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

TL;DR

This paper investigates the Riesz transform on metric cones with potential, establishing endpoint estimates and a sharp reverse inequality that characterizes the boundedness range in Lebesgue spaces.

Contribution

It extends previous work by proving Lorentz endpoint estimates and deriving a precise reverse inequality for the Riesz transform on metric cones with potential.

Findings

Lorentz endpoint estimates for the Riesz transform at both ends of the boundedness range

A sharp reverse inequality relating $H^{1/2}f$ to $ abla f$ and $f/r$

Characterization of the $L^p$ range where the reverse inequality holds

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/>$H=\Delta+V_0/r^2$ 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/>\[<br/>\frac{d}{\min\big((d+4)/2+\mu_0,\,d\big)}<br/> < p <<br/>\frac{d}{\max\big((d-2)/2-\mu_0,\,0\big)}.\]