Quaternionic inner and outer functions

We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.The first author is a member of INDAM-GNAMPA and is partially supported by the 2015 PRIN grant Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis of the Italian Ministry of Education (MIUR). The second author is partially supported by INDAM-GNSAGA, by the 2014 SIR grant Analytic Aspects in Complex and Hypercomplex Geometry and by Finanziamento Premiale FOE 2014 Splines for accUrate NumeRics: adaptive models for Simulation Environments of the Italian Ministry of Education (MIUR). The third author is grateful for the financial support by the Severo Ochoa Programme for Centers of Excellence in R&D (SEV-2015-0554) at ICMAT, and by the Spanish Ministry of Economy and Competitiveness, through grant MTM2016-77710-P. Part of this project was carried out during a visit of the first and third author at the University of Firenze, and we wish to thank the Department of Mathematics and Computer Sciences for the financial support and the warm hospitality