Approximation numbers of critical Besov embeddings
Abstract
We determine the approximation numbers of critical embeddings between isotropic Besov spaces on nonempty bounded Lipschitz domains in any dimension. For 0 < p_1 < 2 < p_2 <= infinity and s_1 - s_2 = d max{1/p_1, 1 - 1/p_2} > d(1/p_1 - 1/p_2), the approximation numbers have order N^{-1/2}(log(e+N))^Gamma for every 0 < q_1, q_2 <= infinity, with an explicit four-case formula for Gamma. This resolves the strictly compact critical Besov approximation-number problem identified in Vybiral's overview. For q_1 < q_2, the logarithmic exponent depends on the ordering of p_1' and p_2; it can vanish for one pair (p_1,p_2) and remain positive for another with the same fine indices. A row estimate for the approximation error gives lower bounds when the target fine index is below one. Gaussian finite-rank approximations acting on several levels give the sharp upper bounds when Gamma < 1/2. For q_1 < q_2, matching lower bounds for the two strict orderings of the inner indices follow from Malykhin's finite-window estimates and norm comparisons. The results include quasi-Banach indices and the full spaces of bounded coefficient sequences at infinite fine indices.
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BibTeX
@misc{guo2026approximation,
title = {{Approximation numbers of critical Besov embeddings}},
author = {Feng Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25910},
eprint = {2610.25910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25910},
}
RIS
TY - UNPB
TI - Approximation numbers of critical Besov embeddings
AU - Feng Guo
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25910
UR - https://arxiv.org/abs/2610.25910
AB - We determine the approximation numbers of critical embeddings between isotropic Besov spaces on nonempty bounded Lipschitz domains in any dimension. For 0 < p_1 < 2 < p_2 <= infinity and s_1 - s_2 = d max{1/p_1, 1 - 1/p_2} > d(1/p_1 - 1/p_2), the approximation numbers have order N^{-1/2}(log(e+N))^Gamma for every 0 < q_1, q_2 <= infinity, with an explicit four-case formula for Gamma. This resolves the strictly compact critical Besov approximation-number problem identified in Vybiral's overview. For q_1 < q_2, the logarithmic exponent depends on the ordering of p_1' and p_2; it can vanish for one pair (p_1,p_2) and remain positive for another with the same fine indices. A row estimate for the approximation error gives lower bounds when the target fine index is below one. Gaussian finite-rank approximations acting on several levels give the sharp upper bounds when Gamma < 1/2. For q_1 < q_2, matching lower bounds for the two strict orderings of the inner indices follow from Malykhin's finite-window estimates and norm comparisons. The results include quasi-Banach indices and the full spaces of bounded coefficient sequences at infinite fine indices.
ER -