The Reshetnyak formula and Natterer stability estimates in tensor tomography

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Abstract
The Reshetnyak formula (also known as the Plancherel formula for the Radon transform) states that the Radon transform R is an isometry between L 2 ( R n ) and H ( n − 1 ) / 2 , e ( n − 1 ) / 2 ( S n − 1 × R ) , the latter being the Hilbert space of even functions on S n − 1 × R furnished by some special norm. We generalize this statement to Sobolev spaces: R is an isometry between H s ( R n ) and H ( n − 1 ) / 2 , e s + ( n − 1 ) / 2 ( S n − 1 × R ) for every real s . Moreover, with the help of Riesz potentials, we define some new Hilbert spaces H t s ( R n )   ( t > − n / 2 ) and prove that R is an isometry between H t s ( R n ) and H t + ( n − 1 ) / 2 , e s + ( n − 1 ) / 2 ( S n − 1 × R ) . The generalized Reshetnyak formula closely relates to the Natterer stability estimates: c ∥   f ∥ H s ( R n ) ⩽ ∥ R f ∥ H s + ( n − 1 ) / 2 ( S n − 1 × R ) ⩽ C ∥   f ∥ H s ( R n ) for functions f supported in a fixed ball. Then we obtain analogs of these statements for the x-ray transform of symmetric tensor fields.