Fourier Restriction Problem

When does an inequality of the form $ | \hat{f} |_{L^q(S^{n-1})} \leq C | f |_{L^p(\mathbb{R}^n)} $ hold for functions $ f \in L^p(\mathbb{R}^n) $, where $ \hat{f} $ denotes the Fourier transform of $ f $ and $ q $ is related to $ p $?

Fourier Transform

The Fourier transform of a function $ f $ defined on a measure space $ (X, \mu) $ is defined as: $$ \hat{f}(\xi) = \int_X f(x) e^{-i \langle x, \xi \rangle} \, d\mu(x) $$ where $ \xi \in \mathbb{R}^n $, $ \langle x, \xi \rangle $ denotes the inner product between $ x $ and $ \xi $, and $ d\mu(x) $ is the measure on $ X $.

Stein-Tomas Inequality

For $ f \in L^p(\mathbb{R}^n) $ with $ 1 \leq p \leq \frac{2(n+1)}{n+3} $, the Fourier transform $ \hat{f} $ restricted to the unit sphere $ S^{n-1} $ belongs to $ L^2(S^{n-1}) $, and there exists a constant $ C $ such that $$ \left( \int_{S^{n-1}} |\hat{f}(\xi)|^2 \, d\sigma(\xi) \right)^{1/2} \leq C | f |_p $$ where $ d\sigma $ is the surface measure on $ S^{n-1} $.