We study the boundedness properties, on Lebesgue and Sobolev spaces, of Fourier integral operators associated with canonical relations such that at least one of the projections is a simple (Whitney) cusp. In the process, we obtain decay estimates for oscillatory integral operators whose symplectic relations have the same singular structure. Such singularities occur generically for averages over lines and curves in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. On L2, we show that the operators lose 1/3 derivative. To obtain sharp results off of L2, we need to impose an additional transversality condition, satisfied by many geometric averaging operators, which leads to the notion of a strong cusp. These estimates can be further improved if we impose curvature conditions on the cusp surface. One application is the L2compL21/6,loc and L12/7compL2loc boundedness of restrictions of the X-ray transform on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] to four-dimensional families of lines satisfying a natural curvature and torsion condition.


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pp. 1077-1119
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