Relative volume comparison theorem under Kato type conditions

Let $(M,g)$ be an $n$ $(\ge 3)$ dimensional, non-collapsed compact Riemannian manifold and ${\mathrm{Ric}}^{-}$ be the negative part of the Ricci curvature and $\beta \in (\frac{2n}{n+2},2)$. We prove a relative volume comparison theorem when $\mathrm{\mid }{\mathrm{Ric}}^{-}{\mathrm{\mid }}^{\beta }$ is in the Kato class (cf. Definition 1.1), which results from a new integral Laplace comparison theorem in the spirit of \cite{PW} for a suitable conformal metric. This partly addresses an expectation in \cite{TZZZZ}, where the same result was proven when $\mathrm{\mid }{\mathrm{Ric}}^{-}{\mathrm{\mid }}^2$ is in the class.