Abstract : The "positivity phenomenon" for Bessel sequences, frames and Riesz bases $\left(u_k\right)$ are studied in $L^2$ spaces over the compacts of homogeneous (Coifman-Weiss) type $\Omega=(\Omega, \rho, \mu)$. Under some relations between three basic metric-measure dimensions of $\Omega$, we obtain asymptotics for the mass moving norms $\left\|u_k\right\|_{K R}$ (Kantorovich-Rubinstein), as well as for singular numbers of the Lipschitz and Hajlasz-Sobolev embeddings. Our main observation shows that, quantitatively, the rate of the convergence $\left\|u_k\right\|_{K R} \longrightarrow 0$ depends on an interplay between geometric doubling and measure doubling/halving exponents. The "more homogeneous" is the space, the sharper are the results.

Keywords : Metric-measure spaces; sign interlacing; KR-metrics; Riesz Bases; doubling/halving measures; p-Schatten classes