This note is a continuation of [ 7 ]. We shall show that an ancient Ricci flow with uniformly bounded Nash entropy also has uniformly bounded |$\nu $| -functional. Consequently, on such an ancient solution, there are uniform logarithmic Sobolev and Sobolev inequalities. We emphasize that the main theorem in this paper is true so long as the theory in [ 3 ] is valid, and in particular, when the underlying manifold is closed.