广义Greiner算子的几类Hardy型不等式

对构成广义Greiner算子的向量场$X_j = \frac{\partial }{\partial x_j} + 2ky_j \vert z\vert ^{2k - 2}\frac{\partial }{\partialt}$, $Y_j = \frac{\partial }{\partial y_j } - 2kx_j \vert z\vert^{2k - 2}\frac{\partial }{\partial t}$, j = 1,... ,n, x,y∈ Rⁿ, $z = x + \sqrt { - 1} \,y$, t ∈ R, k ≥1, 得到了拟球域内和拟球域外的Hardy型不等式;建立了广义Picone型恒等式,并由此导出比文献3]更一般的全空间上的Hardy型不等式;并在$p = 2$时建立了具最佳常数的Hardy型不等式.

Several Hardy Type Inequalities of Generalized Greiner Operator

The vector fields $X_j = \frac{\partial}{\partial x_j } + 2ky_j \vert z\vert ^{2k - 2}\frac{\partial}{\partial t}$, $Y_j = \frac{\partial }{\partial y_j } - 2kx_j\vert z\vert ^{2k - 2}\frac{\partial }{\partial t}$, j = 1,...,n, x,y ∈ Rⁿ, $z = x + \sqrt { - 1} y$, t ∈ R, k ≥ 1 are considered. Hardy type inequalities in the pseudo ball and outside the pseudo ball are obtained. The generalized Picone type identity and then Hardy type inequalities on the whole space containing the known results in 3] are established. When p = 2 the sharp constant in the Hardy type inequality is discussed.