A distinction between the paraboloid and the sphere in weighted restriction
arxiv(2023)
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.
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