How to use van der Corput's lemma to get the following estimates?

Question

Recently, I'am reading Tao's article Spherically Averaged Endpoint Strichartz Estimates for the Two-Dimensional Schrodinger equation, in which there are some problems that I can't solve by myself.

We define Bessel function by

\begin{equation} J_n(\lambda)=\frac{1}{2 \pi} \int_{0}^{2\pi} e^{i\lambda \cos\theta} e^{in\theta} d\theta \end{equation}

and decompose $J_n$ smoothly by \begin{equation} J_n(\lambda)=m_0(\lambda)+m_1(\lambda)+\sum_{2^j \gg n} m_j(\lambda), \end{equation}

where $m_0,m_1,m_j$ are supported on $|r| \ll n , |r|\sim n$ and $|r| \sim 2^j \gg n$ respectively.

As for $m_1$ and its derivative, Tao says using van der Corput's lemma, we can get the following estimates:

• \begin{equation}|m_1(\lambda)|\lesssim n^{-1/3}(1+n^{-1/3}|\lambda-n|)^{-1/4},\end{equation}
• \begin{equation}|m_1'(\lambda)| \lesssim n^{-1/2}. \end{equation}

Can someone present details of the argument to the two estimates above?

Answer 1

For the first expression, to get the first two terms in the asymptotics when $r=n$, use the alternative integral expression for first-order Bessel functions, $\displaystyle J_{n}(r)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{i(r\sin \theta -n\theta )}\,d\theta. $

Proposition 3, on page 334 of Stein's Harmonic Analysis, then applies

Proposition. Let $\displaystyle I(\lambda)=\int\limits_{a}^{b}e^{i\lambda \phi} \psi dx \displaystyle$. Suppose $k\geq 2$, and $$ \displaystyle \phi(x_0)=\phi'(x_0)=\dots=\phi^{(k-1)}(x_0)=0 \displaystyle,$$ while $\phi^{(k)}(x_0)\neq 0$. If $\psi$ is supported in a sufficiently small neighborhood of $x_0$, then $$\displaystyle I(\lambda)=\int\limits e^{i\lambda \phi}\psi dx \sim \lambda^{-1/k}\sum\limits_{j=0}^\infty a_j\lambda^{-j/k} \displaystyle.$$

On pg 256, in section 5.2 of the same chapter, it is shown that for $n=r$ one can get the estimate

$$J_{n}(n)=cn^{-1/3}+O(n^{-2/3})\ \text{ as } n\rightarrow \infty $$ simply by noting that $\phi(\theta)=\sin{\theta}-n\theta/r$ satisfies $\phi(0)=\phi'(0)=\phi''(0)=0$ but $\phi'''(0)\neq 0$. This can be adjusted to the case that $|\lambda|\sim n$ to get a bound agreeing (asymptotically) with that in the first question.

The second estimate should follow from a similar argument.