Sharp estimates for a class of hyperbolic pseudo-differential equations (Q1411551)

Let \(X\) be a real analytic compact manifold and let \[ P=D_t^m+\sum_{j=1}^m P_j(t,D_x)D_t^{m-j},\,\,\,\,t\in {\mathbb R}, x\in X, \] be a strictly hyperbolic operator of order \(m\) with principal symbol real analytic in \(\xi\), where the \(P_j\) are classical pseudodifferential operators of order \(j\) independent of \(x,\) but possibly dependent on \(t.\) In this paper, the author establishes sharp \(L^p-L^q\), Lipschitz and other estimates, for the solution of the Cauchy problem associated with \(P.\) In the case of \({\mathbb R}^{1+n}\), \(n\leq 4,\) obtained by considering compactly supported Cauchy data and localizing the spaces, he gives a complete list of \(L^p-L^q\) properties.