On the Reverse Inequality of Riesz transform on metric cone with potential

Let $M=(0,\infty)_r\times Y$ be a $d$-dimensional ($d\ge 3$) metric cone with metric<br/>$g=dr^2+r^2h$, where $(Y,h)$ is a closed Riemannian manifold. Let<br/>$H=\Delta+V_0/r^2$ be the associated Schrodinger operator, with<br/>$V_0\in C^\infty(Y)$ satisfying the positivity condition<br/>$\Delta_Y+V_0+(d-2)^2/4>0$. First, we complement previous results by proving<br/>Lorentz-type endpoint estimates for the Riesz transform $\nabla H^{-1/2}$:<br/>it is of restricted weak type at both endpoints of its $L^p$-boundedness range.<br/>Second, we establish the sharp reverse inequality<br/>$\|H^{1/2}f\|_{L^p}\le C\big(\|\nabla f\|_{L^p}+\|f/r\|_{L^p}\big)$<br/>which holds if and only if<br/>$$<br/>\frac{d}{\min\big((d+4)/2+\mu_0,\,d\big)}<br/> < p <<br/>\frac{d}{\max\big((d-2)/2-\mu_0,\,0\big)}.$$