  We use the polynomial partitioning method of Guth to prove weighted Fourier
restriction estimates in $\Bbb R^3$ with exponents $p$ that range between $3$
and $3.25$, depending on the weight. As a corollary to our main theorem, we
obtain new (non-weighted) local and global restriction estimates for compact
$C^\infty$ surfaces $S \subset \Bbb R^3$ with strictly positive second
fundamental form. For example, we establish the global restriction estimate $\|
Ef \|_{L^p(\Bbb R^3)} \leq C \, \| f \|_{L^q(S)}$ in the full conjectured range
of exponents for $p > 3.25$ (up to the sharp line), and the global restriction
estimate $\| Ef \|_{L^p(\Omega)} \leq C \, \| f \|_{L^2(S)}$ for $p>3$ and
certain sets $\Omega \subset \Bbb R^3$ of infinite Lebesgue measure. As a
corollary to our main theorem, we also obtain new results on the decay of
spherical means of Fourier transforms of positive compactly supported measures
on $\Bbb R^3$ with finite $\alpha$-dimensional energies.
