Estimates for Hyperbolic Equations of Space Dimension 3

We discuss the problem of boundedness fromLp(Rn) toLp′(Rn) (1/p+1/p′=1, 1⩽p⩽2) of operators of the typeM=F−1eiϕ(ξ)a(ξ) F, which is related to the study of hyperbolic equations with constant coefficients. The boundedness is dependent on a geometrical property ofΣ=ϕ−1(1), and its dependence has been exactly determined in the casesn=2, 1⩽p⩽2 andn⩾3,p=1, 2 (M. Sugimoto,Math. Z.215(1994), 519–531;222(1996), 521–531). This paper is devoted to the unsolved case 1<p<2, and a strange phenomenon is exhibited in the simplest casen=3.