This is a cursory remark in the text Oscillatory Integrals in Classical Analysis that I am unable to justify.
Let $f \in L^p(\mathbb{R}^n)$. Then, the Fourier Restriction theorem states that $$\left( \int_{S^{n-1}} | \hat{f}(\xi)|^2 d \sigma(\xi) \right)^{\frac{1}{2}} \leq C ||f||_{L^p}, \quad 1 \leq p \leq \frac{2(n+1)}{n + 3}$$
The author then claims that this is equivalent to the inequality $$\left( \int_{\{1 \leq|\xi| \leq 1 + \varepsilon\}}| \hat{f}(\xi)|^2 d \xi \right)^{\frac{1}{2}} \leq C \varepsilon^{\frac{1}{2}}||f||_{L^p}, \quad 0 < \varepsilon < \frac{1}{2}$$
I have tried to justify the forward direction by rewriting the integral in polar coordinates and using the fact that $$\hat{f}(r \xi) = r^{-n} \widehat{f(\frac{x}{r})}$$ but I wasn't able to get the right power of $\varepsilon$. I do not know how to prove the reverse direction at all though. How can both directions be shown?