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

Abstract

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