Big (\sum \limits _{N\gg 1}N^{2\alpha }\left| P_Nu_0\right| ^2\Big )^{1/2}\bigg \Vert ^2_{L^q_{x}}\\ \simeq&\Vert u_0\Vert ^2_{W^{\alpha ,q}_{x}({\mathbb {R}}^n)}. \end{aligned}$$
Here we use Hölder’s inequality for the third inequality, Sobolev imbedding for the fourth one. Hence we have
$$\begin{aligned} \Vert e^{it\Delta }u_0\Vert _{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)}\lesssim \Vert (1-\Delta _\theta )^{s/2}u_0\Vert _{W^{\alpha ,q}_{x}({\mathbb {R}}^n)}. \end{aligned}$$
Now we are left to prove claim (5.1). Assume \(\mathrm {supp}~{\hat{f}} \subset [0, 1]\). Note that
$$\begin{aligned} e^{it\Delta }f=\frac{1}{(it)^{n/2}}\int \limits _{{\mathbb {R}}^n} e^{i|x-y|^2/t}f(y)dy,\quad \forall ~t\in {\mathbb {R}}\backslash \{0\}. \end{aligned}$$
On the other hand, we have for \(t\ne 0\)
$$\begin{aligned} e^{it\Delta }f=&\int \limits _{{\mathbb {R}}^n}e^{i(t|\xi |^2+x\cdot \xi )}{\hat{f}}(\xi )d\xi =e^{-\frac{i|x|^2}{4t}}\int \limits _{{\mathbb {R}}^n}e^{it|\xi +\frac{x}{2t}|^2}{\hat{f}}(\xi )d\xi \\ =&\frac{1}{(it)^{n/2}}e^{-\frac{i|x|^2}{4t}}\left( e^{i\frac{\Delta }{t}}{\hat{f}}\right) \left( -\frac{x}{2t}\right) . \end{aligned}$$
So we have for every dyadic number N
$$\begin{aligned} \Vert e^{it\Delta }f\Vert _{L^q_{t,x}(|t|\sim N^2; |x|\lesssim N^2)}\lesssim & {} N^{-n}\left\| \left( e^{i\frac{\Delta }{t}}{\hat{f}}\right) \left( -\frac{\bullet }{2t}\right) \right\| _{L^q_{t,x}(|t|\sim N^2; |x|\lesssim N^2)}\\&\lesssim N^{-n+\frac{2n+4}{q}}\left\| e^{it\Delta }{\hat{f}}\right\| _{L^q_{t,x}(|t|\sim N^{-2}; |x|\lesssim 1)}. \end{aligned}$$
By making use of Theorem 1.1, we obtain for \(q>2(n+1)/n\) and \(\frac{n+2}{q}=\frac{n}{p'}\)
$$\begin{aligned} \left\| e^{it\Delta }{\hat{f}}\right\| _{L^q_{t,x}(|t|\sim N^{-2}; |x|\lesssim 1)}\lesssim \Vert f\Vert _{L^p_{\mu (r)}({\mathbb {R}}^+;H^{s,p}_\theta ({\mathbb {S}}^{n-1}))}. \end{aligned}$$
(5.2)
This yields
$$\begin{aligned} \Vert e^{it\Delta }f\Vert _{L^q_{t,x}(|t|\sim N^2; |x|\lesssim N^2)}\lesssim N^{-n+\frac{2n+4}{q}}\Vert f\Vert _{L^p_{\mu (r)}({\mathbb {R}}^+;H^{s,p}_\theta ({\mathbb {S}}^{n-1}))}. \end{aligned}$$
This implies that
$$\begin{aligned} \Vert e^{it\Delta }(1-\Delta _\theta )^{-s/2}f\Vert _{L^q_{t,x}(|t|\sim N^2; |x|\lesssim N^2)}\lesssim N^{-n+\frac{2n+4}{q}}\Vert f\Vert _{L^p_{x}}. \end{aligned}$$
(5.3)
For the sake of convenience, we recall [20, Lemma 8]

Lemma 5.1

Let \(q\ge p_1\ge p_0\), \(r\ge 1\) and \(I\subset [0,R^2]\). If one has
$$\begin{aligned} \Vert e^{it\Delta }f\Vert _{L^q_x(B_{R^2};L^r_t(I))}\le CR^s\Vert f\Vert _{L^{p_0}({\mathbb {R}}^n)} \end{aligned}$$
where \(R\gg 1\), and f is frequency supported in unite ball \({\mathbb {B}}^n\). Then for all \(\epsilon >0\)
$$\begin{aligned} \Vert e^{it\Delta }f\Vert _{L^q_x({\mathbb {R}}^n;L^r_t(I))}\le C_\epsilon R^{s+2n(\frac{1}{p_0}-\frac{1}{p_1})+\epsilon }\Vert f\Vert _{L^{p_1}({\mathbb {R}}^n)}. \end{aligned}$$
Since \(q>p\) when \(q>2(n+1)/n\), for any \(0<\epsilon \ll 1\), we have by this lemma
$$\begin{aligned}&\Vert e^{it\Delta }(1-\Delta _\theta )^{-s/2}f\Vert _{L^q_{t,x}(|t|\sim N^2; x\in {\mathbb {R}}^n)}\\&\quad \lesssim N^{-n+\frac{2n+4}{q}+2n(\frac{1}{p}-\frac{1}{q}) +\epsilon }\Vert f\Vert _{L^q_{x}}\\&\quad \lesssim N^{n(1-\frac{2}{q})+\epsilon }\Vert f\Vert _{L^q_{x}}. \end{aligned}$$
Using the scaling argument, if
$$\begin{aligned} \mathrm{supp}\widehat{f_{k,N}}\subset B_{2^{k/2}N}:=\big \{\xi : |\xi |\in [0, 2^{k/2}N]\big \},\quad \; \forall \; k\ge 0, \end{aligned}$$
then
$$\begin{aligned} \Vert e^{it\Delta } (1-\Delta _\theta )^{-\frac{s}{2}}f_{k,N}\Vert _{L^q_{t,x}([2^{-k},2^{-k+1}]\times {\mathbb {R}}^n)} \lesssim N^{n(1-\frac{2}{q})+\epsilon }(2^{\frac{k}{2}}N)^{-\frac{2}{q}}\left\| f_{k,N}\right\| _{L^q_{x}}. \end{aligned}$$
(5.4)
Since
$$\begin{aligned} \mathrm{supp}{\hat{h}} \subset \{\xi : |\xi |\in [N/2, N]\}\subset B_{2^{k/2}N},\quad \; \forall k\ge 2, \end{aligned}$$
we replace \((1-\Delta _\theta )^{-s/2}f_{k,N}\) by h to obtain
$$\begin{aligned} \Vert e^{it\Delta }h\Vert _{L^q_{t,x}([0,1]\times {\mathbb {R}}^n)}=&\bigg (\sum \limits _{k\ge 0}\Vert e^{it\Delta } (1-\Delta _\theta )^{-s/2}u_0\Vert ^q_{L^q_{t,x} ([2^{-k},2^{-k+1}]\times {\mathbb {R}}^n)}\bigg )^{1/q}\nonumber \\&\lesssim \Big (\sum \limits _{k\ge 0}2^{-k}\Big )^{1/q}N^{(2n(1/2-1/q)-2/q)_+}\left\| u_0\right\| _{L^q_{x}}. \end{aligned}$$
(5.5)
This proves inequality (5.1).

Notes

Acknowledgements

The authors would like to express their great gratitude to S. Shao for his helpful discussions. The authors were supported by the NSFC under grants 11771041, 11831004 and H2020-MSCA-IF-2017(790623).

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Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.Department of MathematicsBeijing Institute of TechnologyBeijingChina
  3. 3.Cardiff UniversityCardiffUK

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