A restriction theorem for the Fourier transform

Abstract

Let ƒ be a Schwartz function on R n , and let ƒ(0) denote the restriction of the Fourier transform of ƒ to the unit sphere S n ~x in R n .We prove THEOREM.Iff is in L p (R n ) for some p with 1 < p < 2(n +1)/(« + 3), then 5 s n-i\Ke)\ 2 de<c p \\f\\ 2 p .PROOF.Jl/(0)l 2 d0 = ƒƒ* f(x)fà(x)dx = fmdè*f(x)dx<\\f\\ p \\âd *f\\ p , for conjugate indices p and p .Thus it suffices to prove that the operator given by convolution with *3$ is bounded from LP to LP for p in the appropriate range.Let K(x) be a radial Schwartz function with K(x) = 1 for \x\ < 100, and let).It suffices to show there exists e = e(p) > 0 such that \\T k * ƒ \\ p , < C2~€ fc || ƒ || p .This follows from interpolating the estimates \\T k * ƒ IL < C2-( "~1>*/ 2 || f\\ x and ||r fc *f\\ 2 < 2 k \\f\\ 2 .Professor E. M. Stein has extended the range of this result to include p = 2(n + l)/(n + 3).His proof uses complex interpolation of the operators given by convolution with the functions B a (x) = J 0 (27t\x\)/\x\°.Then A great deal was previously known about such restriction theorems.E. M. Stein originally established the theorem for 1 </? < 4n/(3n + 1).For n = 2, this was extended by Fefferman and Stein [2] to the range 1 < p < 6/5.P. Sjolin (see [1]) proved the theorem for n = 3 and 1 <p < 4/3.Finally, A. Zygmund [3] determined for two dimensions all p and q such that the Fourier transform of an LP function restricts to L q (S x ).Since a

Locations

• Project Euclid (Cornell University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF