Uniform Estimates for the Fourier Transform of Surface Carried Measures in ℝ3 and an Application to Fourier Restriction

Abstract

Let S be a hypersurface in R 3 which is the graph of a smooth, finite type function φ, and let μ = ρ dσ be a surface carried measure on S, where dσ denotes the surface element on S and ρ a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transformμ of μ, which are sharp except for the case where the principal face of the Newton polyhedron of φ, when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp L p -L 2 Fourier restriction theorem for S in the case where the original coordinates are adapted to φ. This improves on earlier joint work with M. Kempe.

Linear and bilinear restriction to certain rotationally symmetric hypersurfaces

“…[20]) and for arbitrary homogeneous polynomials [9]. Ikromov-Müller [15] have proved the sharp unweighted L 2 ξ restriction estimates for hypersurfaces in R 3 expressed in adapted coordinates.…”

Chapter Thirteen. Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra

“…For this case, a complete answer had been given in [40] (for analytic hypersurfaces, partial results had been obtained before by Magyar [48]) :…”

“…In [40], we proved, by a quite different method, that Karpushkin's result remains valid for smooth, finite type functions φ, at least for linear perturbations, which led to the following: Theorem 3.1. Let S = graph(φ) be as before, and assume that φ is smooth and of finite type.…”

“…We remark [40] that the first condition is equivalent to the following one: If y is any adapted local coordinate system at the origin, then either π(φ a ) is a vertex, or a compact edge and m(φ a pr ) = d(φ a ). Varchenko [66] has shown that the leading exponent in (3.1) is given by r 0 = 1/ h(φ), and ν(φ) is the maximal j for which a j,k (φ) = 0.…”

“…All these results are based on joint work with Ikromov, and in parts also with Kempe [37][38][39][40][41][42].…”