# Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

## TL;DR

This paper investigates Sobolev spaces on p.c.f. self-similar sets, identifying critical orders and providing atomic decompositions, with implications for Besov space characterizations.

## Contribution

It introduces a comparison of different Sobolev spaces on fractals and establishes a general atomic decomposition theorem highlighting critical orders.

## Key findings

- Identification of critical orders of Sobolev spaces on fractals
- Development of an atomic decomposition theorem for Sobolev spaces
- Analytic characterizations of Sobolev spaces via Besov spaces

## Abstract

We consider the Sobolev type spaces $H^\sigma(K)$ with $\sigma\geq 0$, where $K$ is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces $H^\sigma_N(K),H^\sigma_D(K)$ and $H^\sigma(K)$, {and observe} a sequence of critical orders of $\sigma$ in our comparison theorem. Secondly, We present a general atomic decomposition theorem of Sobolev spaces $H^\sigma(K)$, where the same critical orders play an important role. At the same time, we provide purely analytic approaches for various Besov type characterizations of Sobolev spaces $H^\sigma(K)$.