Uniform estimates for the Fourier transform of surface carried measures in R³ and an application to Fourier restriction

الأصل · EN

Let S be a hypersurface in R³ which is the graph of a smooth, finite type function ϕ, and let μ=ρ d be a surface carried measure on S, where d denotes the surface element on S and ρ a smooth density with suffiently small support. We derive uniform estimates for the Fourier transform μ 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 Lᵖ-L² 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.