The Laplacian on the Sierpinski Gasket via the Method of Averages

A direct construction of a Laplacian on the Sierpinski gasket as a limit of difference quotients was given by Kigami [Ki1], who later extended the method to a class of self-similar fractals called post critically finite (p.c.f.) [Ki2, Ki3]. At about the same time, Kusuoka and Zhou [KZ] introduced what we will call the method of averages for defining a Laplacian on the Sierpinski carpet, a fractal that is not p.c.f. The method of averages uses average values of functions over basic sets rather than pointwise values in defining all operations. In this paper we will show how the method of averages can be used to define the Laplacian on the Sierpinski gasket. (Since there are many different Laplacians obtainable by the method of [Ki2], we should call this the symmetric Laplacian.) It would be nice to be able to use the method for all Laplacians on p.c.f. fractals, but it is not clear at present how to do this. Ultimately, the goal is to use the method of averages to define Laplacians on wider classes of fractals. To advance these goals it is worthwhile to have a basic example worked out in detail. As we will see, the formulas involved are a bit more complicated than the analogous ones for pointwise values. Also, since average values play such an important role in the usual theory of harmonic functions, it is of independent interest to understand the properties of average values of harmonic functions on the Sierpinski gasket. For the convenience of the reader, we summarize the results from [Ki1] that we will use. In principle we should establish all properties of the theory directly in terms of the average quantities, and then show that the theory is equivalent to the standard pointwise one. However, in the interest of brevity, we will make the connection between average and pointwise quantities at