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

2019/03/31 by Shiping Cao, Cao, Shiping, Hua Qiu +1 · 1 citation
Mathematics · #28A80 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1904.00342

openalex publication_date 2019/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Sobolev type spaces H^σ(K) with σ≥ 0, where K is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces H^σ_N(K),H^σ_D(K) and H^σ(K), and observe a sequence of critical orders of σ in our comparison theorem. Secondly, We present a general atomic decomposition theorem of Sobolev spaces H^σ(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^σ(K).

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