On the Schauder inequality for linear partial differential operators in \(\mathbb R^n\) (Q1874906)

Consider the following operator \[ P=\sum_{| \alpha| \leq m} A_\alpha(x) D^\alpha, \] where \(A_\alpha\in C(\mathbb{R}^n, \Hom(X, X))\) are coefficients and \(\Hom(X, X)\) is the space of bounded linear operators from \(X\) to \(X\) equipped with the uniform operator topology. In this paper, the author improves the well known Schauder inequality which was established in the case of second order partial differential operator in a bounded domain to a class of differential operators \(P:C^{2+r}\to C^r (0<r<1)\) in \(\mathbb{R}^n\). More precisely, the author proves that the following Schauder equality \[ | | u| | _{C^{2+r}}\leq \Lambda(| | u| | _C+| | u| | _{C^r}) \] holds in \(\mathbb{R}^n\) if and only if \(P\) is uniformally elliptic for all \(u\in C^{2+r}\). Where \(\Lambda>0\) is a constant.