Introduction

Abstract

Let Γ be a compact d-set in ${\mathbb{R}^n}$ where n∈ $\mathbb{N}$ and 0 <d < n, according to (9.1) or (9.67). By our previous considerations, d-sets are especially well adapted to the function spaces $B_{pq}^s\left( {{\mathbb{R}^n}} \right)$ and $F_{pq}^s\left( {{\mathbb{R}^n}} \right)$ as treated in this book. In particular, by 9.19 and 9.18 such sets satisfy the ball condition with the consequences for traces as described in Theorem 9.21. Furthermore, there is a natural way of introducing function spaces on P as mentioned in (9.125) with a reference to [Triδ], Definition 20.2, p. 159. Quarkonial representations of such spaces, in the larger context of more general fractals, have been given in 9.29-9.33. These results pave the way to a substantial theory for fractal elliptic (pseudo)differential operators in continuation of [Triδ], Chapter V. We refer especially to [Triδ], Section26, where we discussed in detail our point of view, compared with what has been done in the literature. In particular, there are different interpretations of what is called a fractal drum. The physical background of our approach in [Triδ] and also here, as far as spectral theory is concerned, may be found in [Triδ], 30.1, and will be briefly repeated in 19.1. As indicated one could continue the studies started in [Triδ], Chapters IV and V, based now on the theory developed in Chapter I of the present book on the level of (degenerate) fractal elliptic pseudodifferential operators, including fractional powers of elliptic differential operators. But this is not our aim. As far as elliptic operators are concerned we stick to the Laplacian as an outstanding example.