Estimates for Pseudo-differential and Hyperbolic Differential Equations via Fourier Integrals with Complex Phases
In this paper we present LP and LP— Lq estimates for solutions of the Cauchy problem for some classes of pseudo-differential equations. First, we give estimates for operators with simple complex characteristic roots with non-negative imaginary parts. This class contains general strictly hyperbolic equations with variable coefficients. Then, we give sharp estimates for strictly hyperbolic differential operators with time dependent coefficients. The analysis is based on the corresponding LP and LP— Lq properties of Fourier integral operators with complex phase functions, which are also presented.
KeywordsCauchy Problem Phase Function Hyperbolic Equation Fourier Integral Operator Lipschitz Space
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