On The Vector Field Of The Hardy Inequality, The Rellich Inequality And The Only Extension,

In Chapter one, we prove some Hardy type inequalities related to a class of degenerate elliptic operators with double- weight. This class of operators is more extensive than the generalized Baouendi- Grushin operators. Thus the known results on generalized Baouendi-Grushin operators become particular cases here.In Chapter two, we prove some Rellich type inequalities related to the generalized Baouendi-Grushin operators and the generalized Greiner operators with Picone type identitie. After that ,we established a type of Rellich inequality using the method developed by D'Ambrosio ,these inequalities are generalizations of the classical inequality, obtained earlier in the Euclidean setting by Rellich. Next, a type of Rellich inequality with remainder terms are established by the method of D'Ambrosio.In the last Chaper, unique continuation propeties for generalized Greiner operators are established by studying the solutions of the sub-elliptic equation-Lu+b|-▽u+Vu=0...