On the Reverse Inequality of Riesz transform on metric cone with potential
Dangyang He
TL;DR
This paper analyzes the Riesz transform associated with a Schrödinger operator $H=\Delta+\frac{V_0}{r^2}$ on a metric cone $M=(0,\infty)_r\times Y$ under the positivity condition $\Delta_Y+V_0+(d-2)^2/4>0$. It first establishes Lorentz-type endpoint estimates: the Riesz transform $R=\nabla H^{-1/2}$ is of restricted weak type at the endpoints of its $L^p$-range, with explicit ranges in terms of $d$ and the spectral parameter $\mu_0$. The main focus is the sharp reverse inequality $\|H^{1/2}f\|_{L^p} \le C(\|\nabla f\|_{L^p}+\|f/r\|_{L^p})$, proven to hold if and only if $p$ lies in a concrete interval determined by $d$ and $\mu_0$; the endpoints are sharp in the Lorentz scale. The authors develop two harmonic-annihilation approaches—exterior and interior—to prove the reverse inequality and provide endpoint estimates, while also showing that the thresholds cannot be improved via carefully constructed counterexamples. A duality analysis links forward and reverse estimates, clarifying the precise $p$-range dictated by the cone geometry and the inverse-square potential. These results sharpen the understanding of Riesz transforms on singular geometric settings and highlight the delicate balance between geometry, potential, and functional-analytic bounds.
Abstract
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/>$H=Δ+V_0/r^2$ be the associated Schrodinger operator, with<br/>$V_0\in C^\infty(Y)$ satisfying the positivity condition<br/>$Δ_Y+V_0+(d-2)^2/4>0$. 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/>\[<br/>\frac{d}{\min\big((d+4)/2+μ_0,\,d\big)}<br/> < p <<br/>\frac{d}{\max\big((d-2)/2-μ_0,\,0\big)}.\]