A priori estimates for higher order hyperbolic equations (Q1323464)

The solution of the Cauchy problem of a strictly hyperbolic equation \[ P(D_ t, D_ x)u=f, \qquad D_ t^ j u\biggl|_{t=0}=0 \qquad (j=0,1,\dots,m-1) \] can be expressed as \(u(t)= \int_ 0^ t E(t-\tau) f(\tau) d\tau\). The subject of this paper is to give an \(L^ p\)-\(L^{p'}\)-estimate for the operator \(E(t)\). This type of estimate, which is well known for the wave equation, is intricately related to geometrical properties of the characteristics of the operator \(P(D_ t,D_ x)\). The author makes this relation clear on a convexity condition for the characteristics, and succeeds in obtaining an optimal estimate.