  In this paper we investigate the validity of first and second order $L^{p}$
estimates for the solutions of the Poisson equation depending on the geometry
of the underlying manifold. We first present $L^{p}$ estimates of the gradient
under the assumption that the Ricci tensor is lower bounded in a local integral
sense and construct the first counterexample showing that they are false, in
general, without curvature restrictions. Next, we obtain $L^p$ estimates for
the second order Riesz transform (or, equivalently, the validity of $L^{p}$
Calder\'on--Zygmund inequalities) on the whole scale $1<p<+\infty$ by assuming
that the injectivity radius is positive and that the Ricci tensor is either
pointwise lower bounded or non-negative in a global integral sense. When $1<p
\leq 2$, analogous $L^p$ bounds on even higher order Riesz transforms are
obtained provided that also the derivatives of Ricci are controlled up to a
suitable order. In the same range of values of $p$, for manifolds with lower
Ricci bounds and positive bottom of the spectrum, we show that the $L^{p}$ norm
of the Laplacian controls the whole $W^{2,p}$-norm on compactly supported
functions.
