The local Nash inequality is introduced as a natural extension of the classical Nash inequality yielding a space-homogeneous upper heat kernel estimate. The local Nash inequality contains local information of the heat kernel and is a necessary condition for the space-inhomogeneous heat kernel estimate involving the volume of balls like the one obtained by Li and Yau for a complete Riemannian manifold with non-negative Ricci curvature. Under the volume doubling property, the local Nash inequality combined with the exit time estimate is shown to be equivalent to a sub-Gaussian off-diagonal upper estimate of the heat kernel allowing space-inhomogeneity.
