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In this paper we exhibit some sufficient conditions that imply general weighted \begin{document}$L^{p}$\end{document} 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 \begin{document}$a$\end{document} and \begin{document}$b$\end{document} , if there exists a positive supersolution \begin{document}$\vartheta $\end{document} of the Greiner operator \begin{document}$Δ _{k}$\end{document} such that \begin{document}${\Delta _k}\left( {a|{\Delta _k}\vartheta {|^{p - 2}}{\Delta _k}\vartheta {\rm{ }}} \right) \ge b{\vartheta ^{p - 1}}$ \end{document} almost everywhere in \begin{document}$\mathbb{R}^{2n+1}, $\end{document} then \begin{document}$a$\end{document} and \begin{document}$b$\end{document} satisfy a weighted \begin{document}$L^{p}$\end{document} Rellich type inequality. Here, \begin{document}$p>1$\end{document} and \begin{document}$Δ _{k} = \sum\nolimits_{j = 1}^n {} \left(X_{j}^{2}+Y_{j}^{2}\right) $\end{document} is the sub-elliptic operator generated by the Greiner vector fields \begin{document}${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, $ \end{document} where \begin{document}$\left( z, l\right) = \left( x, y, l\right) ∈\mathbb{R}^{2n+1} = \mathbb{R}^{n}×\mathbb{R}^{n}×\mathbb{R}, $\end{document} \begin{document}$|z{\rm{|}} = \sqrt {\sum\nolimits_{j = 1}^n {} \left( {x_j^2 + y_j^2} \right)} $\end{document} and \begin{document}$k≥ 1$\end{document} . The method we use is quite practical and constructive to obtain both known and new weighted Rellich type inequalities. On the other hand, we also establish a sharp weighted \begin{document}$L^{p}$\end{document} Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight \begin{document}$L^{p}$\end{document} Rellich type inequalities associated to the Greiner operator \begin{document}$Δ _{k}$\end{document} on smooth bounded domains \begin{document}$Ω $\end{document} in \begin{document}$\mathbb{R}^{2n+1}$\end{document} . |
