  We address some fundamental questions concerning geometric analysis on
Riemannian manifolds. It has been asked whether the $L^p$-Calder\'{o}n-Zygmund
inequalities extend to a reasonable class of non-compact Riemannian manifolds
without the assumption of a positive injectivity radius. In the present paper,
we give a positive answer for $1<p<2$ under the natural assumption of a lower
bound on the Ricci curvature. For $p>2$, we complement the study in
G\"{u}neysu-Pigola (2015) and derive sufficient geometric criteria for the
validity of the Calder\'{o}n-Zygmund inequality by adding Kato class bounds on
the Riemann curvature tensor and the covariant derivative of Ricci curvature.
Probabilistic tools, like Hessian formulas and Bismut type representations for
heat semigroups, play a significant role throughout the proofs.
