Approximation order of Kolmogorov, Gel'fand, and linear widths for Sobolev embeddings in euclidian measure spaces

TL;DR

This paper determines the exact approximation orders of Kolmogorov, Gel'fand, and linear widths for Sobolev space embeddings into measure spaces, linking these orders to the measure's $L^{q}$-spectrum and fractal dimensions.

Contribution

It provides a complete solution for the approximation orders in Sobolev embeddings with arbitrary measures, relating them to spectral and fractal properties.

Findings

Exact approximation orders expressed via $L^{q}$-spectrum.

Conditions on $L^{q}$-spectrum regularity for existence of approximation orders.

Connections established between approximation order and Minkowski dimensions.

Abstract

In this paper we completely solve the problem of finding the (upper) approximation order with respect to the Kolmogorov, Gel'fand, and linear widths for the embedding of the Sobolev spaces $W^{\alpha,p}$ and $W_{0}^{\alpha,p}$ in the euclidian measure spaces $L_{\nu}^{q}$ for an arbitrary Borel probability measure $\nu$ with support contained in the open $m$-dimensional unit cube and for all possible choices of $1\leq p,q\leq\infty$. We will determine the exact values for the various upper approximation orders in terms of the $L^{q}$-spectrum of $\nu$ only and finally give sufficient conditions imposed on the regularity of the $L^{q}$-spectrum for the approximation orders to exist. We also elucidate some intrinsic connections between the concept of approximation order and the fractal geometric notion of the upper and lower Minkowski dimension of the support of $\nu$.