Three dimensions of metric-measure spaces, Sobolev embeddings and optimal sign transport

A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases $ (u_{k})$ in $ L^{2}$ spaces over the spaces of homogeneous type $ \Omega =(\Omega, \rho, \mu )$ satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of $ \Omega $, we obtain asymptotics for the mass moving norms $ \| u_{k}\| _{KR}$ in the sense of Kantorovich–Rubinstein, as well as for the singular numbers of the Lipschitz and Hajlasz–Sobolev embeddings. Our main observation shows that, quantitatively, the rate of convergence $ \| u_{k}\| _{KR}\to 0$ mostly depends on the Bernstein–Kolmogorov $n$-widths of a certain compact set of Lipschitz functions, and the widths themselves mostly depend on the interplay between geometric doubling and measure doubling/halving numerical parameters. The “more homogeneous” is the space, the sharper are the results.