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?


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.

  • $\begingroup$ Thanks for your help. Before you posted the answer, I turned to prof for help, and he decomposed the interval [0, 2*pi] into [0,1/4*pi], [1/4*pi,3/4*pi], [3/4*pi,5/4*pi]... and considered the 2-nd or 3-rd derivative of phase phi(theta) on each intervals, then we could use Van der Corput's lemma directly to get the bound n^{-1/3}. But we had no idea of how to adjust the estimate to get the precise answer of the first question? Therefore, is it convinient for you to present some more details of how to get the bound agreeing with the first question? Sorry to bother you. $\endgroup$ – Tao.Zhou Apr 10 at 14:07

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