Abstract: 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.

Key words: Generalized Greiner operator, Hardy-type inequalities, Generalized Picone-type identity, Best constant