Radon Transform on Sobolev Spaces

The Radon transform R maps a function f on R-n to the family of the integrals of f over all hyperplanes. The classical Reshetnyak formula (also called the Plancherel formula for the Radon transform) states that parallel to f parallel to L-2(R-n) = parallel to Rf parallel to(H(n-1)/2(n-1)/2(Sn-1 x R),) where parallel to.parallel to H(n- 1)/2(n-1)/2(Sn-1 x R) is some special norm. The formula extends the Radon transform to the bijective Hilbert space isometry R : L-2(R-n) -> H-(n-1)/2,e((n-1)/2)(Sn-1 x R). Given reals r, s, and t > - n/2, we introduce the Sobolev type spaces H-t((r,s)) (R-n) and H-(r,s)(t,e) (Sn-1 x R) and prove the version of the Reshetnyak formula: parallel to f parallel to((r,s))(Ht) (R-n) = parallel to Rf parallel to(Ht+(n-1)/2(r,(s+n- 1)/2)(Sn-1 x R)). The formula extends the Radon transform to the bijective Hilbert space isometry R : H-t((r,s)) (R-n) -> Ht+(n-1)/2,(e(r,s+(n-1)/2)) (Sn-1 x R). If r >= 0 and s >= 0 are integers then H-0,e((r,s)) (Sn-1 x R) consists of the even functions phi(xi, p) with square integrable derivatives of order <= r with respect to xi and order <= s with respect to p.