Weighted restriction estimates using polynomial partitioning

Abstract

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 $C^{\infty }$ surfaces $S\subset {\mathbb{R}}^3$ with strictly positive second fundamental form. For example, we establish the global restriction estimate $\Vert Ef{\Vert }_{L^p({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^q(S)}$ in the full conjectured range of exponents for $p>3.25$ (up to the sharp line), and the global restriction estimate $\Vert Ef{\Vert }_{L^p(\Omega )}\le C\Vert f{\Vert }_{L^2(S)}$ for $p>3$ and certain sets $\Omega \subset {\mathbb{R}}^3$ 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.

55 pages. Revised following the suggestions of the referee. Accepted for publication in Proceedings of the London Mathematical Society

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