Some stability properties for the Bishop-Phelps-Bollobás property for Lipschitz maps Chiclana Vega, Rafael Martín Suárez, Miguel Lipschitz function Lipschitz map Lipschitz-free space Norm-attaining operators Bishop-Phelps-Bollobás property Metric space The authors would like to thank A. Aviles, L. C. Garcia-Lirola, V. Kadets, P. Koszmider, and A. Rueda Zoca for kindly answering some inquiries about the contents of the paper. They also thank G. Choi, Y. S. Choi, and M. Jung for some comments. Finally, they thank the anonymous referee for the careful checking of the manuscript and multiple suggestions which have improved the final form of the paper. This research has been partially supported by Spanish AEI Project PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE) and by Junta de Andalucia/FEDER, UE project FQM-185. We study the stability behavior of the Bishop-Phelps-Bollobas property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobas property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very close to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of Gamma-flat operators and ACK structure. We get sharper results for the case of Lipschitz compact maps. The behavior of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above for the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps. 2022-02-02T08:31:14Z 2022-02-02T08:31:14Z 2020-04-22 info:eu-repo/semantics/article Published version: Chiclana, R. & Martín, M. (2022). Some stability properties for the Bishop-Phelps-Bollobás property for Lipschitz maps. Studia Mathematica. DOI: [10.4064/sm200427-7-4] http://hdl.handle.net/10481/72599 eng http://creativecommons.org/licenses/by-nc-nd/3.0/es/ info:eu-repo/semantics/openAccess Atribución-NoComercial-SinDerivadas 3.0 España Instytut Matematyczny