In this well-written paper, the authors study operators of the form $L=-\\Delta -µd^{-2}$, where $d(x)={\\rm dist}(x,\\Sigma)$, $µ\\in R$ and $\\Sigma \\subset R^{n}$. More precisely, they study inequalities which suggest that the operator $L$ has a positive first eigenvalue. Such inequalities, with many variants of the parameters, have already been studied in the literature. However, the authors establish them under fewer assumptions. One of their main results reads as: Let $\\Omega\\subset R^{n}$ be an open bounded set and $\\Sigma \\subset \\Omega $ be a compact smooth manifold without boundary of codimension $k\\neq 2$. Let $H=(k-2)/2$. Then there exist $C>0$ and $\\gamma >0$ independent of $u$ such that for any $u\\in C_{c}^{\\infty }(\\Omega \\sbs \\Sigma)$, $$ \\gamma \\left( \\int_{\\Omega }|u|^{p}\\right) ^{2/p}+H^{2}\\int_{\\Omega }\\frac{u^{2}}{d^{2}}\\leq \\int_{\\Omega }\\left| \\nabla u\\right| ^{2}+C\\int_{\\Omega}u^{2},$$ where $d(x)={\\rm dist}(x,\\Sigma)$, $1\\leq p2$$ and $$\\frac{1}{p_{1}}=\\frac{1}{2}-\\frac{1}{n+1}\\quad \\text{ if }\\quad k=1. $$ As an application of this result, it is shown that the quantity $$J_{\\lambda }\\coloneq \\inf_{u\\neq 0}\\frac{\\int_{\\Omega }\\left|\\nabla u\\right| ^{2}-\\lambda \\int_{\\Omega }u^{2}}{\\int_{\\Omega }u^{2}/d^{2}}$$ is achieved if and only if $J_{\\lambda }-\\infty,$$ and $$\\gamma \\left( \\int_{\\Omega }\\left| u\\right| ^{r}\\right) ^{2/r}+\\int_{\\Omega }a(x)u^{2}\\leq \\int_{\\Omega }\\left| \\nabla u\\right| ^{2}+M\\int_{\\Omega }u^{2},$$ for some $r>2$, $\\gamma >0$, and $M>0$.