Sharp estimates for a class of hyperbolic pseudo-differential equations
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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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- M. Beals, Lp boundedness of Fourier integrals, Mem. Amer. Math. Soc., 264 (1982).Google Scholar
- L. Hörmander, The analysis of linear partial differential operators. Vols. III–IV, Springer-Verlag, New York, Berlin, 1985.Google Scholar
- D.H. Phong, Regularity of Fourier integral operators, Proc. Int. Congress Math., 862–874 (1994), Zürich, Switzerland.Google Scholar
- M. Ruzhansky, Sharp estimates for a class of hyperbolic differential equations, preprint, 1999.Google Scholar
- M. Ruzhansky, Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, CWI Tracts, to appear.Google Scholar
- M. Ruzhansky, On the failure of the factorization condition for non-degenerate Fourier integral operators, to appear in Proc. Amer. Math. Soc.Google Scholar
- C.D. Sogge, Fourier integrals in classical analysis, Cambridge University Press, 1993.Google Scholar
- F. Treves, Introduction to pseudodifferential and Fourier integral operators, Vol. 2, Plenum Press, 1982.Google Scholar