  Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly
positive second fundamental form, $E$ be the Fourier extension operator on $S$,
and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball
of each radius, then the problem of determining the range of exponents $(p,q)$
for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is
equivalent to the restriction conjecture. In this paper, we study the estimate
under the following assumption on the set $X$: there is a number $0 < \alpha
\leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in
$\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are
integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our
approach consists of replacing the characteristic function $\chi_X$ of $X$ by
an appropriate weight function $H$, and studying the resulting estimate in
three different regimes: small values of $\alpha$, intermediate values of
$\alpha$, and large values of $\alpha$. In the first regime, we establish the
estimate by using already available methods. In the second regime, we prove a
weighted H\"{o}lder-type inequality that holds for general non-negative
Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result
from the first regime. In the third regime, we borrow a recent fractal Fourier
restriction theorem of Du and Zhang and combine it with the result from the
second regime. In the opposite direction, the results of this paper improve on
the Du-Zhang theorem in the range $0 < \alpha < n/2$.
