Distribution theory on p.c.f. fractals

TL;DR

This paper develops a distribution theory for analysis on p.c.f. fractals, establishing foundational properties and structure theorems that facilitate future research on differential operators in fractal settings.

Contribution

It introduces a novel distribution framework on p.c.f. fractals and related structures, including structure theorems and analysis of point-supported distributions.

Findings

Distributions are sums of Laplacian powers applied to continuous functions.

Basic properties of test functions and distributions are established.

Analysis of distributions with point support is provided.

Abstract

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.