On the Riesz transform and its reverse inequality on manifolds with quadratically decaying curvature

We study Riesz and reverse Riesz inequalities on manifolds whose Ricci curvature decays quadratically. First, we refine existing results on the boundedness of the Riesz transform by establishing a Lorentz-type endpoint estimate. Next, we explore the relationship between the Riesz and reverse Riesz transforms, proving that the reverse Riesz, Hardy, and weighted Sobolev inequalities are essentially equivalent. Finally, we apply our methods to Grushin spaces, which exhibit a quadratic decay in 'Ricci curvature', verifying that the reverse inequality holds for all $p\in (1,\infty)$ and that the Riesz transform is bounded on $L^p$ for $p\in (1,n)$. Our approach relies on an asymptotic formula for the Riesz potential combined with an extension of the so-called harmonic annihilation method.