Isotropic fractals and related function spaces

Abstract

Entropy numbers and eigenvalues may be considered as two sides of the same coin. Entropy numbers are the geometrical twins of eigenvalues. This opinion is based on Carl’s observation (19.129) at an abstract level. It is also supported by the equivalences in (19.73), Theorem 19.17 combined with (19.109), and the assertions about the negative spectrum in Theorem 21.7 and Corollary 21.11 where estimates for entropy numbers play an important role. In 19.18(vii) we remarked that in case of Hilbert spaces, entropy numbers and eigenvalues might be even equivalent. The decisive link between the abstract inequality (19.129) on the one hand, and the more concrete sharp estimates from above of related eigenvalue distributions on the other hand, is the possibility to calculate the entropy numbers of compact embeddings between function spaces and the consequences for respective operators. Theorem 21.3 and Corollary 21.9 may serve as typical examples. We followed this path in [ET96], Chapter 5, based on the earlier papers mentioned there, for regular and degenerate elliptic pseudodifferential operators in ${\mathbb{R}^n}$and on domains Ω in ${\mathbb{R}^n}$. At that time estimates of entropy numbers of compact embeddings between function spaces were based on the original Fourier-analytical definitions of $B_{pq}^s\left( {{\mathbb{R}^n}} \right)$ and $F_{pq}^s\left( {{\mathbb{R}^n}} \right)$ as described in (2.37), (2.38), and Theorem 2.9. These tools (including atomic decompositions) proved to be inadequate when switching from ${\mathbb{R}^n}$and domains Ω in ${\mathbb{R}^n}$to fractals, preferably d-sets according to (9.148).