Generalized Levi Conditions for Weakly Hyperbolic Equations - An Attempt to Treat the Degeneracy with Respect to the Space Varlables

In the celebrated paper [3], Ivrii and Petkov showed that the effective hyperbolicity of an operator is necessary for the Cauchy problem to be C°°-wellposed with arbitrary lower order terms. In the same paper, they also gave general necessary conditions on lower order terms for the C°°-well-posedness. (Conditions of this kind are sometimes called "generalized Levi conditions".) The individual behavior of each characteristic root, however, is not reflected in these conditions. Hence, when some characteristic roots coincide with one another in a variety of ways, their results fail to give the conditions expected to be necessary and sufficient. In [8], we treated each characteristic root separately, restricting our consideration to the degeneracy with respect to the time variable. We gave some necessary conditions for C°°-well-posedness and some estimates for the regularityloss of solutions. These results were " micro-local" in the sense that conditions were stated in terms of the characteristic root 0 in a fixed cotangential direction. By applying these results after appropriate coordinate transformations, however, we got a necessary and sufficient condition in case of the operators with only finite-order degeneracy w. r. t. the time variable ([9]). This paper is an attempt to treat the degeneracy with respect to the space variables in a similar way. Though the results in this paper hold for the operators with singular coefficients of the same kind as discussed in [8] and [9], we restrict ourselves to the case of operators with C°°-coefficients for simplicity.