Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

摘要

We consider a class of ramified bidimensional domains? with a self-similar fractal boundary Γ~∞, which is supplied with a probability measure μ called the selfsimilar measure. Emphasis is put on the case when the domain is not a ε ? δ domain as defined by Jones and the fractal set is not totally disconnected. We compare two notions of trace on Γ~∞ for functions in W~(1,q)(?): the classical one, see for instance the book by Jonnson and Wallin, 1984, using the strict definition of a function at a point of ?, and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide μ-almost everywhere on Γ~∞. As a corollary, we characterize the critical number q for which for all q < q (resp. q > q) there is a (resp. no) continuous extension operator from W~(1,q)(?) to W~(1,q)(?~2).