Results in Math., 41 (2002), 361-368. Sharp estimates for a class of hyperbolic pseudo-differential equations Michael Ruzhansky Abstract In this paper we consider the Cauchy problem for a class of hyperbolic pseudo-differential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L p − L q , Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R 1+n with n ≤ 4 (total dimension ≤ 5), we give a complete list of L p − L q properties. In particular, this contains the very important case R 1+3 . 1 Introduction The solutions of the Cauchy problem for the hyperbolic partial differential and pseudo-differential operators have been under study for a very long time. We will consider the situation when the j -th Cauchy data is in the Sobolev spaces L p m j , and we will study the question for which m j the fixed time solutions of the Cauchy prob- lem belong to L p . Using standard methods one gets estimates for solutions in more general Sobolev spaces L p α , as well as Lipschitz and other function spaces. For p = 2, L 2 estimates correspond to the conservation of energy law and are relatively easy to obtain. However, for p = 2, the problem becomes more subtle even for the wave equation. Some earlier estimates for the wave equation with variable coefficients in L p spaces can be found in [1], [5], [6], [7], [18], [20]. The case of compact Riemannian manifold is treated in [2]. The general approach to the L p estimates for solutions of hyperbolic Cauchy problems is to write them as a sum of time dependent Fourier integral operators applied to Cauchy data. This method is described in [3], [4] for differential operators, and in [24] for operators differential in time, but pseudo-differential in the space variables. In this way the estimates reduce to the corresponding L p estimates for time dependent Fourier integral operators. General regularity properties of F ourier 0 Mathematics Subject Classification (1991): 35A20, 35S30, 58G15, 32D20. 1