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

Nikolski, N.

doi:10.1090/spmj/1752

Three dimensions of metric-measure spaces, Sobolev embeddings and optimal sign transport
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by N. Nikolski

St. Petersburg Math. J. 34 (2023), 221-245

DOI: https://doi.org/10.1090/spmj/1752

Published electronically: March 22, 2023

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Abstract:

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 $\Omega$, asymptotics is obtained 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. The 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.

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