Fourier restriction to convex surfaces of revolution in R3

If Γ is a C3 hypersurface in Rn and dσ is induced Lebesgue measure on Γ, then it is well known that a Tomas-Stein Fourier restriction estimate on Γ implies that Γ has a nowhere vanishing Gaussian curvature. In a recent paper, Carbery and Ziesler observed that if induced Lebesgue measure is replaced by affine surface area, then a Tomas-Stein restriction estimate on Γ implies that Γ satisfies the affine isoperimetric inequality. Since the only property needed for a hypersurface to satisfy the affine isoperimetric inequality is convexity, this raised the question of whether a TomasStein restriction estimate can be obtained for flat but convex hypersurfaces in Rn such as Γ(x) = (x, e−1/ $m ), m = 1, 2, . . . . We prove that this is indeed the case in dimension n = 3

A class of nonhomogeneous singular integrals in Rn

AbstractLet {δt}t>0 be a nonisotropic dilation group on Rn, let ρ be a distance function on Rn which is homogeneous with respect to {δt}t>0, and for f∈C0∞(Rn) define Tf=[p.v.ρ(·)−αei/|·|β]∗f, where α and β are positive parameters. We give necessary and sufficient conditions on p, α and β for which T extends to a bounded linear operator on Lp(Rn)

Fourier restriction to convex surfaces of revolution in R3

If Γ is a C3 hypersurface in Rn and dσ is induced Lebesgue measure on Γ, then it is well known that a Tomas-Stein Fourier restriction estimate on Γ implies that Γ has a nowhere vanishing Gaussian curvature. In a recent paper, Carbery and Ziesler observed that if induced Lebesgue measure is replaced by affine surface area, then a Tomas-Stein restriction estimate on Γ implies that Γ satisfies the affine isoperimetric inequality. Since the only property needed for a hypersurface to satisfy the affine isoperimetric inequality is convexity, this raised the question of whether a TomasStein restriction estimate can be obtained for flat but convex hypersurfaces in Rn such as Γ(x) = (x, e−1/ $m ), m = 1, 2, . . . . We prove that this is indeed the case in dimension n = 3