Trace Theorems for a Class of Ramified Domains with Self-Similar Fractal Boundaries

This work deals with trace theorems for a class of ramified bidimensional domains Ω with a self-similar fractal boundary Γ. The fractal boundary Γ is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a − δ domain as defined by Jones and the fractal set is not totally disconnected. In this case, the classical trace results cannot be used. Here, the Lipschitz spaces with jumps recently introduced by Jonsson play a crucial role. Indeed, it is proved in particular that if the Hausdorff dimension d of Γ is not smaller than one, then the space of the traces of functions in W(Ω), m ∈ N, 1 < q <∞ is JLip(m+ 1− 2−d q , q, q;m; Γ). The proof is elementary; a main step is a strengthened trace inequality in the norm Lμ(Γ ).