Weighted restriction estimates using polynomial partitioning

We use the polynomial partitioning method of Guth to prove weighted Fourier restriction estimates in ${\mathbb{R}}^3$ 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 $C^{\mathrm{\infty }}$ surfaces $S\subset {\mathbb{R}}^3$ 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 $\mathrm{\parallel }Ef{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(S)}$ for $p>3$ and certain sets $\mathrm{\Omega }\subset {\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$ with finite $\alpha$-dimensional energies.