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

Abstract

A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases ( u k ) (u_{k}) in L 2 L^2 spaces over the spaces of homogeneous type Ω = ( Ω , ρ , μ ) \Omega =(\Omega ,\rho ,\mu ) satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of Ω \Omega , asymptotics is obtained for the mass moving norms ‖ u k ‖ K R \|u_k\|_{KR} in the sense of Kantorovich–Rubinstein, as well as for the singular numbers of the Lipschitz and Hajlasz–Sobolev embeddings. The main observation shows that, quantitatively, the rate of convergence ‖ u k ‖ K R → 0 \|u_k\|_{KR}\to 0 mostly depends on the Bernstein–Kolmogorov n 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.