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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 (uk) in L2 spaces over the spaces of homogeneous type Ω=(Ω,ρ,μ) satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of Ω, we obtain asymptotics for the mass moving norms ‖uk‖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 ‖uk‖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.
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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Abstract page: | 240 | Full-text PDF : | 6 | References: | 61 | First page: | 41 |
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