Sharp estimates for a class of hyperbolic pseudo-differential equations
In this paper we consider the Cauchy problem for a class of hyperbolic pseudodifferential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L p − Lp, 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 R1+n with n ≤ 4 (total dimension ≤ 5), we give a complete list of L p − Lp properties. In particular, this contains the very important case R1+3.
Mathematics Subject Classification (1991)35A20 35S30 58G15 32D20
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