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

Recently, I'am reading Tao's article Spherically Averaged Endpoint Strichartz Estimates for The Two-dimensional Schrodinger Equation, and the link of the article is there

in which there are some problems that I can't solve by myself.

We define Bessel function by

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

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

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:

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

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

• I think this is better suited to MathOverflow. Include a link to the arXiv of the paper you are reading. Apr 4, 2019 at 10:25