FOURIER INTEGRAL-OPERATORS WITH CUSP SINGULARITIES

We study the boundedness properties, on Lebesgue and Sobolev spaces, o f Fourier integral operators associated with canonical relations such that at least one of the projections is a simple (Whitney) cusp. In th e process, we obtain decay estimates for oscillatory integral operator s whose symplectic relations have the same singular structure. Such si ngularities occur generically for averages over lines and curves in R- 4. On L-2, we show that the operators lose 1/3 derivative. To obtain s harp results off of L-2, we need to impose an additional transversalit y condition, satisfied by many geometric averaging operators, which le ads to the notion of a strong cusp. These estimates can be further imp roved if we impose curvature conditions on the cusp surface. One appli cation is the L-comp(2) --> L-1/6,loc(2) and L-comp(12/7) --> L-loc(2) boundedness of restrictions of the X-ray transform on R-4 to four-dim ensional families of lines satisfying a natural curvature and torsion condition.