On certain generalized Hardy’s inequalities and applications

Introduction Let P (x1, . . . , xn) be a homogeneous polynomial of degree d in n-real variables belonging to the class H described below. Let V (P ) = {x ∈ R/P (x) = 0} be the algebraic set that it defines. We prove the following inequality c1(P ).P − 2 d ≤ (−∆), c1(P ) > 0 in the operator sense on the domain C 0 (R n \ V (P )). This inequality while it is elementary to prove when the algebraic variety V (P ) is smooth away from the origin, it is rather cumbersome when the variety is singular. The above inequality may be viewed as direct generalization of Hardy’s inequalities