Riesz and reverse Riesz 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. We then explore the relationship between the Riesz and reverse Riesz transforms, proving that the reverse Riesz, Hardy, and weighted Sobolev inequalities are essentially equivalent. Our approach relies on an asymptotic formula for the Riesz potential, combined with an extension of the so-called harmonic annihilation method.