Lectures on Analysis on Metric Spaces

Heinonen, Juha

doi:10.1007/978-1-4613-0131-8

Cited by 1,644 publications

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“…For a discussion and definition of line integration, see [19,Chapter 7]. Theorem 5.2 has also a consequence that will be proved later: One should compare the metrics D µ in Theorems 5.4 and 5.5 to those studied in [5] and [29].…”

Section: Strong a ∞ -Weightsmentioning

confidence: 99%

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Costea¹

2007

Rev. Mat. Iberoam.

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This article studies strong A ∞ -weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space R n with n ≥ 2, whenever n − 1 < s ≤ n, a function u yields a strong A ∞ -weight of the form w = e nu if the distributional gradient ∇u has sufficiently small || · || L s,n−s (R n ; R n )-norm. Similarly, it is proved that if 2 ≤ n < p < ∞, then w = e nu is a strong A ∞ -weight whenever the BesovLower estimates of the Besov B p -capacities are obtained in terms of the Hausdorff content associated with gauge functions h satisfying the condition 1 0 h(t) p −1 dt t < ∞.

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Section: Strong a ∞ -Weightsmentioning

confidence: 99%

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Costea¹

2007

Rev. Mat. Iberoam.

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“…In this section we summarize basic facts on quasiconformal and related maps (see [20,23,33] for general background). Let f : C → C be a homeomorphism, and for x ∈ C and small r > 0 define…”

Section: Quasiconformal and Related Mapsmentioning

confidence: 99%

“…Under mild extra assumptions on the spaces weak quasisymmetry of a map implies its quasisymmetry [20,Theorem 10.19]. A metric space is called proper if every closed ball in the space is compact.…”

Section: Quasiconformal and Related Mapsmentioning

confidence: 99%

Uniformization of Sierpiński carpets in the plane

Bonk

2011

Invent. math.

120

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Let S i , i ∈ I , be a countable collection of Jordan curves in the extended complex plane C that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map f : C → C such that f (S i ) is a round circle for all i ∈ I . This implies that every Sierpiński carpet in C whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.

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“…Hyperconvex spaces may also be considered within this class of intriguing metric structures which allow us to obtain results which one would only expect to be possible under certain linear structures. For references on the study of analysis on metric spaces or nonsmooth analysis the interested reader may check [3,30,31].…”

Section: Introductionmentioning

confidence: 99%

Metric fixed point theory on hyperconvex spaces: recent progress

Espínola

Lorenzo

2012

Arab. J. Math.

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In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown. Mathematics Subject Classification

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Lectures on Analysis on Metric Spaces

Cited by 1,644 publications

References 0 publications

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Uniformization of Sierpiński carpets in the plane

Metric fixed point theory on hyperconvex spaces: recent progress

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