Happy fractals and some aspects of analysis on metric spaces

Semmes, Stephen

doi:10.5565/publmat_47203_01

Cited by 3 publications

(1 citation statement)

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“…In particular, we show that there is a self-similar fractal in R 2 with Hausdorff dimension d > 1 and a Lipschitz function defined on it with exponent d. Thus, unlike the Euclidean case, Lipschitz spaces of order greater than 1 may be non-trivial on fractals or metric-measure spaces. This fact is well known (see [30,32,39]) and one simple example in the one-dimensional case is provided in [30] by considering X consisting of two disjoint intervals in R n . In § 3, we discuss the relationships between our Lipschitz-type spaces and Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.…”

Section: Definition 11 (See [23])mentioning

confidence: 99%

Spaces of Lipschitz Type on Metric Spaces and Their Applications

Yang¹,

Lin²

2004

Proceedings of the Edinburgh Mathematical Society

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New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel-Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Haj lasz-Sobolev spaces, Korevaar-Schoen-Sobolev spaces, Newtonian Sobolev space and Cheeger-Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy-Littlewood-Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

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