Approximation order of Kolmogorov diameters via $L^{q}$-spectra and applications to polyharmonic operators

TL;DR

This work links the $L^{q}$-spectrum of a measure to the approximation order of Kolmogorov diameters for Sobolev spheres, providing a unified framework that translates mass distribution into quantitative rates of adaptive approximation. By introducing $s_{\varrho}$ through the $L^{q}$-spectrum and exploiting adaptive dyadic partitions together with piecewise polynomial approximations, the authors derive a sharp bound $\overline{\mathbf{ord}}(\mathscr{S}W_p^{\ell},L_{\nu}^{q})\le -1/(q s_{\varrho})$, recovering classical rates when the spectrum matches the homogeneous bound and improving them for many singular measures. The paper then applies this theory to polyharmonic operators, showing that eigenvalue decay is governed by the same spectrum via $\sqrt{\lambda_{n+1}^{\nu}}=d_n(\mathscr{S}H_0^{\ell},L_{\nu}^{2})$ and obtaining enhanced decay bounds; exact orders are established for measures with AC parts and for self-similar measures under OSC. In the one-dimensional Krein–Feller setting ($\ell=m=1$, $p=q=2$), a constructive mapping demonstrates spectral equivalence with polyharmonics, yielding a clean sub-/superadditivity framework for eigenvalue counts. Overall, the results provide a robust bridge between fractal-type measure complexity and both approximation theory and spectral asymptotics, with implications for adaptive algorithms and stochastic/quantitative spectral analysis.

Abstract

We establish a connection between the $L^{q}$-spectrum of a Borel measure $ν$ on the $m$-dimensional unit cube and the approximation order of Kolmogorov diameters of the unit sphere with respect to Sobolev norms in $L_{ν}^{p}$. This leads to improvements of classical results of Borzov and Birman/Solomjak for a broad class of singular measures. As an application, we consider spectral asymptotics of polyharmonic operators and obtain improved upper bounds of the decay rate of their eigenvalues. For measures with non-trivial absolutely continuous parts as well as for self-similar measures the exact approximation orders are stated.

Figures (1)

• Figure 1.1: For $m=3$ the solid line illustrates the $L^{q}$-spectrum $\beta _{\nu}$ for the self-similar measure $\nu$ supported on the Sierpiński tetraeder with all four contraction ratios equal $1/2$ and with probability vector $\left (0.659,0.28,0.001,0.06\right )$; $\beta _{\nu}\left (0\right )=\overline{\dim}_{M}\left (\nu \right )=2$. For $\varrho =2$ (slope of the dashed line) the intersection of the spectrum and the dashed line determines $s_{\varrho }$. The dotted line $s\mapsto 3\left (1-s\right )$, which coincides with the graph of $\beta _{\Lambda |_{\textbf{Q}}}$, intersects the dashed line in $m/\left (m+\varrho \right )$ giving the upper bound for $s_{\varrho }$.

Theorems & Definitions (40)

• Theorem 1.1
• Remark 1.2
• Remark 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• Remark 2.5
• proof