Remarks on hyperbolic systems fo first order with constant coefficient characteristic polynomials (Q2471360)

The author is interested in the Cauchy problem for hyperbolic systems of first-order, where the characteristic polynomials have constant coefficients. For \(t> 0\) he considers the Cauchy problem \[ L(x,D)u= f(x),\quad \text{supp\,}u,\;\text{supp\,}f\subset\{x\in \mathbb{R}^n: x_1\geq t\}\quad\text{in }\mathbb{R}^n \] in \(C^\infty\) or \(D'\), where \[ L(x,\xi)= L_1(x,\xi)+ L_0(x),\quad L_1(x,\xi)= \sum^n_{j=1} \xi_j A_j(x). \] The following two assumptions are of importance: Hyperbolicity assumption: \(A_1(x)= I_m\), \(\text{det\,}L_1(x, \xi)\) is independent of \(x\) and \(p(\xi):= \text{det\,}L_1(x,\xi)\) is hyperbolic with respect to the direction \(\vartheta= (1,0,\dots, 0)\). Maximal rank condition: rank \(L_1(x,\xi)= m-1\) for any \((x,\xi)\in\mathbb{R}^n\times S^{n-1}\) with \(dp(\xi)= 0\). Then the author proves that under these assumptions the condition \[ |Q(x, \xi; x^0 \xi^{0\prime})|\leq C|p(\xi- i\vartheta)| \] for \(x\) from a neighborhood of \(x_0\) and \(\xi\) from a conic neighborhood of \(\xi^{0\prime}\) is necessary and sufficient for the \(C^\infty\) well-posedness. The matrix \(Q\) is related to a suitable microlocal block-diagonalization procedure.