2 0 Fe b 20 02 OSCILLATORY INTEGRAL OPERATORS WITH LOW – ORDER DEGENERACIES

We prove sharp L estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ΩL × T ∗ΩR projects to T ∗ΩL and to T ∗ΩR with corank one singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line complexes in five dimensions. Introduction Let ΩL, ΩR be open sets in R . This paper is concerned with L bounds for oscillatory integral operators Tλ of the form (1.1) Tλf(x) = ∫ eσ(x, z)f(z)dz where Φ ∈ C(ΩL × ΩR) is real-valued, σ ∈ C∞ 0 (ΩL × ΩR) and λ is large. We shall also write Tλ ≡ Tλ[σ] to indicate the dependence on the symbol σ. The decay in λ of the L operator norm of Tλ is determined by the geometry of the canonical relation (1.2) C = {(x,Φx, z,−Φz) : (x, z) ∈ ΩL × ΩR} ⊂ T ΩL × T ΩR, specifically by the behavior of the projections πL : C → T ΩL and πR : C → T ΩR , (1.3) πL : (x, z) 7→ (x,Φx(x, z)) πR : (x, z) 7→ (z,−Φz(x, z)); here Φx and Φz denote the partial gradients with respect to x and z. Note that rank DπL = rank DπR is equal to d+ rank Φxz and that the determinants of DπL and DπR are equal to (1.4) h(x, z) := detΦxz(x, z). 1991 Mathematics Subject Classification. 35S30 (primary), 42B99, 47G10 (secondary).