A restriction theorem for space curves

Abstract:

Let $\alpha$ be a ${C^k}$ curve $(k \geqslant 3)$ in ${R^3}$ with nonvanishing curvature and torsion. It is proved that the restriction operator $T:f \to \hat f{|_\alpha }$ is bounded from ${L^p}({R^3})$ to ${L^q}(\alpha )$ if $1 \leqslant p < 15/13$ and $1/q > 6(1 - 1/p)$, and that T is not bounded if $p \geqslant 6/5$ or $1/q < 6(1 - 1/p)$.