Sharp Local $L^{p}$ estimates for the Hermite eigenfunctions

Koch-Tataru [45, Corollary 3.2] proved the following global $L^{p}$ eigenfunction bounds

(1.4)

$$\|e_{\lambda}\|_{L^{p}(\mathbb{R}^{n})}\lesssim\lambda^{\rho(p)}\|e_{\lambda}\|_{L^{2}(\mathbb{R}^{n})},$$

where for $n\geq 2$,

(1.5)

$\displaystyle\rho(p)=\begin{cases}-\frac{1}{2}+\frac{1}{p},\ \ \ \ \ 2\leq p<\frac{2n+6}{n+1}\\ \frac{n-2}{6}-\frac{n}{3p},\ \ \ \frac{2n+6}{n+1}<p\leq\frac{2n}{n-2}\\ \frac{n-2}{2}-\frac{n}{p},\ \ \ \ \frac{2n}{n-2}<p\leq\infty\end{cases}$

and for $n=1$,

(1.6)

$\displaystyle\rho(p)=\begin{cases}\lambda^{-\frac{1}{2}+\frac{1}{p}},\ \ \ \ 2\leq p<4\\ \lambda^{-\frac{1}{6}-\frac{1}{3p}},\ \ \ 4<p\leq\infty.\end{cases}$

See Figure 1. These results strengthen those of Karadzhov [44] and Thangavelu [62]. It is interesting to observe that the exponent $\rho(p)$ is decreasing when $p<\frac{2n+6}{n+1}$, and increasing when $p>\frac{2n+6}{n+1}$. It is due to the special concentration features of the eigenfunctions, see the discussion after Theorem 3. At the kink point $p=\frac{2n+6}{n+1}$, it is known that [47]

(1.7)

$$\|e_{\lambda}\|_{L^{\frac{2n+6}{n+1}}(\mathbb{R}^{n})}\lesssim\lambda^{-\frac{1}{n+3}}(\log\lambda)^{\frac{n+1}{2n+6}}\|e_{\lambda}\|_{L^{2}(\mathbb{R}^{n})}.$$

The log factor is necessary when $n=1$. Recently, the log loss has been removed by Jeong-Lee-Ryu [41] for $n\geq 3$. Their significant improvements are due to a new phenomenon concerning the asymmetric localization near the sphere $\lambda S^{n-1}$, see [41, Theorem 1.2]. One may also refer to [40, 39, 42] for their related works on Bochner-Riesz means and $L^{p}-L^{q}$ bounds for the Hermite spectral projection operator.

The Hermite eigenfunctions are essentially concentrated in the ball $\{|x|\leq\lambda\}$ and have an exponential Airy type decay beyond this threshold. As was observed in [45], the behavior of eigenfunctions inside the ball $\{|x|\leq\lambda\}$ is not very different from (a rescaling of) what happens in a bounded domain. But considerable care is required near the boundary $\{|x|=\lambda\}$, where the concentration scales are different. Consequently, Koch-Tataru [45] split the space $\mathbb{R}^{n}$ into overlapping dyadic parts with respect to the distance to the boundary

$$D_{j}^{int}=\{\lambda-|x|\approx\lambda 2^{-2j}\},\ 1\leq 2^{j}\leq\lambda^{\frac{2}{3}}$$

$$D^{bd}=\{||x-\lambda|\leq\lambda^{-\frac{1}{3}}\}$$

$$D^{ext}=\{|x|>\lambda+\frac{1}{2}\lambda^{-\frac{1}{3}}\}.$$

Note that the thickness of the annulus $D_{j}^{int}$ is comparable to its distance to the boundary.

To state the main theorem in [45], we define the spaces $l_{\lambda}^{q}L^{p}$ of functions in $\mathbb{R}^{n}$ with norm

$$\|f\|_{l_{\lambda}^{q}L^{p}}^{q}=\|f\|_{L^{p}\left(D^{ext}\right)}^{q}+\|f\|_{L^{p}\left(D^{bd}\right)}^{q}+\sum_{1\leq 2^{j}\leq\lambda^{2/3}}\|f\|_{L^{p}\left(D_{j}^{int}\right)}^{q}$$

with the usual modification when $q=\infty$. For $x\in\mathbb{R}^{n}$ we let

$$y=\lambda^{-\frac{2}{3}}\left(\lambda^{2}-|x|^{2}\right),\quad\langle y\rangle_{-}=1+y_{-},\quad\langle y\rangle_{+}=1+y_{+}$$

Koch-Tataru [45] proved that for $2\leq p\leq\frac{2n+2}{n-1}$,

(1.8)

$$\left\|\lambda^{\frac{1}{3}-\frac{n}{3}\left(\frac{1}{2}-\frac{1}{p}\right)}\langle y\rangle_{+}^{-\frac{1}{4}+\frac{n+3}{4}\left(\frac{1}{2}-\frac{1}{p}\right)}\langle y\rangle_{-}^{1-\frac{n}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}e_{\lambda}\right\|_{l_{\lambda}^{\infty}L^{p}}\lesssim\|e_{\lambda}\|_{L^{2}},$$

and for $\frac{2n+2}{n-1}\leq p\leq\infty$,

(1.9)

$$\left\|\lambda^{\frac{1}{3}-\frac{n}{3}\left(\frac{1}{2}-\frac{1}{p}\right)}\langle y\rangle_{+}^{\frac{1}{2}-\frac{n}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\langle y\rangle_{-}^{N}e_{\lambda}\right\|_{l_{\lambda}^{\infty}L^{p}}\lesssim\|e_{\lambda}\|_{L^{2}},\ \forall N.$$

As a corollary, one can obtain $L^{p}$ bounds over these dyadic annuli. For $2\leq p\leq\frac{2n+2}{n-1}$

(1.10)

$$\|e_{\lambda}\|_{L^{p}(D_{j}^{int})}\lesssim\lambda^{\frac{1}{p}-\frac{1}{2}}2^{j(\frac{n+1}{4}-\frac{n+3}{2p})}\|e_{\lambda}\|_{2},\ \ 1\leq 2^{j}\leq\lambda^{\frac{2}{3}},$$

(1.11)

$$\|e_{\lambda}\|_{L^{p}(D^{bd})}+\|e_{\lambda}\|_{L^{p}(D^{ext})}\lesssim\lambda^{-\frac{1}{3}+\frac{n}{3}(\frac{1}{2}-\frac{1}{p})}\|e_{\lambda}\|_{2}$$

and for $\frac{2n+2}{n-1}\leq p\leq\infty$

(1.12)

$$\|e_{\lambda}\|_{L^{p}(D_{j}^{int})}\lesssim(\lambda 2^{-j})^{\frac{n-2}{2}-\frac{n}{p}}\|e_{\lambda}\|_{2},\ \ 1\leq 2^{j}\leq\lambda^{\frac{2}{3}},$$

(1.13)

$$\|e_{\lambda}\|_{L^{p}(D^{bd})}+\|e_{\lambda}\|_{L^{p}(D^{ext})}\lesssim\lambda^{-\frac{1}{3}+\frac{n}{3}(\frac{1}{2}-\frac{1}{p})}\|e_{\lambda}\|_{2}.$$

One may observe that the local $L^{2}$ estimate over the dilated ball $D_{0}^{int}=\{|x|<\lambda/4\}$ has no improvement on the trivial bound. Similarly, the local $L^{\frac{2n+6}{n+1}}$ bounds in (1.10) cannot essentially improve the global estimate (1.7) over the whole space. Moreover, the local $L^{p}$ bounds (1.12) over $D_{0}^{int}$ strengthen Thangavelu’s estimates [62, Theorem 2]

(1.14)

$$\|e_{\lambda}\|_{L^{p}(B)}\leq C\lambda^{\frac{n-2}{2}-\frac{n}{p}}\|e_{\lambda}\|_{L^{2}(\mathbb{R}^{n})},\ \ \tfrac{2n+2}{n-1}\leq p\leq\infty.$$

where $B$ is any fixed compact set in $\mathbb{R}^{n}$. This means that replacing $B$ by a much larger dilated ball $D_{0}^{int}$ does not affect the bounds. This interesting phenomenon can be explained by the existence of the Hermite eigenfunctions with point concentration near the origin. See Section 5 and [45, Example 5.2].

Now let’s go back to the starting point of this study. Bohr’s correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large, see [11, 63]. For the classical harmonic oscillator, when a particle’s potential energy is equal to its energy level, it moves at its slowest speed and has the highest probability of being detected around those points. We call these points turning points, which actually correspond to the boundary $\{|x|=\lambda\}$. By the correspondence principle, we would expect that for a quantum harmonic oscillator, for eigenstates with large energy, their probability density should also peak near these turning points in some sense. For the one-dimensional case, let

$$s^{-}(x)=\int_{0}^{x}\sqrt{|t^{2}-\lambda^{2}|}dt,\ \ \ \ s^{+}(x)=\int_{\lambda}^{x}\sqrt{|t^{2}-\lambda^{2}|}dt.$$

As in Szegö [58, p. 201] and Koch-Tataru [45, Lemma 5.1], the normalized eigenfunctions $\tilde{h}_{k}=h_{k}/\|h_{k}\|_{2}$ satisfy

	$\displaystyle\tilde{h}_{2k}$	$\displaystyle=\begin{cases}a_{2k}^{-}\left(\lambda^{2}-x^{2}\right)^{-\frac{1}{4}}\left(\cos s^{-}(x)+\text{ error }\right)&|x|<\lambda-\lambda^{-\frac{1}{3}}\\ O(\lambda^{-\frac{1}{6}})&\lambda-\lambda^{-\frac{1}{3}}\leq x\leq\lambda+\lambda^{-\frac{1}{3}}\\ a_{2k}^{+}e^{-s^{+}(x)}\left(\lambda^{2}-x^{2}\right)^{-\frac{1}{4}}(1+\text{ error })&|x|>\lambda+\lambda^{-\frac{1}{3}}\end{cases}$
	$\displaystyle\tilde{h}_{2k+1}$	$\displaystyle=\begin{cases}a_{2k+1}^{-}\left(\lambda^{2}-x^{2}\right)^{-\frac{1}{4}}\left(\sin s^{-}(x)+\text{ error }\right)&|x|<\lambda-\lambda^{-\frac{1}{3}}\\ O(\lambda^{-\frac{1}{6}})&\lambda-\lambda^{-\frac{1}{3}}\leq x\leq\lambda+\lambda^{-\frac{1}{3}}\\ a_{2k+1}^{+}e^{-s^{+}(x)}\left(\lambda^{2}-x^{2}\right)^{-\frac{1}{4}}(1+\text{ error })&x>\lambda+\lambda^{-\frac{1}{3}}\end{cases}$

where $|a_{k}^{\pm}|\sim 1$, $\text{ error }=O((\left|x^{2}-\lambda^{2}\right|^{-\frac{1}{2}}||x|-\lambda|^{-1})$. Obviously the maximum values of Hermite eigenfunctions occur near the turning points $x=\pm\lambda$, see Figure 2 by Wolfram Mathematica.

However, the eigenfunctions become more complicated in higher dimensions. Koch-Tataru [45] showed that there are eigenfunctions that attain maximal $L^{\infty}$ growth near the origin. This phenomenon seems to violate Bohr’s correspondence principle, and suggests that it may be not suitable to measure only by the local $L^{\infty}$ norm. A more reliable choice should be the local $L^{2}$ norm, i.e. local probability, since the square of the amplitude is interpreted as a probability density. As we will see in Theorem 1, the local probability over a compact set, such as the unit ball with arbitrary center, decreases as it moves away from the boundary $\{|x|=\lambda\}$. This satisfies Bohr’s correspondence principle.

Let $B$ be any fixed compact set in $\mathbb{R}^{n}$, and $\nu\in\mathbb{R}^{n}$. For $r>0$, let

(1.15)

$$B(\nu,r)=\{\nu+rx:x\in B\}$$

be the compact set with dilation rate $r$ and translation vector $\nu$. We shall restrict $|\nu|\leq\lambda$ and $r\leq\lambda$, since the Hermite eigenfunctions are essentially supported in the ball $\{|x|\leq\lambda\}$. The $L^{p}$ bounds (1.14) of the Hermite eigenfunctions over the fixed compact sets $B$ for $p\geq\frac{2n+2}{n-1}$ have been established by Thangavelu [62], motivated by the local version of the Bochner-Riesz conjecture. Koch-Tataru [45] strengthened those of Thangavelu [62] as well as Karadzhov [44] by obtaining global bounds over $\mathbb{R}^{n}$ and dyadic annuli (including the dilated ball $D_{0}^{int}$), and they also observed that eigenfunctions may concentrate in some small compact subsets, e.g. balls and tubes.

To our best knowledge, compared to the Laplace eigenfunctions on compact manifolds, the picture of the Hermite eigenfunction estimates are still far from complete. See Section 6 for a list of related open problems. So we aim to investigate these problems in a series of works. In this paper, we first establish sharp estimates over the compact sets $B(\nu,r)$ with arbitrary dilations and translations, which extend the local estimates of Thangavelu [62] and improve those derived from Koch-Tataru [45]. Our main theorem (Theorem 1) gives sharp local $L^{2}$ bounds (local probabilities) over the compact sets with arbitrary dilations and translations.

See Figures 3, 4, 5, 6.

These results improve the bounds derived from Koch-Tataru’s estimates (1.8) and (1.9) when $(\lambda\mu^{\frac{1}{2}})^{-1}\ll r\ll\lambda\mu$. The endpoints $(\lambda\mu^{\frac{1}{2}})^{-1}$ and $\lambda\mu$ correspond to the sizes of two different kinds of eigenfunction concentrations: point concentration and tube concentration respectively. These are similar to the eigenfunctions of the spherical Laplacian, e.g. zonal functions and Gaussian beams. Indeed, Koch-Tataru [45] used the eigenbasis (1.3) and linear combinations to construct two kinds of eigenfunctions concentrated in the dyadic annulus $D_{j}^{int}$. One concentrates in the ball of radius $(\lambda 2^{-j})^{-1}$, while the other concentrates in the tube with length $\lambda 2^{-2j}$ and radius $2^{-j/2}$. We shall exploit the strategy in [45] to construct new intermediate examples between these two extreme cases, and prove the sharpness of local $L^{p}$ estimates, see Section 5 for the detailed construction.

For compact sets with fixed size $r$, we shall discuss the relation between their locations and the local $L^{2}$ bounds. It is interesting to note that the $L^{2}$ bounds (1.16) with $r$ larger than the threshold $\lambda^{-\frac{1}{3}}$ are decreasing in $\mu$ with the decay rate $\sim\mu^{-\frac{1}{4}}$ when $\mu\gg r/\lambda$. See Figure 4. This suggests that the local probabilities are decreasing away from the boundary and then satisfy Bohr’s correspondence principle. When the size $r$ is smaller than the threshold $\lambda^{-\frac{1}{3}}$, there are some significant differences between one dimension and higher dimensions. See Figures 5, 6. It is because in the sense of $L^{\infty}$ norm, higher dimensional eigenfunctions can concentrate away from the boundary, while one dimensional eigenfunctions only concentrate near the boundary.

An important special case of Theorem 1 is the $\mu=r=1$ case, which extends Thangavelu’s local estimates (1.14) to all $p\geq 2$.

Theorem 2 is new for $2\leq p<\frac{2n+2}{n-1}$. It is enlightening to compare (1.17) with Sogge’s $L^{p}$ estimates [52] on compact manifolds

(1.18)

$$\|e_{\lambda}\|_{L^{p}(M)}\leq C\lambda^{\sigma(p)}\|e_{\lambda}\|_{L^{2}(M)},\ 2\leq p\leq\infty.$$

This phenomenon suggests that the Hermite eigenfunctions locally resemble the rescaled Laplace eigenfunctions.

Furthermore, optimal local $L^{p}$ estimates for all $p\geq 2$ can be obtained by the interpolation between Theorem 1 and Koch-Tataru’s $L^{p}$ bounds.

Here we use the convention that the endpoints $\frac{2n+2}{n-1}=\infty$ when $n=1$, and $\frac{2n}{n-2}=\infty$ when $n=1,2$. At the kink point $p=\frac{2n+6}{n+1}$, the local $L^{p}$ bounds for $\lambda\mu\lesssim r\leq\lambda$ coincide with the global estimates (1.7). The number $\lambda\tilde{\mu}$ is essentially the distance between the compact $B(\nu,r)$ and the boundary $\{|x|=\lambda\}$, and obviously $\lambda\tilde{\mu}\leq\lambda\mu$.

Theorem 3 improves the local bounds derived from Koch-Tataru’s estimates (1.8) and (1.9) when $(\lambda\mu^{\frac{1}{2}})^{-1}\ll r\ll\lambda\mu$ and $2\leq p\leq\frac{2n+2}{n-1}$. Moreover, for any fixed $r$, we can use Theorem 3 to determine the locations of $\nu$ (i.e. conditions on $\mu$) that can attain the maximal local $L^{p}$ norms

$$\sup_{\nu\in\mathbb{R}^{n}}\|e_{\lambda}\|_{L^{p}(B(\nu,r))}.$$

See the following Tables 1, 2, 3 with $n\geq 2$.

These tables demonstrate some new concentration features of the eigenfunctions. For $2\leq p<\frac{2n+6}{n+1}$, the maximal local $L^{p}$ bounds significantly improve the global estimates $\lambda^{\frac{1}{p}-\frac{1}{2}}$ whenever $r\ll\lambda$. However, the maximal local $L^{p}$ bounds $cannot$ improve the global estimates (1.4) for $p>\frac{2n+6}{n+1}$ and $r\gtrsim\lambda^{-\frac{1}{3}}$. This phenomenon suggests that the Hermite eigenfunctions $cannot$ highly concentrate in any “small” compact set with size $r\ll\lambda$, in terms of the $L^{p}$ norm with $p$ less than $\frac{2n+6}{n+1}$. But this kind of concentration is possible for all larger $p$. These features of the eigenfunctions explain why the exponent $\rho(p)$ in (1.4) is decreasing in $p$ when $p<\frac{2n+6}{n+1}$, and increasing when $p>\frac{2n+6}{n+1}$. See Figure 1.

The paper is structured as follows. In Section 2, we presents the proof of the stationary phase lemma with a precise remainder term. In Section 3, we review the representation of the kernel of the Hermite spectral projection operator. In Section 4, we prove Theorem 1. In Section 5, we show the sharpness of local $L^{p}$ bounds. In Section 6, we discuss some related questions for the Hermite eigenfunctions.

Throughout this paper, $X\lesssim Y$ means $CX\leq Y$ for some positive constants $C$ that depend only on dimension $n$ and the number of times we take derivatives and integrate by parts. In particular, if $C>1$ is a large constant and $CX\leq Y$, then we denote $X\ll Y$. If $X\lesssim Y$ and $Y\lesssim X$, we denote $X\approx Y$. The notation $\|f\|_{p}$ means the Lebesgue norm of $f$ in $\mathbb{R}^{n}$. Sometimes we abbreviate the phase function $\psi(t,x,y)$ as $\psi(t)$ when $x,y$ are fixed, and denote its partial derivative with respect to $t$ by $\psi^{\prime}(t)$.

Let $a\in C_{0}^{\infty}(\mathbb{R})$. Let $\phi\in C^{\infty}(\mathbb{R})$ be real-valued. We estimate

$$I_{\lambda}=\int e^{i\lambda\phi(t)}a(t)dt.$$

Let $N\geq 1$, $b\in C^{\infty}(\mathbb{R})$ and $\partial=\frac{d}{dt}$. We define

$$(\partial\circ b)(a)=\partial(ab)=a\partial b+b\partial a.$$

Note that

$$(\partial\circ b)^{N}(a)=\sum_{|\alpha|=N}c_{\alpha}\partial^{\alpha_{0}}a\partial^{\alpha_{1}}b\cdot\cdot\cdot\partial^{\alpha_{N}}b,$$

where $\alpha=(\alpha_{1},...,\alpha_{N})\in\mathbb{N}^{N}$, and $|\alpha|=\alpha_{0}+...+\alpha_{N}$. For example,

$$(\partial\circ b)^{2}(a)=(\partial\circ b)(a\partial b+b\partial a)=b^{2}\partial^{2}a+3b\partial a\partial b+a\partial b\partial b+ab\partial^{2}b.$$

Moreover, for $j=1,...,N$,

$$\partial^{\alpha_{j}}(f^{-1})=f^{-\alpha_{j}-1}\sum_{|\gamma^{j}|=\alpha_{j}}c_{\gamma^{j}}\partial^{\gamma^{j}_{1}}f\cdot\cdot\cdot\partial^{\gamma^{j}_{\alpha_{j}}}f,$$

where $\gamma^{j}=(\gamma_{1}^{j},...,\gamma^{j}_{\alpha_{j}})$. Here we use the convention that the sum is 1 if $\alpha_{j}=0$.

Suppose that $\phi^{\prime}\neq 0$ on $\text{supp }a$. Integrating by parts $N$ times, we have

	$\displaystyle I_{\lambda}$	$\displaystyle\lesssim\lambda^{-N}|\text{supp }a|\|(\partial\circ(\phi^{\prime-1}))^{N}(a)\|_{\infty}$
		$\displaystyle\lesssim\lambda^{-N}|\text{supp }a|\sum_{|\alpha|=N}\|(\partial^{\alpha_{0}}a)(\phi^{\prime})^{\alpha_{0}-2N}\prod_{j=1}^{N}\sum_{|\gamma^{j}|=\alpha_{j}}|\partial^{\gamma_{1}^{j}}(\phi^{\prime})\cdot\cdot\cdot\partial^{\gamma_{\alpha_{j}}^{j}}(\phi^{\prime})|\|_{\infty}$
		$\displaystyle\lesssim\lambda^{-N}|\text{supp }a|\sum_{\alpha_{0}=0}^{N}\sum_{|\beta|=N-\alpha_{0}}\|(\partial^{\alpha_{0}}a)(\phi^{\prime})^{\alpha_{0}-2N}\partial^{\beta_{1}}(\phi^{\prime})\cdot\cdot\cdot\partial^{\beta_{N-\alpha_{0}}}(\phi^{\prime})\|_{\infty}$
		$\displaystyle\lesssim\lambda^{-N}|\text{supp }a|\sum_{\alpha_{0}=0}^{N}\sum_{\sigma}\|(\partial^{\alpha_{0}}a)(\phi^{\prime})^{\alpha_{0}-2N+\sigma_{0}}(\phi^{\prime\prime})^{\sigma_{1}}\cdot\cdot\cdot(\phi^{(N-\alpha_{0}+1)})^{\sigma_{N-\alpha_{0}}}\|_{\infty}$

where $\beta=(\beta_{1},...,\beta_{N-\alpha_{0}})$, and $\sigma=(\sigma_{0},...,\sigma_{N-\alpha_{0}})$ satisfies $\sum k\sigma_{k}=N-\alpha_{0}=\sum\sigma_{k}$.

Suppose that $\phi\in C^{\infty}(\mathbb{R})$ is real-valued and satisfies $\phi(0)=\phi^{\prime}(0)=0,\phi^{\prime\prime}(0)\neq 0$. Let

(2.1)

$$g(t)=\phi(t)-\frac{1}{2}\phi^{\prime\prime}(0)t^{2},$$

(2.2)

$$\phi_{\theta}(t)=\frac{1}{2}\phi^{\prime\prime}(0)t^{2}+\theta g(t),\ 0\leq\theta\leq 1.$$

Then $\phi_{1}=\phi$. Suppose that

$$|t|\leq|\phi^{\prime\prime}(0)|/\|\phi^{\prime\prime\prime}\|_{\infty},\ \ \forall t\in\text{supp }a.$$

Then

$$|g(t)/t^{3}|\leq\frac{1}{6}\|\phi^{\prime\prime\prime}\|_{\infty},$$

$$|g^{\prime}(t)/t^{2}|\leq\frac{1}{2}\|\phi^{\prime\prime\prime}\|_{\infty},$$

$$|g^{\prime\prime}(t)/t|\leq\|\phi^{\prime\prime\prime}\|_{\infty},$$

$$|g^{(k)}(t)|\leq\|\phi^{(k)}\|_{\infty},\ k\geq 3.$$

$$|t/\phi_{\theta}^{\prime}(t)|=\frac{1}{|\phi^{\prime\prime}(0)+\theta g^{\prime}(t)/t|}\leq\frac{1}{|\phi^{\prime\prime}(0)|-\frac{1}{2}|t|\|\phi^{\prime\prime\prime}\|_{\infty}}\leq\frac{2}{|\phi^{\prime\prime}(0)|},$$

$$|t/\phi_{\theta}^{\prime}(t)|=\frac{1}{|\phi^{\prime\prime}(0)+\theta g^{\prime}(t)/t|}\geq\frac{1}{|\phi^{\prime\prime}(0)|+\frac{1}{2}|t|\|\phi^{\prime\prime\prime}\|_{\infty}}\geq\frac{2}{3|\phi^{\prime\prime}(0)|}.$$

Thus

$$|\phi_{\theta}^{\prime}(t)|\approx|t||\phi^{\prime\prime}(0)|,$$

$$|\phi_{\theta}^{\prime\prime}(t)|=|\phi^{\prime\prime}(0)+\theta g^{\prime\prime}(t)|\leq 2|\phi^{\prime\prime}(0)|,$$

$$|\phi_{\theta}^{(k)}(t)|\leq\|\phi^{(k)}\|_{\infty},\ k\geq 3.$$

Let

$$I_{\lambda}(\theta)=\int e^{i\lambda\phi_{\theta}(t)}a(t)dt,\ \ a\in C_{0}^{\infty}(\mathbb{R}).$$

For $m\geq 1$, we need to estimate

$$I_{\lambda}(1)=\sum_{k=0}^{2m-1}I^{(k)}_{\lambda}(0)/k!+\frac{1}{(2m-1)!}\int_{0}^{1}I_{\lambda}^{(2m)}(\theta)(1-\theta)^{2m-1}d\theta.$$

For the last term,

	$\displaystyle|I_{\lambda}^{(2m)}(\theta)|\lesssim\lambda^{2m}\Big{|}\int e^{i\lambda\phi_{\theta}(t)}g(t)^{2m}a(t)dt\Big{|}$
	$\displaystyle\lesssim\lambda^{2m}\lambda^{-3m}|\text{supp }a|\sum_{\alpha_{0}=0}^{3m}\sum_{\sigma}\|(\partial^{\alpha_{0}}(g^{2m}a))(\phi_{\theta}^{\prime})^{\alpha_{0}-6m+\sigma_{0}}(\phi_{\theta}^{\prime\prime})^{\sigma_{1}}\cdot\cdot\cdot(\phi_{\theta}^{(3m-\alpha_{0}+1)})^{\sigma_{3m-\alpha_{0}}}\|_{\infty}$
	$\displaystyle\lesssim\lambda^{-m}|\text{supp }a|\sum_{\alpha_{0}=0}^{3m}\sum_{\gamma,\sigma}\|(\partial^{\gamma^{\prime}}a)g^{\gamma_{0}}(g^{\prime})^{\gamma_{1}}\cdot\cdot\cdot(g^{(\alpha_{0})})^{\gamma_{\alpha_{0}}}(\phi_{\theta}^{\prime})^{\alpha_{0}-6m+\sigma_{0}}(\phi_{\theta}^{\prime\prime})^{\sigma_{1}}\cdot\cdot\cdot(\phi_{\theta}^{(3m-\alpha_{0}+1)})^{\sigma_{3m-\alpha_{0}}}\|_{\infty}$
	$\displaystyle\lesssim\lambda^{-m}|\text{supp }a|\sum_{\alpha_{0}=0}^{3m}\sum_{\gamma,\sigma}\|a^{(\gamma^{\prime})}\|_{\infty}|\phi^{\prime\prime}(0)|^{2\alpha_{0}-12m+3\gamma_{0}+2\gamma_{1}+\gamma_{2}+2\sigma_{0}+\sigma_{1}}\|\phi^{\prime\prime\prime}\|_{\infty}^{6m-\alpha_{0}-2\gamma_{0}-\gamma_{1}+\gamma_{3}-\sigma_{0}+\sigma_{2}}$
	$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\prod_{k\geq 4}\|\phi^{(k)}\|_{\infty}^{\gamma_{k}+\sigma_{k-1}}$
	$\displaystyle\lesssim\lambda^{-m}|\text{supp }a|(\|a\|_{\infty}|\phi^{\prime\prime}(0)|^{-3m}\|\phi^{\prime\prime\prime}\|_{\infty}^{2m}+......)$

where $\gamma=(\gamma_{0},...,\gamma_{\alpha_{0}})$ satisfies $\sum\gamma_{k}=2m$ and $\gamma^{\prime}+\sum k\gamma_{k}=\alpha_{0}$, and $\sigma=(\sigma_{0},...,\sigma_{3m-\alpha_{0}})$ satisfies $\sum k\sigma_{k}=3m-\alpha_{0}=\sum\sigma_{k}$.