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.