  We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schr\"odinger operators
$P=\Delta+V$ with short range potential $V$ on asymptotically conic manifolds
$(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$.
We make the assumption that the dimension is greater or equal to 3 and that $P$
has no $L^2$ null space and no resonance at 0. In particular, we show that the
Schwartz kernel of $R(k)$ is a conormal polyhomogeneous distribution on a
desingularized version of $M\times M\times [0,1]$. Using this, we show that the
Riesz transform of $P$ is bounded on $L^p$ for $1<p<n$ and that this range is
optimal if $V$ is not identically zero or if $M$ has more than one end. We also
analyze the case V=0 with one end. In a follow-up paper, we shall deal with the
same problem in the presence of zero modes and zero-resonances.
