Fourier restriction estimates to mixed homogeneous surfaces

FERRERYRA, ELIDA, URCIUOLO MARTA

Journal of inequalities in pure and applied mathematics

Victoria University

Año: 2009 vol. 10 p. 1 - 11

1443-5756

Let a,b be real numbers such that 2≤a<b, and let ϕ:R²→R a mixed homogeneous funcion. We consider polynomial functions ϕ and also functions of the type ϕ(x₁,x₂)=A|x₁|^{a}+B|x₂|^{b}. Let Σ={(x,ϕ(x)):x∈B} with the Lebesgue induced measure. For f∈S(R³) and x∈B, let (Rf)(x,ϕ(x))=f(x,ϕ(x)), where f denotes the usual Fourier transform.For a large class of functions ϕ and for 1≤p<(4/3) we characterize, up to endpoints, the pairs (p,q) such that R is a bounded operator from L^{p}(R³) on L^{q}(Σ). We also give some sharp L^{p}→L² estimates.