Restriction of the Fourier transform to bidimensional anisotropically homogeneous surfaces

FERRERYRA, ELIDA; URCIUOLO, MARTA

Serie A de los Tarbajos de Matemática de la FaMAF

Año: 2006 vol. 72

For x=(x₁,x₂)∈ R² and β₁,β₂>1, let ϕ:R²→R be defined by ϕ(x)=|x₁|^{β₁}+|x₂|^{β₂}, let B be the open unit ball in R² and let Σ={(x,ϕ(x)):x∈B}. For f∈S(R³), let Rf:Σ→C be defined by    (Rf)(x,ϕ(x))=f(x,ϕ(x))   x∈B,where f denotes the usual Fourier transform of f. Let σ be the Borel measure on Σ defined by σ(A)=∫_{B}χ_{A}(x,ϕ(x))dx and let E be the type set for the operator R, i.e, the set of the pairs ((1/p),(1/q))∈[0,1]×[0,1] for which there exists c>0 such that ‖f‖_{L^{q}(Σ)}≤c‖f‖_{L^{p}(R³)} for all f∈S(R³). In this paper we give necessary conditions for ((1/p),(1/q))∈E. We also obtain new points in E.