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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.

A Uniform Sobolev Inequality for Ancient Ricci Flows with Bounded Nash Entropy