\(JLip\) versus Sobolev spaces on a class of self-similar fractal foliages
The paper deals with special self-similar \(d\)-sets \(\Gamma\) in the plane \(\mathbb R^2\) which originate from two peculiar similitudes. Roughly speaking \(\Gamma\) is the boundary of a domain \(\Omega\). Let \(W^1_p (\Omega)\) be the corresponding Sobolev space and \(B^s_{p,p} (\Gamma)\) its trace, \(s= 1 - \frac{2-d}{p}\), \(1\leq p <\infty\), normed by \[ \| f \, | L_p (\Gamma, \mu) \| + \Big( \int_{\Gamma \times \Gamma} \frac{|f(x)-f(y)|^p}{|x-y|^{d + sp}} \, d \mu (x) \, d \mu (y) \Big)^{1/p} \] where \(\mu\) is the related Hausdorff measure on \(\Gamma\). Let \(\mathrm{Lip} (t,p,\Gamma)\) be Lipschitz spaces, defined via expansions by Haar functions on \(\Gamma\). The paper deals with the trace of \(W^1_p (\Omega)\) on \(\Gamma\) in terms of Besov spaces and a comparison of these Besov spaces with corresponding Lipschitz spaces.
scientific article; zbMATH DE number 6005648
scientific article