Extremal length and functional completion

Fuglede, Bent

doi:10.1007/bf02404474

Cited by 371 publications

(223 citation statements)

References 22 publications

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“…The concept of the modulus of the path family was introduced by Ahlfors and Beurling [1] and developed by Fuglede [30]. This theory constitutes a basic tool for the theory of quasiregular mappings, see Rickman [55] and Väisälä [65].…”

Section: Historical Notesmentioning

confidence: 99%

Sobolev spaces on metric-measure spaces

Hajłasz¹

2003

Contemporary Mathematics

182

192

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Section: Historical Notesmentioning

confidence: 99%

Sobolev spaces on metric-measure spaces

Hajłasz¹

2003

Contemporary Mathematics

182

192

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“…Absolute continuity along n-a.e. curve follows from a theorem of Fuglede [27], since quasiconformal maps of domains in R n lie in the local Sobolev space W 1,n loc . These results were extended by Väisälä [72] to locally quasisymmetric maps from a domain Ω ⊂ R n to any set Y ⊂ R N , N < ∞, with locally finite Hausdorff n-measure and then by the author [70] to the case of general metric spaces Y with locally finite Hausdorff n-measure.…”

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confidence: 99%

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

Tyson

2001

Conform. Geom. Dyn.

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Abstract. Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu's generalized modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension Q > 1, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality.We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper Q-regular Loewner space, Q > 1, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.

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“…The p-modulus (as the module of order p of a system of measures) was defined and studied by Fuglede [17]. Heinonen-Koskela [22] defined the p-modulus of a curve family in a metric measure space and observed that the corresponding results by Fuglede carried over directly.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

Hansevi¹

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p, p)-Poincaré inequality, 1 < p < ∞, and the existence of a unique solution is proved. Furthermore, if the measure is doubling, then it is shown that a continuous obstacle implies that the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.

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Extremal length and functional completion

Cited by 371 publications

References 22 publications

Sobolev spaces on metric-measure spaces

Sobolev spaces on metric-measure spaces

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

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