An analytic study on the self-similar fractals: Differentiation of integrals (Q752214)

Let E be a self-similar fractal in \({\mathbb{R}}^ n\) with positive and finite s-dimensional Hausdorff measure \(H^ s(E)\). The self-similar structure of E induces natural differentiation bases on E; one consists of the pairs \((x,S_{i_ 1}\circ...\circ S_{i_ k}(E)),\) where \(x\in S_{i_ 1}\circ...\circ S_{i_ k}(E)\) and \(S_{i_ j}'s\) belong to the finite set of similitudes generating E. The author studies covering properties of these bases and shows that they differentiate \(L^ 1(E,H^ s)\).