# Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate:

there is a constant $C > 0$ such that, for any $R \ge 1$, $\alpha \ge 0$ and test function $f$ supported on $B(x_0, R)$, $||\hat{f}||_{L^q(S^{n-1})} \le C R^{\alpha}||f||_{L^p(B(x_0, R))}$

is equivalent to the estimate

$||\hat{f}||_{L^q(N_{\frac{1}{R}}(S^{n-1})} \le CR^{\alpha - \frac{1}{q}} ||f||_{L^p(B(x_0, R))}$

where $N_{\frac{1}{R}}(S^{n-1})$ is the $1/R$ neighborhood about $S^{n-1}$. The actual proof of this is made as an exercise, and his suggestion for the forward implication here is to translate the estimate by a factor of $O(1/R)$ and then average the estimate over all such translations. However, it's easy to see that if the first estimate holds, then

$||\hat{f}||_{L^q(S^{n-1} + v)} \le CR^{\alpha} ||f||_{L^p(B(x_0, R))}$ because the translation will simply apply a modulation in physical space which won't affect the magnitude of $f$ anyway. I'm not really sure what he means by averaging the translations, but I assume he means something like integrating

$\int_{B(0; \frac{1}{R})} ||\hat{f}||_{L^q(S^{n-1} + v)} dv$ and relating this term to $||\hat{f}||_{L^q(N_{\frac{1}{R}}(S^{n-1}))}$, but I don't really see how to proceed. Any advice would be appreciated.

• A quick comment in case it helps. I haven't looked through the paper yet. The average of a function $f$ over a set $A$ usually means something like $\frac{1}{\mu(A)}\,\int_{\mathrm space}f\,d\mu$. In your case it could mean $\frac{1}{\mu({\mathrm space})}\,\int_{\mathrm space}f\circ \tau_x\,d\mu(x)$. $\tau_x$ being the translation by $x$. – InTransit Jul 21 '14 at 8:25

I do not think that it is what Tao means. If you have this inequality for the unit sphere, you also have it for the sphere that is centered at $0$ of radius $1+\varepsilon$. Just use the result for the function obtained from $f$ by the dilation that transforms this sphere into the unit sphere. Next you integrate in $\varepsilon$ in the interval $(-1/R, +1/R)$ to find the required estimate: the integral on $N_{R^{-1}}(S^{n-1})$ is bounded by $R^{-1}$ times the sup of the integrals over these spheres.