We define and study pseudo-differential operators on a class of fractals that include post-critically finite (p.c.f.) self-similar sets and Sierpiński carpets. Using sub-Gaussian estimates for the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our results are applicable to a larger class of metric measure spaces with Laplacian, we use them to study elliptic, hypoelliptic, and quasi-elliptic operators on p.c.f. fractals, answering a few open questions posed in a series of recent papers. We extend our class of operators to include the so called Hörmander hypoelliptic operators and we initiate the study of wavefront sets and microlocal analysis on p.c.f. fractals.
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Marius Ionescu, Luke G. Rogers, Robert S. Strichartz, Pseudo-differential operators on fractals and other metric measure spaces. Rev. Mat. Iberoam. 29 (2013), no. 4, pp. 1159–1190