Thin Loewner carpets and their quasisymmetric embeddings in \(S^2\). (English) Zbl 1542.30056
A metric space is called planar if it is homeomorphic to a subset of the plane \(\mathbb{R}^2\). A \(Q\)-Loewner space is a metric space whose \(Q\)-Hausdorff measure is \(Q\)-Ahlfors regular and that supports a \((1,Q)\)-Poincaré inequality. A carpet is a metric space that is homeomorphic to the standard Sierpiński carpet. A planar \(Q\)-Loewner carpet is a metric space satisfying all three of these properties.
The paper under review shows that any planar \(Q\)-Loewner carpet with \(1<Q<2\) can be quasisymmetrically embedded in the plane. Moreover, it provides the first published examples of Loewner carpets with Hausdorff dimension less than 2. Infinitely many quasisymmetrically distinct such examples are constructed for any \(1<Q<2\). The method of construction is an extension of inverse limits that the authors call admissible quotiented inverse systems.
Moreover, the examples admit explicit quasisymmetric embeddings in the plane. These constructions also provide examples of strong \(A_\infty\) weights whose associated metrics cannot be bi-Lipschitzly embedded in any Banach space possessing the Radon-Nikodym property.
The paper under review shows that any planar \(Q\)-Loewner carpet with \(1<Q<2\) can be quasisymmetrically embedded in the plane. Moreover, it provides the first published examples of Loewner carpets with Hausdorff dimension less than 2. Infinitely many quasisymmetrically distinct such examples are constructed for any \(1<Q<2\). The method of construction is an extension of inverse limits that the authors call admissible quotiented inverse systems.
Moreover, the examples admit explicit quasisymmetric embeddings in the plane. These constructions also provide examples of strong \(A_\infty\) weights whose associated metrics cannot be bi-Lipschitzly embedded in any Banach space possessing the Radon-Nikodym property.
Reviewer: Scott Zimmerman (Marion)
MSC:
| 30L10 | Quasiconformal mappings in metric spaces |
| 28A80 | Fractals |
| 46E36 | Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
| 54E35 | Metric spaces, metrizability |
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