Microlocal Energy Estimates and Well-Posedness

Abstract

Naturally the structure of the principal symbol p(x, ξ) changes if (x, ξ) varies in the phase space and so does “microlocal” energy estimates. Having proved microlocal energy estimates , the usual next procedure would be to obtain “local” energy estimates by partition of unity. Then one must get rid of the errors caused by the partition of unity. Sometimes it happens that the microlocal energy estimates is too weak to control such errors. In this chapter we propose a new energy estimates for second order operators, much weaker than strictly hyperbolic ones, energy estimates with a gain of H ^κ norm for a small κ > 0. We show that if for every | ξ′ | = 1 one can find P _ξ′ which coincides with P in a small conic neighborhood of (0, 0, ξ′) for which the proposed energy estimates holds then the Cauchy problem for P is locally solvable in C ^∞, which is crucial for our approach to the well-posedness of the Cauchy problem.