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?