Some estimates in \(\mathbb R^n\) for linear partial differential operators (Q5942108)

The paper concerns abstract linear partial differential operators \[ P= \sum_{|\alpha|\leq m} A_\alpha(x) D^\alpha, \] where \(A_\alpha(x)\) are linear bounded operators on a Banach space \(X\) with continuous dependence on \(x\in\mathbb R^n\). Results are given, connecting the ellipticity of \(P\) with standard a priori estimates of the form \[ \|u\|_{W^m(F)}\leq C_1\|u\|_F+ C_2\|Pu\|_F, \] relevant examples for \(F\) being the spaces \(L^p(\mathbb R^n, X)\), with \(W^m(F)\) the corresponding Sobolev spaces.