What is not in the domain of the Laplacian on Sierpinski gasket type fractals (Q1304671)

An analog of the Laplacian \(\Delta\) on the Sierpinski gasket (SG for short) and related fractals, constructed by Kigami is considered. A function \(f\) is said to belong to the domain of \(\Delta\) if \(f\) is continuous and \(\Delta f\) is defined as a continuous function. The authors give two different proofs of the assertion that if \(f\) is a nonconstant function in the domain of \(\Delta\), then \(f^2\) is not in the domain of \(\Delta\). A Laplacian \(\Delta\) on a fractal \(F\) can be defined by a Dirichlet form \({\mathcal E}(f,f)\), which is the analog of \(\int|\nabla f|^2dx\), and a measure \(\mu\) on \(F\). The Dirichlet form determines the harmonic functions, which are minimizers of \({\mathcal E}(f,f)\) subject to some boundary conditions. The Laplacian is determined by the analog of \(\int\Delta f\cdot\nabla gdx= -\int g\Delta fdx+\) \{boundary terms\} with \({\mathcal E}(f,g)\) playing the role of the left hand side, and \(d\mu\) substituting for \(dx\) on the right side. An interesting interpretation is \({\mathcal E}(f,g)\) can be the total mass of a signed measure \(\nu_{f,g}\) defined by \(\int hd\nu_{f,g}= {\mathcal E}(fh,g)+ {\mathcal E}(f,gh) -{\mathcal E}(h,fg)\) for \(h\) in the domain of \({\mathcal E}\), but the energy measures \(\nu_{f,g}\) may be unrelated to the measure \(\mu\) used to define the Laplacian. \textit{S. Kusuoka} [Publ. Res. Inst. Math. Sci. 25, No. 4, 659-680 (1989; Zbl 0694.60071)] proved they are singular for many fractals. A new original proof of this fact is given here.