A distinction between the paraboloid and the sphere in weighted restriction

Abstract

For several weights based on lattice point constructions in $\mathbb{R}^d (d \geq 2)$, we prove that the sharp $L^2$ weighted restriction inequality for the sphere is very different than the corresponding result for the paraboloid. The proof uses Poisson summation, linear algebra, and lattice counting. We conjecture that the $L^2$ weighted restriction is generally better for the circle for a wide variety of general sparse weights.