Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds

We address some fundamental questions about geometric analysis on Riemannian manifolds. The L^p-Calderón-Zygmund inequality is one of the cornerstones in the regularity theory of elliptic equations, and it has been asked under which geometric conditions it holds for a reasonable class of non-compact Riemannian manifolds, and to what extent assumptions on the derivative of curvature and on the injectivity radius of the manifold are necessary. In the present paper, for 1<p<2, we give a positive answer for the validity of the L^p-Calderón-Zygmund inequality on a Riemannian manifold assuming only a lower bound on the Ricci curvature. It is well known that this alone is not sufficient for p>2. In this case we complement the study of Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-Zygmund inequality under additional Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Bounds in the Kato class are integral conditions and much weaker than pointwise bounds. Throughout the proofs, probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role.