THREE DIMENSIONS OF METRIC-MEASURE SPACES, SOBOLEV EMBEDDINGS AND OPTIMAL SIGN TRANSPORT

A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases (uk) in L2 spaces over the spaces of homogeneous type S2 = (S2, ⠂, mu) satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of S2, asymptotics is obtained for the mass moving norms IIukiiKR in the sense of Kantorovich-Rubinstein, as well as for the singular num-bers of the Lipschitz and Hajlasz-Sobolev embeddings. The main observation shows that, quantitatively, the rate of convergence Iluk �KR-+ 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.