What Hardy's inequality is this?

Question

During a seminar the other day, the lecturer mentioned a Hardy's inequality $|x|^{-2} \le K(1-\Delta)$, where $K$ is a constant and $\Delta$ is the Laplacian in $\mathbb{R}^{3}$. I searched for Hardy's inequality on google, and I found a lot of inequalities under this name, but none of them seems to match the prescription of mine. What is this Hardy's inequality the lecturer was referring to?

Answer 1

Yes, this is a Hardy type inequality. Notice that here it is written as an operator inequality, but you can of course also write it as an integral inequality just by using the definition of positive operator ($A\leq B\iff \forall \varphi, \langle\varphi, A\varphi \rangle \leq \langle\varphi, B\varphi \rangle$). This gives, for any $\varphi : \Bbb R^3\to \Bbb C$, $$ \int \frac{|\varphi(x)|^2}{|x|^2}\,\mathrm dx \leq K \int |\varphi(x)|^2+|\nabla\varphi(x)|^2\,\mathrm dx. $$ Is it closer to what you know? I think the first term on the right-hand side is actually not needed, but not sure (perhaps it depends on the domain?).