A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator

In this paper we exhibit some sufficient conditions that imply general weighted $L^{p}$ Rellich type inequality related to Greiner operator without assuming a priori symmetric hypotheses on the weights. More precisely, we prove that given two nonnegative functions $a$ and $b$, if there exists a positive supersolution $\vartheta $ of the Greiner operator $Δ _{k}$ such that

${\Delta _k}\left( {a|{\Delta _k}\vartheta {|^{p - 2}}{\Delta _k}\vartheta {\rm{ }}} \right) \ge b{\vartheta ^{p - 1}}$

almost everywhere in $\mathbb{R}^{2n+1}, $ then $a$ and $b$ satisfy a weighted $L^{p}$ Rellich type inequality. Here, $p>1$ and $Δ _{k} = \sum\nolimits_{j = 1}^n {} \left(X_{j}^{2}+Y_{j}^{2}\right) $ is the sub-elliptic operator generated by the Greiner vector fields

${X_j} = \frac{\partial }{{\partial {x_j}}} + 2k{y_j}|z{{\rm{|}}^{2k - 2}}\frac{\partial }{{\partial l}}, \;\;\;\;{Y_j} = \frac{\partial }{{\partial {y_j}}} - 2k{x_j}|z{|^{2k - 2}}\frac{\partial }{{\partial l}}, \;\;\;\;j = 1, ..., n, $

where $\left( z, l\right) = \left( x, y, l\right) ∈\mathbb{R}^{2n+1} = \mathbb{R}^{n}×\mathbb{R}^{n}×\mathbb{R}, $ $|z{\rm{|}} = \sqrt {\sum\nolimits_{j = 1}^n {} \left( {x_j^2 + y_j^2} \right)} $ and $k≥ 1$. The method we use is quite practical and constructive to obtain both known and new weighted Rellich type inequalities. On the other hand, we also establish a sharp weighted $L^{p}$ Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight $L^{p}$ Rellich type inequalities associated to the Greiner operator $Δ _{k}$ on smooth bounded domains $Ω $ in $\mathbb{R}^{2n+1}$.