Garabedian Duality for Hole-Punch Domains
2010; Springer Nature; Linguagem: Inglês
10.1007/978-1-4419-6709-1_3
ISSN2191-6675
Autores Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoIn the last chapter we learned from Painlevé’s Theorem (2.7) that linear Hausdorff measure zero implies removability and from the planar Cantor quarter set that the converse fails in general (Theorem 2.19). However, by one of our early results, Proposition 1.1, the converse does hold if the compact set in question lies in a line. A line, or rather line segment, is the simplest example of a rectifiable curve. One can define arclength measure on a rectifiable curve and show that its zero sets coincide with those of linear Hausdorff measure on the curve (see Sections 4.5 and 5.2). This leads to what is known as Denjoy’s Conjecture Conjecture!Denjoy’s Denjoy’s Conjecture: A compact subset of a rectifiable curve is removable if and only if it has arclength measure, i.e., linear Hausdorff measure, zero. In 1909 Denjoy, Arnaud Arnaud Denjoy claimed to have proved this in [DEN]. His proof however had a gap in it, thus giving rise to the eponymous conjecture which was resolved affirmatively only in 1977 by Calderón, Alberto Alberto Calderón’s famous and justly celebrated paper [CAL] on the L 2-boundedness of the Cauchy integral operator on Lipschitz curves. The interested reader may consult [MARSH] for a proof of Denjoy’s Conjecture from Calderón’s result and more history on this topic.
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