  On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of
the integral kernel of the resolvent $R_q(k):=(\Delta_q+k^2)^{-1}$ of the Hodge
Laplacian $\Delta_q$ on $q$-forms as the spectral parameter $k$ approaches
zero, assuming that 0 is not a resonance. The first application we give is an
$L^p$ Sobolev estimate for $d+\delta$ and $\Delta_q$. Then we obtain a complete
characterization of the range of $p>1$ for which the Riesz transform for
$q$-forms $T_q=(d+\delta)\Delta_q^{-1/2}$ is bounded on $L^p$. Finally, we
obtain an asymptotic formula for the analytic torsion of a family of smooth
compact Riemannian manifolds $(\Omega_\epsilon,g_\epsilon)$ degenerating to a
compact manifold $(\Omega_0,g_0)$ with a conic singularity as $\epsilon\to 0$.
