Fine representation of Hessian of convex functions and Ricci tensor on RCD spaces

Abstract

It is known that on $\mathrm{RCD}$ spaces one can define a distributional Ricci tensor ${\bf Ric}$. Here we give a fine description of this object by showing that it admits the polar decomposition $${\bf Ric}=\omega\,|{\bf Ric}|$$ for a suitable non-negative measure $|{\bf Ric}|$ and unitary tensor field $\omega$. The regularity of both the mass measure and of the polar vector are also described. The representation provided here allows to answer some open problems about the structure of the Ricci tensor in such singular setting. Our discussion also covers the case of Hessians of convex functions and, under suitable assumptions on the base space, of the Sectional curvature operator.