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The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property
- Source :
- Mathematische Zeitschrift. 293:1633-1656
- Publication Year :
- 2019
- Publisher :
- Springer Science and Business Media LLC, 2019.
-
Abstract
- Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples.<br />Comment: 23 pages, 3 figures
- Subjects :
- Mathematics::Number Theory
General Mathematics
010102 general mathematics
Zero (complex analysis)
Boundary (topology)
Metric Geometry (math.MG)
Lipschitz continuity
01 natural sciences
Prime (order theory)
Domain (mathematical analysis)
Combinatorics
Metric space
Mathematics - Analysis of PDEs
Prime end
Mathematics - Metric Geometry
Bounded function
31E05, 31B15, 31B25, 31C15, 30L99
0103 physical sciences
FOS: Mathematics
010307 mathematical physics
0101 mathematics
Analysis of PDEs (math.AP)
Mathematics
Subjects
Details
- ISSN :
- 14321823 and 00255874
- Volume :
- 293
- Database :
- OpenAIRE
- Journal :
- Mathematische Zeitschrift
- Accession number :
- edsair.doi.dedup.....e1def2da96c50e607211a77ee57dece8