Let $\lambda$ denote Lebesgue measure on $\mathbb{R}^d$. The Hausdoff-Young inequality is that $$ \| \widehat{f} \|_{L^{q}(\lambda)} \leq \| f \|_{L^{p}(\lambda)}. $$ when $1 \leq p \leq 2$ and $q=p'$.
Given a Borel measure $\mu$ on $\mathbb{R}^d$, the Fourier restriction inequality is $$ \| \widehat{f} \|_{L^{q}(\mu)} \leq \| f \|_{L^{p}(\lambda)}. $$ The Fourier restriction problem asks for which $p$, $q$, and $\mu$ this inequality holds.
Suppose $\mu$ is absolutely continuous with respect to $\lambda$.
If $\mu$ has bounded (or $L^{\infty}$) density, then the restriction inequality holds with $1 \leq p \leq 2$ and $q=p'$ by Holder's inequality and Hausdorff-Young. Moreover, if the support of the density has finite Lebesgue measure, then $q \leq p'$ is also valid.
What happens to the restriction inequality if $\mu$ does not have bounded density?