Consider the unit sphere $\mathcal{S}^n$. Let $Q$ be a cube contained in the unit circle in $\mathbb{R}^n$. Let $E$ be a subset of $\mathcal{S}^n$ defined by $$ E = \{ (x,\sqrt{1-|x|^2}) : x \in Q \}. $$
Let $ f \in L^2(\mathcal{S}^n) $, then one may define the Fourier transform,
$$ \widehat{f \sigma}(x) = \int_{\mathcal{S}^n} e^{-2 \pi i x \cdot \xi} f(\xi)\ d \sigma(\xi)$$
for every $ x \in \mathbb{R}^{n+1}.$
This is well defined and by Tomas-Stein, $\widehat{f \sigma} \in L^\frac{2(n+2)}{n}(\mathbb{R}^{n+1}). $
Can someone help me understand the support of the Fourier transform of $\widehat{f\chi_E\sigma}$. Any help is really appreciated.