Thanks for reading my post. I am trying to prove the following claim:
If we have \begin{equation*} \left\|\hat{f}\right\|_{L^q(N_{1/R}(S))}\lesssim R^{\alpha-1/q}\left\|f\right\|_{L^p(B(0,R))} \end{equation*} then we have \begin{equation*} \left\|\hat{f}|_{S}\right\|_{L^q(S; d\sigma)}\lesssim R^{\alpha}\left\|f\right\|_{L^p(B(0,R))} \end{equation*} in which $S$ is the standard sphere in $\mathbb{R}^n$, $N_{1/R}(S)$ is the $1/R$ neighborhood of the sphere. $R\gg1$. In particular $f$ is supported in the $R$-ball $B(0,R)$.
This is in fact Problem 2.2 in Tao's Recent Progress on the Restriction Conjecture lecures, see arxiv page 26. What I could do so far, is to take a bump funciton $\psi(z/R)$ such that $\psi(z)=1$ when $|z|\leq 1$. Then we use the fact
\begin{equation*} \hat{f}=\hat{f}\ast\hat{\psi}=\int R^n\hat{\psi}(R(\xi-\eta))\hat{f}(\eta)d\eta\hspace{2cm}(1) \end{equation*} to calculate its $L^q(S)$ norm. It is easy to deal with the part when $|\xi-\eta|<1/R$ in $(1)$. Then using the fact $\psi(z)$ is fast decaying when $|z|>1$ we can also easily get rid of part $|\xi-\eta|>1$ ( or a small power of $1/R$ ) in $(1)$. However I am having difficulties in dealing with the part $1/R\leq|\xi-\eta|\leq1$ and especially when $|\xi-\eta|$ is bigger but closed to $1/R$. Should I use some dyadic decomposition or am I working in the wrong direction?
Thanks.
As user90090 suggested, consider a decomposition of $\hat\psi=\sum_{k=0}^{\infty}\psi_k$ in which $\psi_k(x), k\geq1$ have support $k<|x|<{k+1}$. Notice that this decomposition is slight different from the original dyadic decomposition. There are mainly two reasons, of different level, to explain this. First, from uncertainty principle, if $f$ lives in a ball $B(0,R)$, then its Fourier transform should have frequency "lives" in a band with width $1/R$, so it is a little more natural to break up the frequency variable into $1/R$ annular bands. Second, from technical level, we need to repeatedly use the following estimate
$$ ||f||_{L^q(N_{1/R}(S))}\lesssim R^{\alpha-1/q}||f||_{L^p(B(0,R))}\hspace{2cm} (2)$$
So simply speaking what we need to do is to break up the $(k/R, (k+1)/R)$ annular area into a group of small $1/R$ balls with finite over-lapping. Notice that there are at most $O(k^n)$ such balls, so the number can be controlled by $k^{-N}$ decay coming from the fact that $\psi$ is fast decaying. Then we shift these small balls back into $N_{1/R}(S)$ with an exponential multiple of $f$, which does not affect the support of it. So now we can safely apply $(2)$ to get what we want.