Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 303-324

PRESERVATION OF BOUNDED GEOMETRY UNDER SPHERICALIZATION AND FLATTENING: QUASICONVEXITY AND ∞-POINCARÉ INEQUALITY

Estibalitz Durand-Cartagena and Xining Li

UNED, ETSI Industriales, Departamento de Matemática Aplicada
Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain; edurand 'at' ind.uned.es

Sun Yat-sen University, Department of Mathematics
Guangzhou 510275, P.R. China; lixining3 'at' mail.sysu.edu.cn

Abstract. In this work we explore the preservation of quasiconvexity and ∞-Poincaré inequality under sphericalization and flattening in the metric setting. The results developed in [22] show the preservation of Ahlfors regularity, doubling property and the p-Poincaré inequality for 1 ≤ p < ∞ under the sphericalization and flattening transformations provided the underlying metric space is annularly quasicovex. In this work, we propose a weaker assumption to still preserve quasiconvexity and ∞-Poincaré inequality, called radially star-like quasiconvexity (corresponding to sphericalization) and meridian-like quasiconvexity (corresponding to flattening) extending in particular a result in [8] to a wider class of metric spaces and covering the case p = ∞ in [22].

2010 Mathematics Subject Classification: Primary 31E05; Secondary 30L10, 30L99.

Key words: Sphericalization, flattening, doubling, Poincaré inequality, quasiconvexity, annularly quasiconvex, radially star-like quasiconvex spaces, meridian-like quasiconvex spaces.

Reference to this article: E. Durand-Cartagena and X. Li: Preservation of bounded geometry under sphericalization and flattening: quasiconvexity and ∞-Poincaré inequality. Ann. Acad. Sci. Fenn. Math. 42 (2017), 303-324.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4221