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

Shiping Cao; Hua Qiu

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arXiv · 1904.00342

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

Shiping Cao·Hua Qiu

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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 $H^\sigma(K)$.

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BibTeX RIS

Shiping Cao, Hua Qiu. 2019-03-31. Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions. https://arxiv.org/abs/1904.00342

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