A Restriction Theorem for Space Curves

Abstract

Let a be a Ck curve (/c > 3) in R3 with nonvanishing curvature and torsion.It is proved that the restriction operator T: f-*f\a is bounded from L'iR3) to L^a) if 1 < p < 15/13 and \/q > 6(1 -\/p), and that T is not bounded if p > 6/5 or \/q < 6(1 -\/p).Let a be a Ck curve in R3 (/c > 3).For/ E C0°%R3) define the restriction operator F by Tf-f\a.Then the following result holds:Theorem.If the curve a has nonvanishing curvature and torsion, the inequality /20/Í& // 1 < p < 15/13 and l/q > 6(1 -1/p).F/ie inequality does not hold if p > 6/5 or l/q < 6(1 -1/p).Proof.First we show that (1) does not hold if l/q < 6(1 -1/p).Let/be a smooth function such that its Fourier transform / is supported on a parallelepiped centered at any point P on the curve and whose dimensions are 17, r¡2, tj3 along the tangent, normal and binormal direction at P, respectively (tj denotes a small number).Then up to infinitesimals of higher order, ll/llz.>(*3)= t]6(X~x/p) and ||7y||L,(a) = t\x/q, hence T cannot be a bounded operator unless l/q > 6(1 -1/p).Now we prove (1).We may assume, without loss of generality, that a is defined by the equations (/, (pit), ¡Pit), 0 < f < 17), where n is a small number and (pit) = kt2/2 + £(/), »KO = £t/3/6 + f it).Here k and t denote curvature and torsion, |(r) and £ (?) are infinitesimals of third and fourth order w.r.t./.Then denote by T* the adjoint operator (^)3w=nx^-^.(f^^(?^^3(?H)) ■f(h)f(h)f(h)dtxdt2dt3 withy = (y"y2,y3).

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• Proceedings of the American Mathematical Society - View - PDF