On mix-norms and the rate of decay of correlations

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