ON L CONTINUITY OF SINGULAR FOURIER INTEGRAL OPERATORS

We derive Lp continuity of Fourier integral operators with onesided fold singularities. The argument is based on interpolation of (asymptotics of) L2 estimates and H1 → L1 estimates. We derive the latter estimates elaborating arguments of Seeger, Sogge, and Stein’s 1991 paper. We apply our results to the study of the Lp regularity properties of the restrictions of solutions to hyperbolic equations onto timelike hypersurfaces and onto hypersurfaces with characteristic points. 0. Introduction and results The standard Fourier integral operators F : E′(Y )→ D′(X) treated by Hörmander [Ho71] are associated with (local) symplectomorphisms from T ∗X to T ∗Y . The graph C ⊂ (T ∗X\0) × (T ∗Y \0) of this symplectomorphism is referred to as the canonical relation. The continuity of such operators in standard L-based Sobolev spaces already follows from [Ho71]. The L estimates have been obtained in [SeSoSt91]: Let dimX = dimY = n and let F ∈ I(X,Y,C) be the Fourier integral operator of order μ, with its integral kernel vanishing away from a compact set in X × Y . Given 1 < p <∞, the mapping (0.1) F : Lα(Y )→ L p β(X) is continuous if μ ≤ α − αp − β, where αp = (n − 1) ∣∣∣ 1p − 12 ∣∣∣. This continuity is obtained by Fefferman-Stein interpolation between L-L continuity of Fourier integral operators of order 0 and H-L continuity (where H is the Hardy space) of operators of order −(n− 1)/2. In the present paper, we consider singular Fourier integral operators: the associated canonical relation C is again a smooth Lagrangian submanifold in T ∗X×T ∗Y , but, contrary to the standard case, the projections πL : C→ T ∗X , πR : C→ T ∗Y are allowed to have singularities. The simplest singularities are Whitney folds. The L Sobolev continuity for operators associated to canonical relations with Whitney folds on both sides was derived in [MeTa85]. Such operators “lose a sixth of a derivative”, versus operators associated to (local) symplectomorphisms. L continuity of such operators was derived in [SmSo94] (for some special cases see [PhSt91] and [Se93]). Even though there is a certain loss of smoothness near p = 2 (as we pointed out, it is a sixth Received by the editors September 4, 1998 and, in revised form, June 3, 2001. 2000 Mathematics Subject Classification. Primary 35S30. Both authors were partially supported by grants from the National Science Foundation. c ©2003 American Mathematical Society