Function spaces on fractals
It is the main aim of this paper to introduce on the Sierpinski gasket \(SG\), Hölder-Zygmund spaces \(\Lambda^\alpha (SG)\), (fractional) Sobolev spaces \(L^\alpha_p (SG)\), and Besov spaces \(\Lambda^\alpha_{pq} (SG)\) for all orders of smoothness \(\alpha\) and \(1\leq p \leq \infty\), \(1\leq q \leq \infty\) (in the case of the Besov spaces \(\Lambda^\alpha_{pq} (SG)\) with additional restrictions on the parameters). Embedding, lifting, and interpolation theorems are proved or conjectured. The Hausdorff dimension, the resistance metric and a related Laplacian play a decisive role, but not the natural embedding in the plane. The paper is considered as a first step towards a more comprehensive theory. There are some generalisations to post-critical finite self-similar fractals.
scientific article; zbMATH DE number 1901763