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

Abstract

We establish a connection between the $L^q$-spectrum of a Borel measure $\nu$ on the $m$-dimensional unit cube and the approximation order of Kolmogorov diameters of the unit sphere with respect to Sobolev norms in $L_{\nu }^p$. 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.

21 pages, 1 figure

References in corpus (2)

• Spectral dimensions of Krein--Feller operators and $L^q$-spectra
• Spectral asymptotics of Krein--Feller operators for weak Gibbs measures on self-conformal fractals with overlaps

Cited by in corpus (2)

• Quantization dimensions of compactly supported probability measures via Rényi dimensions
• Quantization dimensions of negative order