On an analogue of the de~Leeuw and Mirkil theorem for operators with variable coefficients (Q2518031)

The authors prove the following Theorem. Let \(\ell\geq2\), \(n\geq 3\), and consider the differential operator \[ P(x,D) = \sum_{| \alpha| \leq \ell} a_\alpha(x)D^\alpha, \quad D=(D_1,\dots,D_n),\quad D_k:=-i{{\partial}\over{\partial x_k}} \] with \(a_\alpha(\cdot)\in L^\infty(\mathbb R^n)\) when \(|\alpha|<\ell-1\) and \(a_\alpha(\cdot)\in C^1(\mathbb R^n)\) for \(|\alpha|=\ell-1\). Assume, moreover, that the principal coefficients of \(P(x,D)\) are constants, that is, \(a_\alpha=\text{const}(\alpha)\) when \(|\alpha| =\ell\). Then the operator \(P(x,D)\) is weakly coercive in the isotropic Sobolev space \(W^\ell_\infty(\mathbb R^n)\), that is, there exist constants \(C_1\) and \(C_2\) such that \[ \sum_{|\alpha|<\ell} \| D^\alpha f\|_{L^\infty(\mathbb R^n)}\leq C_1 \| P(x,D)f\|_{L^\infty(\mathbb R^n)} +C_2 \| f\|_{L^\infty(\mathbb R^n)}\quad\text{for all }f\in C^\infty_0 (\mathbb R^n) \] if and only if it is elliptic.