  By considering a suitable Besov type norm, we obtain refined Sobolev
inequalities on a family of Riemannian manifolds with (possibly exponentially
large) ends. The interest is twofold: on one hand, these inequalities are
stable by multiplication by rapidly oscillating functions, much as the original
ones \cite{GMO}, and on the other hand our Besov space is stable by spectral
localization associated to the Laplace-Beltrami operator (while $ L^p $ spaces,
with $ p \ne 2 $, are in general not preserved by such localizations on
manifolds with exponentially large ends). We also prove an abstract version of
refined Sobolev inequalities for any selfadjoint operator on a measure space
(Proposition \ref{general}).
