Pseudo-differential Operators on Fractals

TL;DR

This paper develops a theory of pseudo-differential operators on fractals, establishing their kernel properties and extending analysis to hypoelliptic operators, wavefront sets, and microlocal analysis on fractals.

Contribution

It introduces a framework for pseudo-differential operators on fractals, including hypoelliptic operators, and explores their kernels and microlocal properties, answering open questions.

Findings

Operators have decaying kernels and are smooth off the diagonal.

Extension to product fractals and broader metric measure spaces.

Inclusion of Hörmander hypoelliptic operators and initial microlocal analysis.

Abstract

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of 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 product 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 H\"ormander hypoelliptic operators and we initiate the study of wavefront sets and microlocal analysis on p.c.f. fractals.