Harmonic embeddings of the stretched Siepinski gasket

Abstract

P. Alonso-Ruiz, U. Freiberg and J. Kigami have defined a large family of resistance forms on the Stretched Sierpinski Gasket $G$. In the present paper we introduce a system of coordinates on $G$ (technically, an embedding of $G$ into $\R^2$) such that \noindent$\bullet$) these forms are defined on $C^1(\R^2,\R)$ and \noindent$\bullet$) all affine functions are harmonic for them. We do this adapting a standard method from the Harmonic Sierpinski Gasket: we start finding a sequence $G_l$ of pre-fractals such that all affine functions are harmonic on $G_l$. After showing that this property is inherited by the stretched harmonic gasket $G$, we use the formula for the Laplacian of a composition to prove that, for a natural measure $\mu$ on $G$, $C^2(\R^2,\R)\subset\dc(\Delta)$ and Teplyaev's formula for the Laplacian of $C^2$ functions holds. Lastly, we use the expression for $\Delta u$ to show that the form we have found is closable in $L^2(G,\mu)$.