  Two quantitative notions of mixing are the decay of correlations and the
decay of a mix-norm -- a negative Sobolev norm -- and the intensity of mixing
can be measured by the rates of decay of these quantities. From duality,
correlations are uniformly dominated by a mix-norm; but can they decay
asymptotically faster than the mix-norm? We answer this question by
constructing an observable with correlation that comes arbitrarily close to
achieving the decay rate of the mix-norm. Therefore the mix-norm is the
sharpest rate of decay of correlations in both the uniform sense and the
asymptotic sense. Moreover, there exists an observable with correlation that
decays at the same rate as the mix-norm if and only if the rate of decay of the
mix-norm is achieved by its projection onto low-frequency Fourier modes. In
this case, the function being mixed is called q-recurrent; otherwise it is
q-transient. We use this classification to study several examples and raise
questions for future investigations.
