Gradients on Fractals

In this paper we define and study a gradient on p.c.f. (post critically finite, or finitely ramified) fractals. We use Dirichlet (energy) form analysis developed for such fractals by Kigami. We consider both nondegenerate and degenerate harmonic structures (where a nonzero harmonic function can be identically zero on an open set). We show that the energy is equal to the integral of a certain seminorm of the gradient if the harmonic structure is weakly nondegenerate. This result was proved by Kusuoka in a different form. We show that for a C1-function on the Sierpiński gasket the gradient considered here and Kusuokaʹs gradient essentially coincide with a gradient considered by Kigami. The gradient at a junction point was studied by Strichartz in relation to the Taylor approximation on fractals. He also proved the existence of the gradient almost everywhere with respect to the Hausdorff (Bernoulli) measure for a function in the domain of the Laplacian. In this paper we obtain certain continuity properties of the gradient for a function in the domain of the Laplacian. As an appendix, we prove an estimate of the local energy of harmonic functions which was stated by Strichartz as a hypothesis.