  In this paper we prove $\textit{localised, weighted}$ curvature integral
estimates for solutions to the Ricci flow in the setting of a smooth four
dimensional Ricci flow or a closed $n$-dimensional K\"ahler Ricci flow. These
integral estimates improve and extend the integral curvature estimates shown by
the second author in an earlier paper. If the scalar curvature is uniformly
bounded in the spatial $L^p$ sense for some $p>2,$ then the estimates imply a
uniform bound on the spatial $L^2$ norm of the Riemannian curvature tensor.
Stronger integral estimates are shown to hold if one further assumes a weak
non-inflating condition. In a sequel paper, we show that in many natural
settings, a non-inflating condition holds.
