The main goal of this note is to improve the Rellich inequalities
\[ \int_{\mathbb R^n} {{| \Delta^m u| ^p }\over{| x| ^\gamma}}\,dx \geq A(2m, \gamma) \int_{\mathbb R^n} {{| u| ^p }\over{| x| ^{\gamma + ( 2mp ) }}}\,dx \tag{1} \]
and
\[ \int_{\mathbb R^n} {{| \nabla^m u| ^p}\over{| x| ^\gamma}}\,dx \geq A(2m, \gamma) \int_{\mathbb R^n} {{| u| ^p}\over{| x| ^{\gamma + (2m+1)p }}}\,dx \tag{2} \]
by adding a sharp non-negative term to the respective right-hand sides.
Precisely, let \(M\) be a Riemannian manifold of dimension \(N \geq 2, \) a domain \(\Omega \subset M,\) a closed, piecewise smooth surface \({\mathcal K}\) of codimension \(k, \) \(1\leq k \leq N\), and the distance function \(d(x)= \text{dist} (x,{\mathcal K}) \), which is assumed to be bounded in \(\Omega.\) The author establishes a Rellich inequality of the form
\[ \int_{\Omega} {{| \Delta^{{m}\over{2} } u| ^p \,}\over{d^\gamma}}\,dx \geq A(m, \gamma) \int_{\Omega} {{| u| ^p \,}\over{d^{\gamma + 2mp }}}\,dx +B(m, \gamma) \sum_{i=1}^{\infty}\int_{\Omega} V_i | u| ^p \,dx \tag{3} \]
for all \(u \in C^\infty_0 (\Omega\!-\!{\mathcal K}),\,\) where at each step he considers an optimal function \(V_i(x) \) and a sharp constant \(B(m,\gamma).\) We point out that the quantity \(| \Delta u| ^{{m}\over{2}}\) is considered as \(| \nabla \Delta u | ^{{{(m-1)}\over{2} }} \) when \(m \) is odd and observe that in \((3) \) instead of \(| x| \) the author puts the distance function \(d(x)= \text{dist} (x,{\mathcal K}).\)
In a technical theorem, the optimality of the constants and exponents is studied.