Remarks on the general Funk transform and thermoacoustic tomography

We discuss properties of a generalized Minkowski-Funk transform de…ned for a family of hypersurfaces. We prove two-side estimates for the integral operator and show that the range conditions can be written in terms of the reciprocal Funk transform. Some applications to the spherical mean transform are considered. 1 Families of hypersurfaces Let X and be smooth manifolds of dimension n > 1 and F be a closed smooth hypersurface in X : We assume that (*) the natural projection : F ! has rank n and the mapping g : F ! G T (X) is a local di¤eomorphism, where g (x; ) = (x; ), is the tangent hyperplane to F ( ) at x and G T (X) denotes the variety of n 1-subspaces in the tangent bundle T (X) of X: It follows that the sets F ( ) = 1 ( ) ; 2 are hypersurfaces in X and for any point x 2 X and any tangent hyperplane Tx (X) there is a (locally unique) hypersurface F ( ) through x tangent to : We shall see below that the property (*) is in fact symmetric with respect to X and : If F is cooriented, we choose a smooth function in X such that = 0 and d 6= 0 on F (the phase function). If F is not cooriented, one can choose a set of local phase functions f g such that = in any domain, where both functions , are de…ned. Proposition 1.1 The condition (*) is equivalent to the inequality det J ( ) 6= 0 in F;