Sobolev extension property for tree-shaped domains with self-contacting fractal boundary

Abstract

In this paper, we investigate the existence of W1,p-extension operators for a class of bidimensional ramified domains with a self-similar fractal boundary previously studied by Mandelbrot and Frame. When the fractal boundary has no self-contact, the domains have the (E , δ)-property, and the extension results of Jones imply that there exist such extension operators for all 1 6 p 6 1. In the case where the fractal boundary self-intersects, this result does not hold. In this work we construct extension operators for 1 < p < p?, where p? depends only on the dimension of the self-intersection of the boundary. The construction of the extension operators is based on a Haar wavelet decomposition on the fractal part of the boundary. It relies mainly on the self-similar properties of the domain. The result is sharp in the sense that W1,p-extension operators fail to exist when p > p?.