Home Local one-sided porosity and pretangent spaces
Article
Licensed
Unlicensed Requires Authentication

Local one-sided porosity and pretangent spaces

  • Maya Altınok , Oleksiy Dovgoshey and Mehmet Küçükaslan EMAIL logo
Published/Copyright: September 22, 2015

Abstract

For subsets of + we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function μ:+ provides an extension of the concept of porosity numbers on subsets of . The main results describe interconnections between porosity numbers of a set, features of the scaling functions, and the geometry of so-called pretangent spaces to this set.

MSC 2010: 28A05; 54E35; 30D40

Funding statement: The research of the second author was supported by a grant received from TUBİTAK within 2221-Fellowship Programme for Visiting Scientists and Scientists on Sabbatical Leave, and as part of the EUMLS project with grant agreement PIRSES-GA-2011-295164.

References

[1] Abdullayev F., Dovgoshey O. and Küçükaslan M., Metric spaces with unique pretangent spaces. Conditions of the uniqueness, Ann. Acad. Sci. Fenn. Math. 36 (2011), 353–392. 10.5186/aasfm.2011.3623Search in Google Scholar

[2] Bilet V. and Dovgoshey O., Investigations of strong right upper porosity at a point, Real Anal. Exchange 39 (2013/14), 175–206. 10.14321/realanalexch.39.1.0175Search in Google Scholar

[3] Bilet V. and Dovgoshey O., Boundedness of pretangent spaces to general metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014), 73–82. 10.5186/aasfm.2014.3902Search in Google Scholar

[4] Bilet V., Dovgoshey O. and Küçükaslan M., Uniform boundedness of pretangent spaces, local constancy of metric derivatives and strong right upper porosity at a point, J. Anal. 21 (2013), 31–55. Search in Google Scholar

[5] Denjoy A., Sur une propriété des séries trigonométriques, Amst. Ak. Versl. 29 (1920), 628–639. Search in Google Scholar

[6] Denjoy A., Leçons sur le calcul des cofficients d’une série trigonométrique, Part II, Métrique et topologie d’ensembles parfaits et de fonctions, Gauthier-Villars, Paris, 1941. Search in Google Scholar

[7] Dolženko E. P., Boundary properties of arbitrary functions (in Russian), Izv. Akad. Nauk SSSR Ser. Math. 31 (1967), 3–14. 10.1070/IM1967v001n01ABEH000543Search in Google Scholar

[8] Dovgoshey O., Tangent spaces to metric spaces and to their subspaces, Ukr. Math. Vins 5 (2008), 470–487. Search in Google Scholar

[9] Dovgoshey O., Abdullayev F. and Küçükaslan M., Compactness and boundedness of tangent spaces to metric spaces, Beitr. Algebra Geom. 51 (2010), 547–576. Search in Google Scholar

[10] Dovgoshey O. and Martio O., Tangent spaces to metric spaces, Reports in Math. 480, University of Helsinki, Helsinki, 2008. Search in Google Scholar

[11] Dovgoshey O. and Martio O., Tangent spaces to general metric spaces, Rev. Roumaine Math. Pures Appl. 56 (2011), 137–155. Search in Google Scholar

[12] Dovgoshey O. and Riihentaus J., Mean value type inequalities for quasinearly subharmonic functions, Glasgow Math. J. 55 (2013), 349–368. 10.1017/S0017089512000602Search in Google Scholar

[13] Karp L., Kilpenläinen T., Petrosyan A. and Shahgholian H., On the porosity of free boundaries in degenerate variational inequalities, J. Differential Equations 164 (2000), 110–117. 10.1006/jdeq.1999.3754Search in Google Scholar

[14] Kelly J. L., General Topology, D. Von Nostrand Company, Princeton, 1965. Search in Google Scholar

[15] Khintchine A., An investigation of the structure of measurable functions (in Russian), Mat. Sbornik 31 (1924), 265–285. Search in Google Scholar

[16] Przytycki F. and Rohde S., Porosity of Collet–Eckmann Julia sets, Fund. Math. 155 (1998), 189–199. Search in Google Scholar

[17] Thomson B. S., Real Functions, Lecture Notes in Math. 1170, Springer, Berlin, 1985. 10.1007/BFb0074380Search in Google Scholar

[18] Väisälä J., Porous sets and quasisymmetric maps, Trans. Amer. Math. Soc. 299 (1987), 525–533. 10.1090/S0002-9947-1987-0869219-8Search in Google Scholar

[19] Yanagihara N., Angular cluster sets and oricyclic cluster sets, Proc. Japan Acad. 45 (1969), 423–428. 10.3792/pja/1195520715Search in Google Scholar

[20] Yoshida H., Tangential boundary properties of arbitrary functions in the unit disc, Nagoya Math. J. 46 (1972), 111–120. 10.1017/S002776300001480XSearch in Google Scholar

[21] Yoshida H., On the boundary properties and the spherical derivatives of meromorphic functions in the unit disc, Math. Z. 132 (1973), 51–68. 10.1007/BF01214033Search in Google Scholar

[22] Zajiček L., On cluster sets of arbitrary functions, Fund. Math. 83 (1974), 197–217. 10.4064/fm-83-3-197-217Search in Google Scholar

Received: 2015-1-28
Revised: 2015-8-25
Accepted: 2015-8-29
Published Online: 2015-9-22
Published in Print: 2016-8-1

© 2016 by De Gruyter

Downloaded on 11.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anly-2015-0011/html
Scroll to top button