Estimates for hyperbolic equations with non-convex characteristics (Q2365035)

This paper is a continuation of the author's previous one [Math. Z. 215, No. 4, 519-531 (1994; Zbl 0790.35063)] which treated \(L^p\)-\(L^{p'}\)-estimates for the solutions to strictly hyperbolic Cauchy problems with convex characteristics. The subject of it the present paper is to extend the result for those with non-convex characteristics. The solution operators to these problems are expressed as linear combinations of Fourier multipliers of the following type (modulo regularizing operator): \(M=F^{-1}e^{i\varphi(\xi)}a(\xi)F\), where \(\varphi(\xi)\in C^\omega(\mathbb{R}^n\backslash 0)\) is one of the characteristic roots of the problems and \(a(\xi)\in C^\infty(\mathbb{R}^n)\) is a homogeneous function for large \(|\xi|\). In loc. cit. it is shown that the \(L^p\)-\(L^{p'}\)-boundedness of the operator \(M\) is affected by a geometrical property of the function \(\varphi(\xi)\) while the \(L^p\)-boundedness is not, and the affection is determined exactly in the case that \(\varphi(\xi)\) is a convex function. This paper shows that we can remove the convexity assumption in the case \(n=2\) and that we cannot remove it for general \(n\) in the same manner.