Restriction theorems for anisotropically homogeneous hypersurfaces of R^n+1

FERRERYRA, ELIDA; URCIUOLO, MARTA

Georgian Mathematical Journal

Año: 2006

For x=(x₁,...,x_{n})∈ Rⁿ and β₁,...,β_{n}>1, let ϕ:Rⁿ→R be defined by ϕ(x)=∑_{j=1}ⁿ|x_{j}|^{βj}, let B be the open unit ball in Rⁿ and let Σ={(x,ϕ(x)):x∈B}. For f∈S(Rⁿ⁺¹), 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}(Rⁿ⁺¹)} for all f∈S(Rⁿ⁺¹). In this paper we obtain a polygonal region contained in E. We also give necessary conditions for a pair ((1/p),(1/q))∈E. In some cases this result is sharp, up to endpoints.