\(L^ p\)-\(L^{p'}\)-estimates for hyperbolic equations (Q1902212)

Let \(P= P(D_t, D_x)\) be a homogeneous, constant-coefficient partial differential operator of degree \(m\) in \(D_t, D_{x_1},\dots, D_{x_n}\) which is strictly hyperbolic, and whose symbol is \(p(\tau, \xi)= (\tau- \varphi_1(\xi)),\dots, \tau- \varphi_m(\xi))\), where \(\varphi_1(\xi)>\cdots> \varphi_m(\xi)\) for \(\xi\neq 0\). Consider the following Cauchy problem: \(Pu= 0\), \(D^k_t u(0, x)= g_k(x)\) for \(k= 0, 1,\dots, m- 1\). In this expository article (based on three previous papers) the author expresses the solution of Cauchy problem as \(u(t)= \sum^{m- 1}_{k= 0} E_k(t) g_k\), and gives \(L^p\)-\(L^{p'}\)-estimates for \(E_k(t)\), where \(1\leq p\leq 2\), \(1/p+ 1/p'= 1\).