Riesz transforms on non-compact manifolds

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform $d{\mathrm{\Delta }}^{-\frac{1}{2}}$ on both Hardy spaces $H^p$ and Lebesgue spaces $L^p$ under two different conditions on the negative part of the Ricci curvature $R^{-}$. First we prove that if $R^{-}$ is $\alpha$-subcritical for some $\alpha \in [0,1)$,thentheRiesztransform$d^*\Delta^{-\frac{1}{2}}$ondifferential$1$-formsisboundedfromtheassociatedHardyspace$H^p_{\overrightarrow{\Delta}}(\Lambda^1T^*M)$to$L^p(M)$forall$p\in [1,2]$. As a consequence, the Riesz transform (on functions) is bounded on $L^p$ for all $p\in (1,p_0)$ where $p_0>2$ depends on $\alpha$ and the constant appearing in the doubling property. Second, we prove that if $${\int }_0^1{\parallel \frac{\mathrm{\mid }{R}^{-}{\mathrm{\mid }}^{\frac{1}{2}}}{v(\cdot ,\sqrt{t}{)}^{\frac{1}{p_1}}}\parallel }_{p_1}\frac{dt}{\sqrt{t}}+{\int }_1^{\mathrm{\infty }}{\parallel \frac{\mathrm{\mid }{R}^{-}{\mathrm{\mid }}^{\frac{1}{2}}}{v(\cdot ,\sqrt{t}{)}^{\frac{1}{p_2}}}\parallel }_{p_2}\frac{dt}{\sqrt{t}}<\mathrm{\infty },$$ for some $p_1>2$ and $p_2>3$, then the Riesz transform $d{\mathrm{\Delta }}^{-\frac{1}{2}}$ is bounded on $L^p$ for all $1<p<p_2$. In the particular case where $v(x, r) \ge C r^D$forall$r \ge 1$and$|R^-| \in L^{D/2 -\eta} \cap L^{D/2 + \eta}$ for some $\eta >0$, then $d{\mathrm{\Delta }}^{-\frac{1}{2}}$ is bounded on $L^p$ for all $1<p<D\mathrm{.}$ Furthermore, we study the boundedness of the Riesz transform of Schr"odinger operators $A=\mathrm{\Delta }+V$ on $L^p$ for $p>2$ under conditions on $R^{-}$ and the potential $V$. We prove both positive and negative results on the boundedness of $d{A}^{-\frac{1}{2}}$ on $L^p$