Hodge theory on the Harmonic Gasket and other fractals

Abstract

S. Kusuoka has proven that, on many fractals $G\subset R^d$, it is possible to build a natural bilinear form on the vector space of Borel fields of one-forms on $G$. A variant of this construction yields a bilinear form on Borel fields of $q$-forms; it is tempting to ask (and several authors have done it) whether some features of Hodge theory survive in this setting. In this paper we define a weak version of the codifferential on fractals and we show that, for one-forms on the Harmonic Sierpinski Gasket, a Hodge decomposition theorem holds. As a further example, we calculate the codifferential of 2-forms and 1-forms on a fractal of $R^3$ which is the product of the harmonic Sierpinski gasket with the interval $[0,1]$.