Research Information System
Communications on Pure and Applied Analysis, vol.18, no.2, pp.869-886, 2019 (SCI-Expanded, Scopus)
In this paper we exhibit some sufficient conditions that imply general weighted Lp 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 ν of the Greiner operator Δκ such that Δκ(a|Δκν|p-2Δκν)≥bνp-1 almost everywhere in R2n+1; then a and b satisfy a weighted Lp Rellich type inequality. Here, p > 1 and Δκ = Σn j=1(x2 j+y2 j) is the sub-elliptic operator generated by the Greiner vector fields xj{equation presented} where (z,l)=(x,y,l)∈ R2n+1=Rn×Rn×R,|Z|={equation presented} 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 Lp Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight Lp Rellich type inequalities associated to the Greiner operator Δκ on smooth bounded domains Ω in R2n+1.