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<br/>$g=d{r}^2+r^{2}h$, 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/>$\Vert H^{1/2}f{\Vert }_{L^p}\le C(\Vert \nabla f{\Vert }_{L^p}+\Vert f/r{\Vert }_{L^p})$<br/>which holds if and only if<br/>\[<br/>\frac{d}{\min\big((d+4)/2+μ_0,\,d\big)}<br/> < p <<br/>\frac{d}{\max\big((d-2)/2-μ_0,\,0\big)}.\]

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