Integrals and potentials of differential 1-forms on the Sierpinski gasket

We provide a definition of the integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form epsilon on K. We show how this tool can be used to study the potential theory on K. In particular, we prove: (i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; (ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; (iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K. (C) 2013 Elsevier Inc. All rights reserved.