2 6 Ju l 2 00 0 OSCILLATORY INTEGRAL OPERATORS WITH LOW – ORDER DEGENERACIES

(1.4) h(x, z) := detΦxz(x, z). If C is locally the graph of a canonical transformation, i.e., h(x, z) 6= 0, then ‖Tλ‖ = O(λ−d/2) (see Hörmander [15], [16]). If the projections have singularities then there is less decay in λ and in various specific cases the decay has been determined. In dimension d = 1 Phong and Stein [21] obtained a complete description of the L mapping properties, for the case of real-analytic phase functions. Similar results for C∞ phases (which however missed the endpoints) and related L estimates for averaging operators in the plane are in [24]. The bounds for oscillatory integral