  Suppose $0 < \alpha \leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable
function, and $A_\alpha(H)$ is the infimum of all numbers $C$ for which the
inequality $\int_B H(x) dx \leq C R^\alpha$ holds for all balls $B \subset \Bbb
R^n$ of radius $R \geq 1$. After Guth introduced polynomial partitioning to
Fourier restriction theory, weighted restriction estimates of the form $\| Ef
\|_{L^p(B,Hdx)} \leq C R^\epsilon A_\alpha(H)^{1/p} \| f \|_{L^q(\sigma)}$ have
been studied and proved in several papers, leading to new results about the
decay properties of spherical means of Fourier transforms of measures and, in
some cases, to progress on Falconer's distance set conjecture in geometric
measure theory. This paper improves on the known estimates when $E$ is the
extension operator associated with the unit paraboloid ${\mathcal P} \subset
\Bbb R^3$, reaching the full possible range of $p,q$ exponents (up to the sharp
line) for $p \geq 3 + (\alpha-2)/(\alpha+1)$ and $2 < \alpha \leq 3$.
