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Research Papers
Three dimensions of metric-measure spaces, Sobolev embeddings and optimal sign transport
N. Nikolski Institut de Mathématiques de Bordeaux, France
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 $, 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.
Keywords:
sign interlacing, Kantorovich–Rubinstein (Wasserstein) metrics, Riesz bases, frames, Bessel sequences, geometric doubling condition, measure halving and doubling conditions, $ p$-Schatten classes, dyadic cubes, Haar-like functions, Hajlasz–Sobolev spaces, Hadamard matrix.
Received: 14.12.2021
Citation:
N. Nikolski, “Three dimensions of metric-measure spaces, Sobolev embeddings and optimal sign transport”, Algebra i Analiz, 34:2 (2022), 118–151; St. Petersburg Math. J., 34:2 (2023), 221–245
Linking options:
https://www.mathnet.ru/eng/aa1803 https://www.mathnet.ru/eng/aa/v34/i2/p118
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