Factorizations and Hardy-Rellich-type inequalities

The principal aim of this note is to illustrate how factorizations of singular, even order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, introducing the two-parameter n-dimensional homogeneous scalar differential expressions T-alpha,T-beta := -Delta+alpha vertical bar x vertical bar(-2)x.del+vertical bar x vertical bar(-2), alpha, beta is an element of R, x is an element of R-n \ {0}, n is an element of N, n >= 2, and its formal adjoint, denoted by T-alpha,beta(+), we show that nonnegativity of T alpha,beta+T alpha,beta on C-0(infinity)(R-n \ {0}) implies the fundamental inequality integral(Rn)[(Delta f)(x)](2) d(n) x >= [(n - 4)alpha - 2 beta] integral(Rn) vertical bar x vertical bar(-2)vertical bar(del f)(x)vertical bar(2) d(n)x - alpha(alpha - 4) integral(Rn) vertical bar x vertical bar(-4)vertical bar x .(del f)(x)vertical bar(2) d(n)x (*) + beta[(n - 4)(alpha - 2) - beta] integral(Rn) vertical bar x vertical bar(-4)vertical bar (x)vertical bar(2) d(n)x, f is an element of C-0(infinity)(R-n \ {0}). A particular choice of values for alpha and beta in (*) yields known Hardy-Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where R-n is replaced by an arbitrary open set Omega subset of R-n for functions f is an element of C-0(infinity)(Omega \ {0}). Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.