\(L^ 2\)-theory of singular perturbation of hyperbolic equations. I: A priori estimates with parameter \(\epsilon\) (Q1118097)

The author considers Cauchy problems of a singularly perturbed linear strictly hyperbolic single equation of order \(m+p\) \((p=1,2)\) with smooth coefficients in \({\mathbb{R}}^{n+1}_{t,x}\), whose reduced equation of order m is also strictly hyperbolic: \(\epsilon^ pLu+Mu=f\) \((0<t<T\), \(x\in {\mathbb{R}}^ n)\) with Cauchy data at \(t=0\). Some \(L^ 2\) estimates with small parameter \(\epsilon\) are given under the separation conditions due to G. B. Whitham and T. T. Wu between characteristic roots of L and M. The derivation of the estimates depends on a pseudo-differential operator version of the classical Leray-Gårding inequality.