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 $d\Delta ^{-\frac{1}{2}}$ 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 $\alpha \in [0,1)$, then the Riesz transform $d^*\Delta^{-\frac{1}{2}}$ on differential $1$-forms is bounded from the associated Hardy space $H^p_{\overrightarrow{\Delta}}(\Lambda^1T^*M)$ to $L^p(M)$ for all $p\in [1,2]$. 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 $$\int_0^1 \left\|\frac{|R^-|^{\frac{1}{2}}}{v(\cdot,\ \sqrt{t})^{\frac{1}{p_1}}}\right\|_{p_1}\frac{dt}{\sqrt{t}}+\int_1^\infty \left\|\frac{|R^-|^{\frac{1}{2}}}{v(\cdot,\ \sqrt{t})^{\frac{1}{p_2}}}\right\|_{p_2}\frac{dt}{\sqrt{t}}<\infty,$$ for some $p_1>2$ and $p_2>3$, then the Riesz transform $d\Delta^{-\frac{1}{2}}$ is bounded on $L^p$ for all $1<p<p_2$. In the particular case where $v(x, r) \ge C r^D$ for all $r \ge 1$ and $|R^-| \in L^{D/2 -\eta} \cap L^{D/2 + \eta}$ for some $\eta > 0$, then $d\Delta^{-\frac{1}{2}}$ is bounded on $L^p$ for all $1<p< D.$ Furthermore, we study the boundedness of the Riesz transform of Schr\"odinger operators $A=\Delta+V$ 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 $dA^{-\frac{1}{2}}$ on $L^p$