Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order \(2m\ge 4\) have been shown so far only when the principal part of L is the polyharmonic operator \((-\Delta )^m\). In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.

从上面的估计和下面由相同的正原型函数在用于椭圆算狄利克雷边界条件下有界光滑域适当地修改格林函数大号高阶\（2M \ GE 4 \）已经显示出到目前为止只有当的主要部分L是多调和算子\（（-\ Delta）^ m \）。在本说明中，表明了当拉普拉斯算子被具有平滑可变系数的发散形式的任何二阶均匀椭圆算子代替拉普拉斯算子时，这种结果仍然成立。对于一般的高阶椭圆算子，其主要部分不能写成二阶算子的幂，最近证明了这种结果通常是错误的。